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Proceed to summarize the following text: in recent years , the rapid increase in both the availability of data on networks ( of all kinds , but especially social ones ) and the demand , from many scientific areas , for analyzing such data has resulted in a surge of generative and descriptive models for network data @xcite . within statistics , this trend has led to a renewed interest in developing , analyzing and validating statistical models for networks @xcite . yet as networks are a nonstandard type of data , many basic properties of statistical models for networks are still unknown or have not been properly explored . in this article we investigate the conditions under which statistical inferences drawn over a sub - network will generalize to the entire network . it is quite rare for the data to ever actually be the _ whole _ network of relations among a given set of nodes or units ; typically , only a sub - network is available . guided by experience of more conventional problems like regression , analysts have generally fit models to the available sub - network , and then extrapolated them to the larger true network which is of actual scientific interest , presuming that the models are , as it were , consistent under sampling . what we show is that this is only valid for very special model specifications , and the specifications where it is _ not _ valid include some of which are currently among the most popular and scientifically appealing . in particular , we restrict ourselves to exponential random graph models ( ergms ) , undoubtedly one of the most important and popular classes of statistical models of network structure . in addition to the general works already cited , the reader is referred to @xcite for detailed accounts of these models . there are many reasons ergms are so prominent . on the one hand , ergms , as the name suggests , are exponential families , and so they inherit all the familiar virtues of exponential families in general : they are analytically and inferentially convenient @xcite ; they naturally arise from considerations of maximum entropy @xcite and minimum description length @xcite , and from physically - motivated large deviations principles @xcite ; and if a generative model obeys reasonable - seeming regularity conditions while still having a finite - dimensional sufficient statistic , it must be an exponential family @xcite . on the other hand , ergms have particular virtues as models of networks . the sufficient statistics in these models typically count the number or density of certain `` motifs '' or small sub - graphs , such as edges themselves , triangles , @xmath0-cliques , stars , etc . these in turn are plausibly related to different network - growth mechanisms , giving them a substantive interpretation ; see , for example , @xcite as an exemplary application of this idea , or , more briefly , section [ secergms ] below . moreover , the important task of edge prediction is easily handled in this framework , reducing to a conditional logistic regression @xcite . since the development of ( comparatively ) computationally - efficient maximum - likelihood estimators ( based on monte carlo sampling ) , ergms have emerged as flexible and persuasive tools for modeling network data @xcite . despite all these strengths , however , ergms are tools with a serious weakness . as we mentioned , it is very rare to ever observe the whole network of interest . the usual procedure , then , is to fit ergms ( by maximum likelihood or pseudo - likelihood ) to the observed sub - network , and then extrapolate the same model , with the same parameters , to the whole network ; often this takes the form of interpreting the parameters as `` provid[ing ] information about the presence of structural effects observed in the network '' @xcite , page 194 , or the strength of different network - formation mechanisms ; @xcite are just a few of the more recent papers doing this . this obviously raises the question of the statistical ( i.e. , large sample ) consistency of maximum likelihood estimation in this context . unnoticed , however , is the logically prior question of whether it is _ probabilistically _ consistent to apply the same ergm , with the same parameters , both to the whole network and its sub - networks . that is , whether the marginal distribution of a sub - network will be consistent with the distribution of the whole network , for all possible values of the model parameters . the same question arises when parameters are compared across networks of different sizes ( as in , e.g. , @xcite ) . when this form of consistency fails , then the parameter estimates obtained from a sub - network may not provide reliable estimates of , or may not even be relatable to , the parameters of the whole network , rendering the task of statistical inference based on a sub - network ill - posed . we formalize this question using the notion of `` projective families '' from the theory of stochastic processes . we say that a model is _ projective _ when the same parameters can be used for both the whole network and any of its sub - networks . in this article , we fully characterize projectibility of discrete exponential families and , as corollary , show that ergms are projective only for very special choices of the sufficient statistic . _ outline_. our results are not specific just to networks , but pertain more generally with exponential families of stochastic processes . in section [ secprojectiblity - and - exp - fam ] , therefore , we lay out the necessary background about projective families of distributions , projective parameters and exponential families in a somewhat more abstract setting than that of networks . in section [ secprojectibility - and - independent - increments ] we show that a necessary and sufficient condition for an exponential family to be projective is that the sufficient statistics obey a kind of additive decomposition . this in turn implies strong independence properties . we also prove results about the consistency of maximum likelihood parameter estimation under these conditions ( section [ secconsistency - of - mle ] ) . in section [ secergms ] , we apply these results to ergms , showing that most popular specifications for social networks and other stochastic graphs can not be projective . we then conclude with some discussion on possible constructive responses . the proofs are contained in the . _ related work_. an early recognition of the fact that sub - networks may have statistical properties which differ radically from those of the whole network came in the context of studying networks with power - law ( `` scale - free '' ) degree distributions . on the one hand , stumpf , wiuf and may @xcite showed that `` subnets of scale - free networks are not scale - free ; '' on the other , achlioptas et al . @xcite demonstrated that a particular , highly popular sampling scheme creates the appearance of a power - law degree distribution on nearly any network . while the importance of network sampling schemes has been recognized since then @xcite , chapter 5 , and valuable contributions have come from , for example , @xcite , we are not aware of any work which has addressed the specific issue of consistency under projection which we tackle here . perhaps the closest approaches to our perspective are @xcite and @xcite . the former considers conditions under which infinite - dimensional families of distributions on abstract spaces have projective limits . the latter , more concretely , addresses the consistency of maximum likelihood estimators for exponential families of dependent variables , but under assumptions ( regarding markov properties , the `` shape '' of neighborhoods , and decay of correlations in potential functions ) which are basically incomparable in strength to ours.=-1 our results about exponential random graph models are actually special cases of more general results about exponential families of dependent random variables , and are just as easy to state and prove in the general context as for graphs . setting this up , however , requires some preliminary definitions and notation , which make precise the idea of `` seeing more data from the same source . '' in order to dispense with any measurability issues , we will implicitly assume the existence of an underlying probability measure for which the random variables under study are all measurable . furthermore , for the sake of readability we will not rely on the measure theoretic notion of filtration : though technically appropriate , it will add nothing to our results . let @xmath1 be a collection of finite subsets of a denumerable set @xmath2 partially ordered with respect to subset inclusion . for technical reasons , we will further assume that @xmath3 has the property of being an ideal : that is , if @xmath4 belongs to @xmath3 , then all subsets of @xmath4 are also in @xmath3 and if @xmath4 , and @xmath5 belongs to @xmath3 , then so does their union . we may think of passing from @xmath6 to @xmath7 as taking increasingly large samples from a population , or recording increasingly long time series , or mapping data from increasing large spatial regions , or over an increasingly dense spatial grid , or looking at larger and larger sub - graphs from a single network . accordingly , we consider the associated collection of parametric statistical models @xmath8 indexed by @xmath1 , where , for each @xmath9 , @xmath10 is a family of probability distributions indexed by points @xmath11 in a fixed open set @xmath12 . the probability distributions in @xmath13 are also assumed to be supported over the same @xmath14 , which are countable sets for each @xmath4 . we assume that the partial order of @xmath1 is isomorphic to the partial order over @xmath15 , in the sense that @xmath16 if and only if @xmath17 . for given @xmath11 and @xmath6 , we denote with @xmath18 the random variable distributed as @xmath19 . in particular , for a given @xmath20 , we can regard the @xmath21 as finite - dimensional ( i.e. , marginal ) distributions . for each pair @xmath22 in @xmath1 with @xmath23 , we let @xmath24 be the natural index projection given by @xmath25 . in the context of networks , we may think of @xmath26 as the set of nodes of a possibly infinite random graph , which without loss of generality can be taken to be @xmath27 and of @xmath3 as the collection of all finite subsets of @xmath26 . then , for some positive integers @xmath28 and @xmath29 , we may , for instance , take @xmath30 and @xmath31 , so that @xmath18 will be the induced sub - graph on the first @xmath28 nodes and @xmath32 the induced sub - graph on the first @xmath33 nodes . the projection @xmath34 then just picks out the appropriate sub - graph from the larger graph ; see figure [ figprojective - structure - illustrated ] for a schematic example . we will be concerned with a natural form of probabilistic consistency of the collection @xmath35 which we call _ projectibility _ , defined below . is contained in the larger set of observables @xmath36 , @xmath18 ( on the right ) can be recovered from @xmath37 ( on the left ) through the projection @xmath34 , which simply drops the extra data . ] the family @xmath38 is _ projective _ if , for any @xmath6 and @xmath5 in @xmath3 with @xmath39 , @xmath40 see @xcite , page 115 , for more general treatment of projectibility . in words , @xmath41 is a projective family when @xmath39 implies that @xmath42 can be recovered by marginalization over @xmath43 , for all @xmath11 . within a projective family , @xmath44 denotes the infinite - dimensional distribution , which thus exists by the kolmogorov extension theorem @xcite , theorem 6.16 , page 115 . projectibility is automatic when the generative model calls for independent and identically distributed ( iid ) observations . it is also generally unproblematic when the model is specified in terms of _ conditional _ distributions : one then just uses the ionescu tulcea extension theorem in place of that of kolmogorov @xcite , theorem 6.17 , page 116 . however , many models are specified in terms of _ joint _ distributions for various index sets , and this , as we show in theorem [ thmprojectible - iff - separable ] , can rule out projectibility . we restrict ourselves to _ exponential family _ models by assuming that , for each choice of @xmath45 and @xmath9 , @xmath19 has density with respect to the counting measure over @xmath14 given by @xmath46 where @xmath47 is the measurable function of minimal sufficient statistics , and @xmath48 is the _ partition function _ given by @xmath49 if @xmath50 , we will write @xmath51 for the random variable corresponding to the sufficient statistic . equation ( [ eqnexponential - family - density ] ) implies that @xmath52 itself has an exponential family distribution , with the same parameter @xmath11 and partition function @xmath53 @xcite , proposition 1.5 . specifically , the distribution function is @xmath54 where the term @xmath55 , which we will call the _ volume factor _ , counts the number of points in @xmath14 with the same sufficient statistics @xmath56 . the moment generating function of @xmath52 is @xmath57 = z_{{a}}(\theta+\phi)/z_{{a}}(\theta ) . \label{eqnmgf - of - exponential - family}\ ] ] if the sufficient statistic is completely additive , that is , if @xmath58 , then this is a model of independent ( if not necessarily iid ) data . in general , however , the choice of sufficient statistics may impose , or capture , dependence between observations . because we are considering exponential families defined on increasingly large sets of observations , it is convenient to introduce some notation related to multiple statistics . fix @xmath59 such that @xmath23 . then @xmath60 , and we will sometimes write this function @xmath61 , where the first argument is in @xmath14 and the second in @xmath62 . we will have frequent recourse to the increment to the sufficient statistic , @xmath63 . the volume factor @xmath64 is defined as before , but we shall also consider , for each observable value @xmath56 of the sufficient statistics for @xmath4 and increment @xmath65 of the sufficient statistics from @xmath4 to @xmath5 , the _ joint volume factor , _ @xmath66 and the _ conditional volume factor , _ @xmath67 as we will see , these volume factors play a key role in characterizing projectibility . in this section we characterize projectibility in terms of the increments of the vector of sufficient statistics . in particular we show that exponential families are projective if , and only if , their sufficient statistics decompose into separate additive contributions from disjoint observations in a particularly nice way which we formalize in the following definition . the sufficient statistics of the family @xmath35 have _ separable increments _ when , for each @xmath23 , @xmath68 , the range of possible increments @xmath65 is the same for all @xmath69 , and the conditional volume factor is constant in @xmath69 , that is , @xmath70 . it is worth noting that the property of having separable increments is an intrinsic property of the family @xmath8 that depends only on the functional forms of the sufficient statistics @xmath71 and not on the model parameters @xmath45 . this follows from the fact that , for any @xmath4 , the probability distributions @xmath72 have identical support @xmath73 . thus , this property holds for all of @xmath11 or none of them . the main result of this paper is then as follows . the exponential family @xmath41 is projective if and only if the sufficient statistics @xmath74 have separable increments . [ thmprojectible - iff - separable ] because projectibility implies separable increments , it also carries statistical - independence implications . specifically , it implies that the increments to the sufficient statistics are statistically independent , and that @xmath75 and @xmath18 are conditionally independent given increments to the sufficient statistic . interestingly , independent increments for the statistic are necessary but not quite sufficient for projectibility . these claims are all made more specific in the propositions which follow . we first show that projectibility implies that the sufficient statistics have independent increments . in fact , a stronger results holds , namely that the increments of the sufficient statistics are independent of the actual sequence . below we will write @xmath76 to signify @xmath77 . if the exponential family @xmath78 is projective , then sufficient statistics @xmath79 have independent increments , that is , @xmath39 implies that @xmath80 under all @xmath11 . [ propproj - to - ii ] in a projective exponential family , @xmath81 . [ propincrement - indep - of - old - observable ] we note that independent increments for the sufficient statistics @xmath52 in no way implies independence of the actual observations @xmath18 . as a simple illustration , take the one - dimensional ising model , where @xmath82 , each @xmath83 , @xmath1 consists of all intervals from @xmath84 to @xmath28 , and the single sufficient statistic @xmath85 . clearly , @xmath86 when @xmath87 , otherwise @xmath88 . since @xmath89 , by theorem [ thmprojectible - iff - separable ] , the model is projective . by proposition [ propproj - to - ii ] , then , increments of @xmath90 should be independent , and direct calculation shows the probability of increasing the sufficient statistic by 1 is @xmath91 , no matter what @xmath92 are . while the sufficient statistic has independent increments , the random variables @xmath93 are all dependent on one another .. ] [ example : ising - model ] the previous results provide a way , and often a simple one , for checking whether projectibility fails : if the sufficient statistics do not have independent increments , then the family is not projective . as we will see , this test covers many statistical models for networks . it is natural to inquire into the converse to these propositions . it is fairly straightforward ( if somewhat lengthy ) to show that independent increments for the sufficient statistics implies that the joint volume factor separates . if an exponential family has independent increments , @xmath94 , then its joint volume factor separates , @xmath95 , and the distribution of @xmath90 is projective . [ propii - to - volume - factor - separation ] however , independent increments for the sufficient statistics do _ not _ imply that separable increments ( hence projectibility ) , as shown by the next counter - example . hence independent increments are a necessary but not sufficient condition for projectibility . suppose that @xmath96 , and @xmath97 . ( thus there are 20 possible values for @xmath37 . ) let @xmath98 so that @xmath99 . further , let @xmath100 it is not hard to verify that @xmath76 is always either @xmath101 or @xmath102 . it is also straightforward to check that @xmath103 for all combinations of @xmath56 and @xmath65 , implying that @xmath104 , and that the joint volume factor separates . on the other hand , the _ conditional _ volume factors are not constant in @xmath69 , as @xmath105 while @xmath106 . thus , the sufficient statistic has independent increments , but does not have separable increments . since projective families have separable increments ( proposition [ propproj - to - si ] ) , this can not be a projective family . ( this can also be checked by a direct and straightforward , if even more tedious , calculation . ) we conclude this section with a final observation . butler , in @xcite , showed that when observations follow from an iid model with a minimal sufficient statistic , the predictive distribution for the next observation can be written entirely in terms of how different hypothetical values would change the sufficient statistic ; cf . this predictive sufficiency property carries over to our setting . [ thmpredictive - sufficiency ] in a projective exponential family , the distribution of @xmath107 conditional on @xmath18 depends on the data only through @xmath76 . the main implications among our results are summarized in figure [ figimplications ] . . probabilistic properties of the models are on the right , and algebraic / combinatorial properties of the sufficient statistic are on the left . ] _ exponential families of time series_. as the example of the ising model in section [ secindependence - properties ] ( page ) makes clear , our theorem applies whenever we need an exponential family to be projective , not just when the data are networks . in particular , they apply to exponential families of time series , where @xmath2 is the natural or real number line ( or perhaps just its positive part ) , and the elements of @xmath1 are intervals . an exponential family of stochastic processes on such a space has projective parameters if , and only if , its sufficient statistics have separable increments , and so only if they have independent increments . _ transformation of parameters_. allowing the dimension of @xmath11 to be fixed , but for its components to change along with @xmath6 , does not really get out of these results . specifically , if @xmath11 is to be re - scaled in a way that is a function of @xmath6 alone , we can recover the case of a fixed @xmath11 by `` moving the scaling across the inner product , '' that is , by re - defining @xmath52 to incorporate the scaling . with a sample - invariant @xmath11 , it is this transformed @xmath90 which must have separable increments . other transformations can either be dealt with similarly , or amount to using a nonuniform base measure ; see below . _ statistical - mechanical interpretation_. it is interesting to consider the interpretation of our theorem , and of its proof , in terms of statistical mechanics . as is well known , the `` canonical '' distributions in statistical mechanics are exponential families ( boltzmann gibbs distributions ) , where the sufficient statistics are `` extensive '' physical observables , such as energy , volume , the number of molecules of various species , etc . , and the natural parameters are the corresponding conjugate `` intensive '' variables , such as , respectively , ( inverse ) temperature , pressure , chemical potential , etc . equilibrium between two systems , which interact by exchanging the variables tracked by the extensive variables , is obtained if and only if they have the same values of the intensive parameters @xcite . in our terms , of course , this is simply projectibility , the requirement that the same parameters hold for all sub - systems . what we have shown is that for this to be true , the increments to the extensive variables must be completely unpredictable from their values on the sub - system . furthermore , notice the important role played in both halves of the proof by the separation of the joint volume factor , @xmath108 . in terms of statistical mechanics , a macroscopic state is a collection of microscopic configurations with the same value of one or more macroscopic observables . the boltzmann entropy of a macroscopic state is ( proportional to ) the logarithm of the volume of those microscopic states @xcite . if we define our macroscopic states through the sufficient statistics , then their boltzmann entropy is just @xmath109 . thus , the separation of the volume factor is the same as the additivity of the entropy across different parts of the system , that is , the entropy is `` extensive . '' our results may thus be relevant to debates in statistical mechanics about the appropriateness of alternative , nonextensive entropies ; cf . @xcite . _ beyond exponential families_. it is not clear just how important it is that we have an exponential family , as opposed to a family admitting a finite - dimensional sufficient statistic . as is well known , the two concepts coincide under some regularity conditions @xcite , but not quite strictly , and it would be interesting to know whether or not the exponential form of equation ( [ eqnexponential - family - density ] ) is strictly required . we have attempted to write the proofs in a way which minimizes the use of this form ( in favor of the neyman factorization , which only uses sufficiency ) , but have not succeeded in eliminating it completely . we return to this matter in the conclusion . _ prediction_. we have focused on the implications of projectibility for parametric inference . exponential families are , however , often used in statistics and machine learning as generative models in applications where the only goal is prediction @xcite , and so ( to quote butler @xcite ) `` all parameters are nuisance parameters . '' but even in then , it must be possible to consistently extend the generative model s distribution for the training data to a joint distribution for training and testing data , with a single set of parameters shared by both old and new data . while this requirement may seem too trivial to mention , it is , precisely , projectibility . _ growing number of parameters_. in the proof of theorem [ thmprojectible - iff - separable ] , we used the fact that @xmath52 , and hence @xmath11 , has the same dimension for all @xmath110 . there are , however , important classes of models where the number of parameters is allowed to grow with the size of the sample . particularly important , for networks , are models where each node is allowed a parameter ( or two ) of its own , such as its expected degree ; see , for instance , the classic @xmath111 model of @xcite , or the `` degree - corrected block models '' of @xcite . we can formally extend theorem [ thmprojectible - iff - separable ] to cover some of these cases including those two particular specifications as follows . assume that @xmath52 has a dimension which is strictly nondecreasing as @xmath6 grows , that is , @xmath112 whenever @xmath23 . furthermore , assume that the set of parameters @xmath113 only grows , and that the meaning of the old parameters is not disturbed . that is , under projectibility we should have @xmath114 for any fixed pair @xmath23 , we can accommodate this within the proof of theorem [ thmprojectible - iff - separable ] by re - defining @xmath52 to be a mapping from @xmath14 to @xmath115 , where the extra @xmath116 components of the vector are always zero . the extra parameters in @xmath117 then have no influence on the distribution of @xmath18 and are unidentified on @xmath6 , but we have , formally , restored the fixed - parameter case . the `` increments '' of the extra components of @xmath118 are then simply their values on @xmath37 , and , by the theorem , the range of values for these statistics , and the number of configurations on @xmath62 leading to each value , must be equal for all @xmath119 . adapting our conditions for the asymptotic convergence of maximum likelihood estimators ( section [ secconsistency - of - mle ] ) to the growing - parameter setting is beyond our scope here . _ nonuniform base measures_. if the exponential densities in ( [ eqnexponential - family - density ] ) are defined with respect to nonuniform base measures different from the counting measures , the sufficient statistics need not have separable increments . in the supplementary material @xcite we address this issue and describe the modifications and additional assumptions required for our analysis to remain valid . we thank an anonymous referee and pavel krivitsky for independently brining up this subtle point to our attention . statistical inference in an exponential family naturally centers on the parameter @xmath11 . as is well known , the maximum likelihood estimator @xmath120 takes a particularly simple form , obtainable using the fact [ which follows from equation ( [ eqnmgf - of - exponential - family ] ) ] that @xmath121 $ ] , @xmath122}{z_{{a}}^2(\theta ) } , \nonumber \\[-8pt ] \\[-8pt ] \nonumber t_{{a}}(x ) & = & \mathbf{e}_{{\widehat{\theta } } } [ t_{{a } } ] .\end{aligned}\ ] ] in words , the most likely value of the parameter is the one where the expected value of the sufficient statistic equals the observed value . assume the conditions of theorem [ thmprojectible - iff - separable ] hold , so that the parameters are projective and the sufficient statistics have ( by lemma [ lemmaseparable - volume - factor - implies - projectible - increments ] ) independent increments . define the logarithm of the partition function @xmath123 . would be the helmholtz free energy . ] suppose that @xmath124 where @xmath125 is some positive - valued measure of the size of @xmath6 , @xmath126 a positive monotone - increasing function of it and @xmath127 is differentiable ( at least at @xmath11 ) . then , by equation ( [ eqnmgf - of - exponential - family ] ) for the moment generating function , the cumulant generating function of @xmath52 is @xmath128 from the basic properties of cumulant generating functions , we have @xmath129 = \nabla_{\phi } \kappa _ { { a},\theta}(0 ) = r_{|{a}| } \nabla a(\theta ) . \label{eqnexpect - of - suff - stat}\ ] ] substituting into equation ( [ eqnimplicit - exp - family - mle ] ) , @xmath130 thus to control the convergence of @xmath120 , we must control the convergence of @xmath131 . consider a growing sequence of sets @xmath6 such that @xmath132 . since @xmath52 has independent increments , and the cumulant generating functions for different @xmath6 are all proportional to each other , we may regard @xmath52 as a time - transformation of a lvy process @xmath133 that is , there is a continuous - time stochastic process @xmath134 with iid increments , such that @xmath135 has cumulant generating function @xmath136 , and @xmath137 . note that @xmath52 itself does not have to have iid increments , but rather the distribution of the increment @xmath138 must only depend on @xmath139 . specifically , from lemma [ corconditional - moment - generating - function ] and equation ( [ eqnscaling - factor ] ) , the cumulant generating function of the increment must be @xmath140 $ ] . the scaling factor homogenizes ( so to speak ) the increments of @xmath90 . writing the sufficient statistic as a transformed lvy process yields a simple proof that @xmath120 is strongly ( i.e. , almost - surely ) consistent . since a lvy process has iid increments , by the strong law of large numbers @xmath141 converges almost surely ( @xmath44 ) to @xmath142 $ ] @xcite . since @xmath143 , it follows that @xmath144 $ ] a.s . ( @xmath44 ) as well ; but this limit is @xmath145 . thus the mle converges on @xmath11 almost surely . we have thus proved suppose that the model @xmath44 is projective , and that the log partition function obeys equation ( [ eqnscaling - factor ] ) for each @xmath9 . then the maximum likelihood estimator exists and is strongly consistent . we may extend this in a number of ways . first , if the scaling relation equation ( [ eqnscaling - factor ] ) holds for a particular @xmath11 ( or set of @xmath11 ) , then @xmath131 will converge almost surely for that @xmath11 . thus , strong consistency of the mle may in fact hold over certain parameter regions but not others . second , when @xmath146 , all components of @xmath52 must be scaled by the _ factor @xmath126 . making the expectation value of one component of @xmath90 be @xmath147 while another was @xmath148 ( e.g. ) would violate equation ( [ eqnexpect - of - suff - stat ] ) and so equation ( [ eqnscaling - factor ] ) as well . finally , while the exact scaling of equation ( [ eqnscaling - factor ] ) , together with the independence of the increments , leads to strong consistency of the mle , ordinary consistency ( convergence in probability ) holds under weaker conditions . specifically , suppose that log partition function or free energy scales in the limit as the size of the assemblage grows , @xmath149 we give examples toward the end of section [ secergms ] below . we may then use the following theorem : suppose that an exponential family shows approximate scaling , that is , equation ( [ eqnasymptotic - scaling ] ) holds , for some @xmath11 . then , for any measurable set @xmath150 , @xmath151 where @xmath152 } , \label{eqnrate - function}\ ] ] and @xmath153 and @xmath154 are , respectively , the interior and the closure of @xmath155 . [ thmldp ] when the limits in equations ( [ eqnldp - lower ] ) and ( [ eqnldp - upper ] ) coincide , which they will for most nice sets @xmath155 , we may say that @xmath156 since @xmath157 is minimized at 0 when @xmath158 , , by a second order taylor expansion , @xmath159 , where @xmath160 acts as the fisher information rate ; cf . equation ( [ eqnldp - for - suff - stat ] ) holds in particular for any neighborhood of @xmath145 , and for the complement of such neighborhoods , where the infimum of @xmath161 is strictly positive . thus @xmath131 converges in probability to @xmath145 , and @xmath162 , for all @xmath11 where equaiton ( [ eqnasymptotic - scaling ] ) holds.=-1 heuristically , when equation ( [ eqnasymptotic - scaling ] ) holds but equation ( [ eqnscaling - factor ] ) fails , we may imagine approximating the actual collection of dependent and heterogeneous random variables with an average of iid , homogenized effective variables , altering the behavior of the global sufficient statistic @xmath90 by no more than @xmath163 . in statistical - mechanical terms , this means using renormalization @xcite . probabilistically , the existence of a limiting ( scaled ) cumulant generating function is a weak dependence condition @xcite , section v.3.2 . while under equation ( [ eqnscaling - factor ] ) we identified the @xmath52 process with a time - transformed lvy process , now we can only use a central limit theorem to say they are close @xcite , section v.3.1 , reducing almost - sure to stochastic convergence ; see @xcite on the relation between central limit theorems and renormalization . in any event , asymptotic scaling of the log partition function implies @xmath120 is consistent . as mentioned in the , our general results about projective structure in exponential families arose from questions about exponential random graph models of networks . to make the application clear , we must fill in some details regarding ergms . given a group of @xmath28 nodes , the network among them is represented by the binary @xmath164 _ adjacency matrix _ @xmath165 , where @xmath166 if there is a tie from @xmath167 to @xmath168 and is 0 otherwise . ( undirected graphs impose @xmath169 . ) we may also have covariates for each node , say @xmath170 . our projective structure will in fact be that of looking at the sub - graphs among larger and larger groups of nodes . that is , @xmath6 is the sub - network among the first @xmath28 nodes , and @xmath171 is the sub - network among the first @xmath33 nodes . the graph or adjacency matrix itself is the stochastic process which is to have an exponential family distribution , conditional on the covariates @xmath172 ( we are only interested in the exponential - family distribution of the graph holding the covariates fixed . ) as mentioned above , the components of @xmath90 typically count the number of occurrences of various sub - graphs or motifs as edges , triangles , larger cliques , `` @xmath0-stars '' ( @xmath0 nodes connected through a central node ) , etc.perhaps interacted with values of the nodal covariates . the definition of @xmath90 may include normalizing the counts of these `` motifs '' by data - independent combinatorial factors to yield densities . a _ dyad _ consists of an unordered pair of individuals . in a dyadic independence model , each dyad s configuration is independent of every other dyad s ( conditional on @xmath134 ) . in an ergm , dyadic independence is equivalent to the ( vector - valued ) statistic @xmath90 adding up over dyads , @xmath173 that is , the statistic can be written as a sum of terms over the information available for each dyad . in particular , in _ block models _ @xcite , @xmath170 is categorical , giving the type of node @xmath167 , and the vector of sufficient statistics counts dyad configurations among pairs of nodes of given pairs of types . dyadic independence implies projectibility : since all dyads have independent configurations , each dyad makes a separate additive contribution to @xmath90 . going from @xmath174 to @xmath28 nodes thus adds @xmath28 terms , unconstrained by the configuration among the @xmath174 nodes . @xmath90 thus has separable increments , implying projectibility by theorem [ thmprojectible - iff - separable ] . ( adding a new node adds only edges between the old nodes and the new , without disturbing the old counts . ) as the distribution factorizes into a product of @xmath175 terms , each of exactly the same form , the log partition function scales exactly with @xmath175 , and the conclusions of section [ secconsistency - of - mle ] imply the strong consistency of the maximum likelihood estimator . thus hold , and these models are projective . ] this result thus applies to the well - studied @xmath176-model @xcite . typically , however , ergms are _ not _ dyadic independence models . in many networks , if nodes @xmath167 and @xmath168 are both linked to @xmath0 , then @xmath167 and @xmath168 are unusually likely to be directly linked . this will of course happen if nodes of the same type are especially likely to be friends ( `` homophily '' @xcite ) , since then the posterior probability of @xmath167 and @xmath168 being of the same type is elevated . however , it can also be modeled directly . the direct way to do so is to introduce the number ( or density ) of triangles as a sufficient statistic , but this leads to pathological degeneracy @xcite , and modern specifications involve a large set of triangle - like motifs @xcite . empirically , when using such specifications , one often finds a nontrivial coefficient for such `` transitivity '' or `` clustering , '' over and above homophily @xcite . it is because of such findings that we ask whether the parameters in these models are projective . sadly , no statistic which counts triangles , or larger motifs , can have the nice additive form of dyad counts , no matter how we decompose the network . take , for instance , triangles . any given edge among the first @xmath28 nodes _ could _ be part of a triangle , depending on ties to the next node . thus to determine the number of triangles among the first @xmath177 nodes , we need much more information about the sub - graph of the first @xmath28 nodes than just the number of triangles among them . indeed , we can go further . the range of possible increments to the number of triangles changes with the number of existing triangles . this is quite incompatible with separable increments , so , by ( [ thmprojectible - iff - separable ] ) , the parameters can not be projective . we remark that the nonprojectibility of markov graphs @xcite , a special instance of ergms where the sufficient statistics count edges , @xmath0-stars and triangles , was noted in @xcite . parallel arguments apply to the count of any motif of @xmath0 nodes , @xmath178 . any given edge ( or absence of an edge ) among the first @xmath28 nodes could be part of such a motif , depending on the edges involving the next @xmath179 nodes . such counts are thus not nicely additive . for the same reasons as with triangles , the range of increments for such statistics is not constant , and nonseparable increments imply nonprojective family . while these ergms are not projective , some of them may , as a sort of consolation prize , still satisfy equation ( [ eqnasymptotic - scaling ] ) . for instance , in models where @xmath90 has two elements , the number of edges and the ( normalized ) number of triangles or of 2-stars , the log partition function is known to scale like @xmath175 as the number of nodes @xmath180 , at least in the parameter regimes where the models behave basically like either very full or very empty erds rnyi networks @xcite . ( we suspect , from @xcite , that similar results apply to many other ergms . ) thus , by equation ( [ eqnldp - for - suff - stat ] ) , if we fix a large number @xmath28 of nodes and generate a graph @xmath165 from @xmath181 , the probability that the mle @xmath182 will be more than @xmath183 away from @xmath11 will be exponentially small in @xmath175 and @xmath184 . since these models are not projective , however , it is impossible to _ improve _ parameter estimates by getting more data , since parameters for smaller sub - graphs just can not be extrapolated to larger graphs ( or vice versa ) . we thus have a near - dichotomy for ergms . dyadic independence models have separable and independent increments to the statistics , and the resulting family is projective . however , specifications where the sufficient statistics count larger motifs can not have separable increments , and projectibility does not hold . such an ergm may provide a good description of a given social network on a certain set of nodes , but it can not be projected to give predictions on any larger or more global graph from which that one was drawn . if an ergm is postulated for the whole network , then inference for its parameters must explicitly treat the unobserved portions of the network as missing data ( perhaps through an expectation - maximization algorithm ) , though of course there may be considerable uncertainty about just how much data is missing . specifications for exponential families of dependent variables in terms of joint distributions are surprisingly delicate ; the statistics must be chosen extremely carefully , in order to achieve separable increments . ( conditional specifications do not have this problem . ) this has , perhaps , been obscured in the past by the emphasis on using exponential families to model multivariate but independent cases , as iid models are always projective . network models , one of the outstanding applications of exponential families , suffer from this problem in an acute form . dyadic independence models are projective models , but are sociologically extremely implausible , and certainly do not manage to reproduce the data well . more interesting specifications , involving clustering terms , never have separable increments . we thus have an impasse which it seems can only be resolved by going to a different family of specifications . one possibility which , however , requires more and different data is to model the evolution of networks over time @xcite . in particular , hanneke , fu and xing @xcite consider situations where the distribution of the network at time @xmath185 conditional on the network at time @xmath56 follows an exponential family . even when the statistics in the conditional specification include ( say ) changes in the number of triangles , the issues raised above do not apply . roughly speaking , the issue with the nonprojective ergm specifications , and with other nonprojective exponential families , is that the dependency structure corresponding to the statistics allows interactions between arbitrary collections of random variables . it is not possible , with those statistics , to `` screen off '' one part of the assemblage from another by conditioning on boundary terms . suppose our larger information set @xmath36 consists of two nonoverlapping and strictly smaller information sets , @xmath39 and @xmath186 , plus the new observation obtained by looking at both @xmath6 and @xmath187 . ( for instance , the latter might be the edges between two disjoint sets of nodes . ) then the models which work properly are ones where the sufficient statistic for @xmath36 partitions into marginal terms from @xmath6 and @xmath187 , plus the interactions strictly between them : @xmath188 . in physical language @xcite , the energy for the whole assemblage needs to be a sum of two `` volume '' terms for its sub - assemblages , plus a `` surface '' term for their interface . the network models with nonprojective parameters do not admit such a decomposition ; every variable , potentially , interacts with every other variable . one might try to give up the exponential family form , while keeping finite - dimensional sufficient statistics . we suspect that this will not work , however , since lauritzen @xcite showed that whenever the sufficient statistics form a semi - group , the models must be either ordinary exponential families , or certain generalizations thereof with much the same properties . we believe that there exists a purely algebraic characterization of the sufficient statistics compatible with projectibility , but must leave this for the future . one reason for the trouble with ergms is that every infinite exchangeable graph distribution is actually a mixture over projective dyadic - independence distributions @xcite , though not necessarily ones with a finite - dimensional sufficient statistic . along any one sequence of sub - graphs from such an infinite graph , in fact , the densities of all motifs approach limiting values which pick out a unique projective dyadic - independence distribution @xcite ; cf . also @xcite . this suggests that an alternative to parametric inference would be nonparametric estimation of the limiting dyadic - independence model , by smoothing the adjacency matrix ; this , too , we pursue elsewhere . for notation in this section , without loss of generality , fix a generic pair of subsets @xmath23 and a value of @xmath11 . we will write a representative point @xmath189 as @xmath190 , with @xmath119 and @xmath191 . also , we abbreviate @xmath192 , for @xmath119 and @xmath191 by @xmath193 . for clarity , we prove the two directions separately . first we show that projectability implies separable increments . if the exponential family @xmath194 is projective , then the sufficient statistics @xmath195 have separable increments , that is , @xmath23 implies that @xmath70 . [ propproj - to - si ] by projectibility , for each @xmath11 , @xmath196 which implies that , for all @xmath119 , @xmath197 re - writing the left - hand side of equation ( [ eqconstant - conditional - partition - function ] ) as a sum over the set @xmath198 of values which the increment @xmath199 to the sufficient statistic might take yields @xmath200 where the joint volume factor is defined in ( [ eqjointvolfactor ] ) . since the right - hand side of equation ( [ eqnconditional - partition - function - as - sum - over - increment - values ] ) is the same for all @xmath69 , so must the left - hand side . observe that this left - hand side is the laplace transform of the function @xmath201 . the latter is a nonnegative function which defines a measure on @xmath202 , whose support is @xmath198 . hence , @xmath203 is the laplace transform of a discrete probability measure in @xmath202 . but the denominator in the inner sum is just @xmath204 , no matter what @xmath69 might be . as a total and not a partial function on @xmath205 . ] so we have that for any @xmath206 , and all @xmath207 , @xmath208 since both sides of equation ( [ eqntwo - laplace - transforms ] ) are laplace transforms of probability measures on a common space , and the equality holds on all of @xmath209 , which contains an open set , we may conclude that the two measures are equal @xcite , theorem 7.3 . this means that they have the same support , @xmath210 , and that they have the same density with respect to counting measure on @xmath211 . as they also have the same normalizing factor ( viz . , @xmath204 ) , we get that @xmath212 . since the points @xmath69 and @xmath213 are arbitrary , this last property is precisely having separable increments . next , we prove the reverse direction , namely that separable increments imply projectibility . this is clearer with some preliminary lemmas . [ lemmaseparable - increments - implies - volume - factor - separation ] if the sufficient statistics have separable increments , then the joint volume factors factorize , that is , @xmath214 for all @xmath215 , @xmath56 and @xmath65 . by definition , @xmath216 when the statistic has separable increments , @xmath217 , so @xmath218 proving the claim . [ lemmaseparable - volume - factor - implies - projectible - increments ] if the joint volume factor factorizes , then the sufficient statistics has independent increments , and the distribution of the sufficient static is projective . without loss of generality , fix a value @xmath56 for @xmath52 and @xmath65 for @xmath76 . by the law of total probability and the definition of the volume factor , @xmath219 if the volume factor factorizes , so that @xmath108 , then we obtain @xmath220 \biggl [ \frac{z_{a}(\theta)}{z_{b}(\theta ) } v_{{{b}\setminus{a}}}(\delta ) e^{\langle\theta,\delta\rangle } \biggr].\ ] ] it then follows that @xmath221 and thus that @xmath90 has independent increments . to establish the projectibility of the distribution of @xmath90 , sum over @xmath65 @xmath222 since @xmath223 , and both distributions must sum to 1 over @xmath56 , we can conclude that @xmath224 , and hence that the distribution of the sufficient statistic is projective . [ lemmamixed - joint - prob - of - obs - and - stat ] if the sufficient statistics of an exponential family have separable increments , then @xmath225 abbreviate @xmath226 by @xmath56 . by the law of total probability , @xmath227 since @xmath118 is sufficient , and @xmath228 for all @xmath229 in the sum , @xmath230 by parallel reasoning , @xmath231 therefore , @xmath232 if the statistic has separable increments , then @xmath233 , and the conclusion follows . the lemma does _ not _ follow merely from the joint volume factor separating , @xmath234 . the conditional volume factor must also be constant in @xmath69 . if the sufficient statistic of an exponential family has separable increments , then the family is projective . [ propsi - to - proj ] we calculate the marginal probability of @xmath18 in @xmath235 , by integrating out the increment to the sufficient statistic . ( the set of possible increments , @xmath211 , is the same for all @xmath69 , by separability . ) once again , we abbreviate @xmath226 by @xmath56 : @xmath236 these steps use , in succession : lemma [ lemmamixed - joint - prob - of - obs - and - stat ] ; the fact that conditional probabilities sum to 1 ; the projectibility of the sufficient statistics ( via lemmas [ lemmaseparable - increments - implies - volume - factor - separation ] and [ lemmaseparable - volume - factor - implies - projectible - increments ] ) ; and the definition of @xmath237 . proof of proposition [ propproj - to - ii ] by proposition [ propproj - to - si ] , a projective family has separable increments , and by lemma [ lemmaseparable - volume - factor - implies - projectible - increments ] , separable increments implies independent increments . proof of proposition [ propincrement - indep - of - old - observable ] by proposition [ propproj - to - si ] , every projective exponential family has separable increments . by lemma [ lemmamixed - joint - prob - of - obs - and - stat ] , in an exponential family with separable increments , @xmath225 therefore , using projectibility , @xmath238 by the definition of @xmath239 , @xmath240 , so @xmath241 but , by lemma [ lemmaseparable - volume - factor - implies - projectible - increments ] , the sufficient statistics have a projective distribution with independent increments , implying @xmath242 therefore , @xmath243 and so @xmath244 . proof of proposition [ propii - to - volume - factor - separation ] below we prove that if the suffiicient statistics of an exponential family have independent increments , then the volume factor separates , and the distribution of the statistic is projective . since @xmath118 is a sufficient statistic , by the neyman factorization theorem ( @xcite , theorem 2.21 , page 89 ) , @xmath245 in light of equation ( [ eqnexponential - family - density ] ) , we may take @xmath246 . abbreviating @xmath226 by @xmath56 and @xmath193 by @xmath65 , it follows that @xmath247 by independent increments , however , @xmath248 whence it follows that , for some functions @xmath249 , @xmath250 and @xmath251 and @xmath252 to proceed , we must identify the new @xmath253 and @xmath0 functions . to this end , recalling that @xmath237 is the number of @xmath254 configurations such that @xmath255 , we have @xmath256 and , at the same time , @xmath257 clearly , then , @xmath258 while @xmath259 . since @xmath260 and @xmath261 , we need @xmath262 , and may take @xmath263 for simplicity . this allows us to write @xmath264 which is exactly the assertion that the volume factor separates . turning to the @xmath253 functions , we sum over @xmath65 again to obtain the marginal distribution of @xmath52 , @xmath265 now , we finally we use the exponential - family form . specifically , we know that @xmath266 so that @xmath267 , @xmath268 . therefore , @xmath269 and normalization now forces @xmath270 as desired . proof of theorem [ thmpredictive - sufficiency ] the conditional density of @xmath107 given @xmath18 is just the ratio of joint to marginal densities ( both with the same @xmath11 , by projectibility ) , @xmath271 which is an exponential family with parameter @xmath11 , sufficient statistic @xmath76 , and partition function @xmath272 . proof of theorem [ thmldp ] under equation ( [ eqnasymptotic - scaling ] ) , the cumulant generating function also scales asymptotically , @xmath273 . since @xmath274 is differentiable , the grtner ellis theorem of large deviations theory @xcite , chapter v , implies that @xmath131 obeys a large deviations principle with rate @xmath126 , and rate function given by equation ( [ eqnrate - function ] ) , which is to say , equations ( [ eqnldp - lower ] ) and ( [ eqnldp - upper ] ) . [ corconditional - moment - generating - function ] the moment generating function of @xmath76 is @xmath275 from the proof of theorem [ thmpredictive - sufficiency ] , @xmath276 has an exponential family distribution with sufficient statistic @xmath76 . thus we may use equation ( [ eqnmgf - of - exponential - family ] ) to find the moment generating function of @xmath76 conditional on @xmath18 , @xmath277 since , however , @xmath244 ( proposition [ propincrement - indep - of - old - observable ] ) , equation ( [ eqnconditional - moment - generating - function ] ) must also give the unconditional moment generating function . we thank luis carvalho , aaron clauset , mark handcock , steve hanneke , brain karrer , sergey kirshner , steffen lauritzen , david lazer , john miller , martina morris , jennifer neville , mark newman , peter orbanz , andrew thomas and chris wiggins , for valuable conversations ; an anonymous referee of an earlier version for pointing out a gap in a proof ; and audiences at the boston university probability and statistics seminar , and columbia university s applied math seminar .

the growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure . typically , however , these are models for the entire network , while the data consists only of a sampled sub - network . parameters for the whole network , which is what is of interest , are estimated by applying the model to the sub - network . this assumes that the model is _ consistent under sampling _ , or , in terms of the theory of stochastic processes , that it defines a projective family . focusing on the popular class of exponential random graph models ( ergms ) , we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models , and that satisfying it drastically limits ergm s expressive power . these results are actually special cases of more general results about exponential families of dependent random variables , which we also prove . using such results , we offer easily checked conditions for the consistency of maximum likelihood estimation in ergms , and discuss some possible constructive responses .

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Proceed to summarize the following text: recent x - ray studies of seyfert 2 galaxies with and have shown that the 0.110 kev spectra of these sources are rich in emission lines at both soft and hard energies ( e.g. , guainazzi et al . 1999 ; turner et al . the interpretation of the emission lines is problematic because of ambiguities about line blending , line profiles , and line flux distribution that are all poorly constrained by and . the data are consistent with emission from gas in photoionization equilibrium ( e.g. , netzer , turner , & george 1998 and references therein ) , but there are also attempts to fit the spectra by a two - temperature gas in collisional equilibrium ( ueno et al . 1994 ) , presumably due to starburst emission . x - ray observations at high resolution both spatially and spectrally are crucial to determining the origin of the x - ray lines in seyfert 2s , a task for which is uniquely suited . here we present a 60 ks hetgs spectrum of the seyfert 2 galaxy . previous and observations of this source revealed several emission lines at both soft and hard energies ( sako et al . 2000a ; guainazzi et al . 1999 ; matt et al . 1996 ) . in a companion paper , focusing on the zeroth - order acis image , we established that several components contribute to the x - ray emission from . in particular , we found that @xmath2 60% of the x - ray flux at 2 kev is due to an extended component on scales @xmath2 2.3 , while at harder energies the contribution from a compact ( 0.8 ) region prevails . importantly , the acis spectrum of the latter component exhibits several emission lines including a prominent ( ew @xmath2 2.5 kev ) line . in this paper , we concentrate on the analysis of the hetgs spectrum and on the implications for the physical conditions of the emitting gas . we also use a simultaneous observation to derive useful constraints on the intrinsic nuclear x - ray continuum . at the distance of the galaxy ( @xmath2 4 mpc ) , 1=19 pc . was observed with the high energy transmission grating spectrometer ( hetgs ; canizares et al . 2000 , in prep . ) on 2000 june 6 , with acis - s ( garmire et al . 2000 ) in the focal plane . the total net exposure was 60,223 s. details on the observation are given in our companion paper . the hetgs carries two mirror assemblies , the high energy grating ( heg ) and medium energy grating ( meg ) . the nuclear heg and meg spectra were extracted in a narrow ( 15 pixel ) rectangular region centered on the zeroth - order position , avoiding contamination from the dispersed spectra of the nearby serendipitous sources . although a few of the serendipitous sources exhibit emission lines in their acis spectra at both soft and hard x - rays , the 0.58 kev flux of the brightest one is a factor 5 weaker than the nucleus . we thus believe that contamination to the hetgs nuclear spectrum ( in the regions of overlap of the dispersed spectra ) is negligible . the heg and meg spectra were gain corrected and flux calibrated using ancillary response files generated with the ` ciao ` software . only the first order heg and meg spectra were used for the analysis , as higher orders contain only a few ( 3 per bin ) counts and are not useful . the spectra were also corrected for cosmological redshift and galactic absorption , @xmath3 ( freeman et al . 1977 ) . we used a simultaneous 30 ks exposure with to constrain the higher - energy x - ray continuum emission from . the data were reduced following standard criteria ; here we use only data from the pca detector and report only on the results most relevant for the modeling of the hetgs data , leaving more details to a future paper . the source = 10.5 cm -0.8 cm was detected with the pca up to @xmath2 30 kev , with a 230 kev count rate of 7.26 @xmath4 0.04 counts s@xmath5 . we fitted the pca data in the energy range 630 kev , where relatively few lines are present ( mainly at 6.4 kev , fek@xmath6 at 7.1 kev , and /nik@xmath1 at 7.9 kev ; guainazzi et al . we find that an excellent description of the pca data ( @xmath7=56 for 62 degrees of freedom ) is obtained with a model including a `` pure '' reflection continuum from neutral gas , plus a heavily absorbed ( @xmath8 ) power law dominant at 10 kev , plus the three gaussians lines . the best - fit model and fitted parameters are in complete agreement with a previous observation of ( matt et al . 1999 , guainazzi et al . we mainly stress here the results for the intrinsic nuclear continuum : power - law photon index @xmath9 ( 90% confidence errors ) and intrinsic ( absorption - corrected ) 210 kev luminosity in the range @xmath10 , in agreement with the data ( matt et al . the first order meg spectra agree well with each other within the resolution of the grating ( @xmath2 0.023 , twice as for the heg ) . the two heg spectra also agree with each other and with the meg spectra . therefore , in order to increase the signal - to - noise ratio , the four spectra were averaged after rebinning the heg data to the meg resolution . while this procedure sacrifices the higher resolution of the heg , in most cases the lines are unresolved and the loss of resolution is thus well compensated by a cleaner detection of the lines . figure [ model - data ] shows a wide wavelength coverage view of the flux - calibrated spectrum , binned at 0.02 . a plethora of emission lines are apparent in the spectrum at all energies . the most prominent one is the line at 6.4 kev , with an ew @xmath11 kev , an unresolved core , and a hint of a broader - base component which will be discussed elsewhere . a detailed view of the = 10.5 cm -0.8 cm fe line complex ( figure [ feline ] ) shows also the fe k@xmath6line at @xmath2 7.1 kev , with an intensity of @xmath12 that of the line . we have identified another feature at around 6.6 kev , consistent with the k@xmath1 line . as shown below , our model can not explain the intensity of this feature . the high absorption column in the direction to , and the relatively faint x - ray flux of the source , do not allow any clear line detections below @xmath13 kev . table 1 gives quantitative information about the detected emission lines , including the line identification ( note some question marks due to uncertain identifications ) , measured fluxes , and ews . many h - like and he - like lines of mg , si , s and possibly ar are detected in the spectrum , together with weaker `` neutral '' lines of si , s , ar , and perhaps ca . the high resolution allows deblending of the forbidden , intercombination , and resonance lines of several of the he - like ions ( figure [ model - data ] ) . the significance of the lines detections can be judged from the corresponding uncertainties on the line fluxes . the poor signal - to - noise ratio did not allow a meaningful line - width determination , and all fwhms are consistent with the instrumental resolution . the ews are calculated with respect to the total observed continuum , which was evaluated using two independent methods . first , we selected line - free regions of the spectrum ( mostly above 23 kev ) , rebinned the data heavily , and fitted them with a smooth curve . second , we used the zeroth - order spectrum from an extraction radius consistent with the extraction width of the hetgs spectrum ( @xmath2 3 ) . the latter was fitted with a power law plus free @xmath14 , plus narrow ( width=0.05 kev ) gaussians representing all the emission lines detected in the hetgs spectrum . both methods gave consistent results . the 3 - 7 kev continuum can be described by an inverted power law with photon index @xmath15 and flux f@xmath16 . this continuum was used in the ew measurements . the uncertainty on the continuum flux is 22% or better at all wavelengths . we modeled the observed spectrum using our new line measurements , as well as the information obtained from the radial flux distribution ( from our companion paper ) and the high en- [ cols="<,^,^,^,^,^ " , ] ergy observations from ( matt et al . 1999 ) and . the modeling is based on the following observations : ( a ) the 210 continuum is flat , in @xmath17 , at long wavelengths ( as expected from an `` ionized mirror '' ) and hardens below about 5 ( as expected from a `` neutral mirror '' ) . this , plus the detection of low ionization species , suggests two distinct components ; ( b ) the ew of the line is very large , indicating iron over abundance ( e.g. netzer et al . 1998 ) ; ( c ) the extended spectrum ( outside of the central 0.8 ) is flatter than the central spectrum , suggesting that the more neutral component contributes less at larger radii . modeling is done using ion00 , the 2000 version of the photoionization code ion ( netzer 1996 ) . this includes the computation of the steady state ionization and thermal structure of the gas , and the emergent spectrum . the main ingredients of the model are : ( a ) central power - law continuum with @xmath18 , extending from 0.1100 kev and normalized to produce @xmath19 , in agreement with the and observations ; ( b ) two component gas with twice - solar metallicity . we have experimented with the density and column density of the two and found the following satisfactory combination : 1 . an _ ionized component _ with a radial density distribution of @xmath20 where @xmath21 pc and @xmath22 @xmath23 . this distribution is consistent with our measured radial flux distribution assuming most of the 0.83 flux is due to scattered continuum . 2 . a _ neutral component _ with a similar ( yet unconstrained ) radial distribution with the same @xmath24 , but with @xmath25 @xmath23 . the first component is allowed to extend all the way to 300 pc while the second is limited by its column density , arbitrarily chosen at @xmath26 @xmath27 , and is thus terminated well inside the inner 8 pc . this component can perhaps be viewed as the wall of the inner torus . the covering factors of the two components are free parameters of the model ; ( c ) the gas turbulent velocity can range from no turbulence ( pure thermal motion ) up to @xmath28 km s@xmath5 . the increased line width results in an increased intensity of all resonance lines due to continuum fluorescence ( krolik and kriss 1995 ; netzer 1996 ) . our best model requires no turbulent motion . several models have been computed , with various covering factors . a satisfactory solution is found for @xmath29 = 0.40.5 , and @xmath30 ( ionized ) = 0.10.2 , consistent with the expected opening angle of the ( hypothetical ) torus , where @xmath31 is the solid angle subtended by the gas to the illuminating source . figure [ model - data ] shows the two components alongside with the observations and figure [ ratio ] shows observed over computed intensities for the strongest lines . note that , given the gas density and location , the only free parameters are the covering factors of the two components . the overall agreement between the model and the observations is good , given the uncertainties . in particular : ( a ) the observed fluxes of most emission lines are reproduced , to within a factor of two . the 1.511 continuum shape is reproduced too . these results support the idea of the two component model , = 10.5 cm with the assumed levels of ionization and metallicity ; ( b ) the covering fractions are consistent with the suggested source geometry . the radial density distribution reproduces well the highly peaked emission of this source and the relative weak flux outside the central 16 pc . the calculated scattered continuum and extended emission lines in the inner 1657 pc are in good agreement with the ( highly uncertain , see our companion paper ) observations and there is no need to assume an additional starburst source ; ( c ) the `` neutral '' iron k@xmath1 and k@xmath6 lines are consistent with the compact , neutral component gas and the assumed metallicity . notable difficulties are the over - prediction of the fe l - shell lines around 912 and the under - prediction of the 6.6 kev feature if due to . all argon lines are also under - predicted by the model . the measured intensity of these lines are highly uncertain and any suggestion for their origin ( e.g. unusual composition ) must await better observations . we also note that the predicted k@xmath1 flux , outside the central 20 pc , is below the observed value . this may be due to the already noted uncertainty in flux measurement . we also note that a `` typical '' nlr , can produce a sizeable fraction of this k@xmath1 emission . regarding earlier x - ray observations of this source , the discovery paper by matt et al . ( 1996 ) reports an spectrum including both neutral and ionized species . netzer et al . ( 1998 ) re - analyzed the data and reported the measurements of six ionized lines plus the iron k - lines , all in reasonable agreement with the present observations . the paper includes a detailed photoionization model of the source and addresses also the k@xmath1/h@xmath6 line ratio . guainazzi et al . ( 1999 ) reported on data that include the measurement of , , and lines , as well as neutral fe - k lines . the observed fluxes are in good agreement except for whose intensity exceeds our estimate by a factor 4 . the present observations , being far superior in terms of the spatial and spectral resolution , yet limited in signal - to - noise ratio , confirm several of the suspected features of this source , such as the gas location , metallicity , and the various components . this , and the recently published observations of mkn 3 ( sako et al . 2000b ) show , for the first time , that centrally illuminated ionized gas in seyfert 2 galaxies can have very different distributions in different sources . while in mkn 3 most of the line emission is spread over several hundred parsecs , this in not the case in , where the ionized gas is highly concentrated near the center and most of its flux originates within the central @xmath32 pc . moreover , the new observations show that the more neutral gas is even more concentrated , and its dimension may be as small as a few parsecs . while we do not want to speculate about the origin of this difference , we note the large difference in luminosity between these two sources . it is therefore possible that the highly ionized gas in seyfert galaxies has dimensions that are regulated by the central source s x - ray luminosity . the new observations of the seyfert 2 galaxy enable the very first detailed analysis of the physical conditions and the gas distribution in the inner 200 pc of this source . our observations show the emission to be highly concentrated within the inner 60 pc and suggest that emission within this volume is entirely due to the reprocessing of the obscured central source s radiation . an even smaller , more neutral component is seen through emission of low ionization iron lines and hard reflected continuum . this is the first determination of the ionized gas distribution in the inner 100 pc region of a seyfert 2 galaxy .

results from a 60 ks hetgs observation of the nearby seyfert 2 are presented . the spectrum shows a wealth of emission lines at both soft and hard x - rays , including lines of ne , mg , si , s , ar , ca , and fe , and a prominent line at 6.4 kev . we identify several of the he - like components and measure several of the lyman lines of the h - like ions . the lines profiles are unresolved at the limited signal - to - noise ratio of the data . our analysis of the zeroth - order image in a companion paper constrains the size of the emission region to be 2060 pc , suggesting that emission within this volume is almost entirely due to the reprocessing of the obscured central source . here we show that a model containing two distinct components can reproduce almost all the observed properties of this gas . the ionized component can explain the observed intensities of the ionized species , assuming twice - solar composition and an @xmath0 density distribution . the neutral component is highly concentrated , well within the 0.8 point source , and is responsible for almost all of the observed k@xmath1 ( 6.4 kev ) emission . seems to be different than mkn 3 in terms of its gas distribution .

You are an expert at summarizing long articles. Proceed to summarize the following text: as exotic natures of unstable nuclei such as the new magic numbers and the neutron halos are disclosed by experiments , microscopic studies based on the nucleon - nucleon ( @xmath4 ) interaction become even more desired in nuclear structure physics . while the fully microscopic @xmath4 ( and @xmath5 ) interaction is still too complicated to cover large volume of nuclei in the periodic table despite significant progress @xcite , the semi - realistic @xmath4 interactions have been developed @xcite by modifying the michigan 3-range yukawa ( m3y ) interaction @xcite , which was originated from brueckner s @xmath6-matrix at the nuclear surface and expressed by the yukawa functions . the modification has been made so that the saturation and the spin - orbit ( @xmath7 ) splitting should be reproduced within the mean - field approximation ( mfa ) . owing to the recently developed numerical methods @xcite , self - consistent calculations in the mfa @xcite and in the random - phase approximation ( rpa ) @xcite have been implemented using the semi - realistic interactions . among the advantages of the m3y - type semi - realistic interactions is that they contain realistic tensor channels as well as correct longest - range central channels originating from the one - pion exchange , which have been pointed out to play significant roles in @xmath8- or @xmath9-dependence of the shell structure @xcite . the semi - realistic interactions are suitable to investigate effects of these channels within the self - consistent mfa and rpa @xcite . while the parameter - sets of the m3y - type interactions in refs . @xcite were adjusted to the data on the nuclear structure , some of them have been applied to the nuclear reactions @xcite and to the neutron stars @xcite as well . in studying structure of the neutron stars , density - dependence of the symmetry energy is crucially important @xcite . it has been pointed out that the symmetry energy at low density is significant in nuclear reactions : _ e.g. _ the charge - exchange reactions @xcite and the multi - fragmentation processes @xcite . the symmetry energy at low density may also affect the so - called pygmy dipole resonance in neutron - rich nuclei @xcite . however , the symmetry energy , particularly its density - dependence , was not sufficiently reliable in the previous parameter - sets in refs . @xcite as pointed out in ref . @xcite , giving rise to instability of the symmetric nuclear matter at the density @xmath10 , which is not consistent with microscopic calculations @xcite . in this article , we shall propose new parameter - sets of the m3y - type semi - realistic @xmath4 interaction . as far as the energy of the symmetric nuclear matter is fixed , the symmetry energy at each density is well connected to the energy of the neutron matter . the new parameters are determined by fitting the neutron - matter energy to microscopic result in ref . @xcite ( fp ) or @xcite ( apr ) . moreover , we additionally take into consideration the binding energy of @xmath0sn . as argued later , the symmetry energy at the saturation point tends to be fixed with good precision by fitting the parameters both to @xmath0sn and @xmath11sn . the symmetry energy is thus constrained to good degree in the new parameter - sets . corresponding to the microscopic results on the neutron - matter energy , we obtain two parameter - sets ` m3y - p6 ' ( fitted to fp ) and ` m3y - p7 ' ( to apr ) . although the parameters are determined from a limited number of data , they will be useful for investigating various aspects of nuclear properties , as will be illustrated by separation energies of the proton- and neutron - magic nuclei and by @xmath8- or @xmath9-dependence of shell structure . we take a non - relativistic isoscalar nuclear hamiltonian of @xmath12 with @xmath13 and @xmath14 representing the indices of individual nucleons . we set @xmath15 throughout this paper , where @xmath16 ( @xmath17 ) is the mass of a proton ( a neutron ) @xcite . for the effective @xmath4 interaction @xmath18 , the following form is considered , @xmath19^{\alpha^{(\mathrm{se } ) } } + t_\rho^{(\mathrm{te } ) } p_\mathrm{te}\cdot [ \rho(\mathbf{r}_i)]^{\alpha^{(\mathrm{te})}}\big ) \,\delta(\mathbf{r}_{ij})\ , , \label{eq : effint}\end{aligned}\ ] ] where @xmath20 is the spin operator of the @xmath13-th nucleon , @xmath21 , @xmath22 , @xmath23 , @xmath24 , and @xmath25 denotes the nucleon density . the tensor operator is defined by @xmath26 $ ] with @xmath27 . the projection operators on the singlet - even ( se ) , triplet - even ( te ) , singlet - odd ( so ) and triplet - odd ( to ) two - particle states are @xmath28 where @xmath29 ( @xmath30 ) expresses the spin ( isospin ) exchange operator . the yukawa function @xmath31 is assumed for all channels except @xmath32 . the density - dependent contact term @xmath32 is added in order to reproduce the saturation properties . physically , @xmath32 may carry effects of the @xmath5 interaction and of the density - dependence that is dropped in the original m3y interaction . we start from the m3y - paris interaction @xcite , which will be denoted by m3y - p0 in this article as in ref . the range parameters @xmath33 of m3y - p0 are maintained in any of @xmath34 , @xmath35 and @xmath36 . as in m3y - p0 , the longest - range part in @xmath34 is kept identical to the central channels of the one - pion exchange potential ( opep ) , @xmath37 . although the @xmath7 splitting plays a significant role in the nuclear shell structure , the @xmath6-matrix is known to underestimate the @xmath7 splitting . even though effects beyond the mfa may cure this problem @xcite , we introduce an overall enhancement factor to @xmath35 in order to describe the shell structure within the mfa . effects of the tensor force on the single - particle ( s.p . ) levels could be relevant to the new magic numbers in unstable nuclei @xcite . we keep @xmath36 without any modification from m3y - p0 . because of this @xmath36 having realistic nature , the present m3y - type interactions are useful to investigate the tensor - force effects within the mfa and rpa , as shown in refs . @xcite with the previous parameter - sets . the parameters in m3y - p6 and p7 are tabulated in table [ tab : param_m3y ] , together with m3y - p0 . .parameters of m3y - type interactions . [ tab : param_m3y ] [ cols="^,^,>,<,>,<,>,<,>,<",options="header " , ] we have developed new parameter - sets of the semi - realistic effective interactions to describe low energy phenomena of nuclei . they are obtained by phenomenologically modifying several parameters in the m3y - paris interaction , while the tensor force and the opep part in the central force are not changed , as before . unlike the previous parameters , the new sets m3y - p6 and p7 are adjusted also to the microscopic ( fp and apr ) results of the neutron - matter energies and to the binding energy of @xmath0sn . we therefore attain improvement on the symmetry energy , up to its density - dependence . in contrast to instability of the spin - saturated symmetric nuclear matter in the sly5 , d1s and d1 m results , neither of m3y - p6 nor p7 predicts such phase transition in the density range of @xmath3 . the new parameter - sets m3y - p6 and p7 have been applied to the doubly magic nuclei in the spherical hf calculations , and to the proton- or neutron - magic nuclei in the spherical hfb calculations . fair agreement with experimental data has been demonstrated for the binding energies of the doubly magic nuclei and for the nucleon separation energies of the proton- or neutron - magic nuclei . owing to the realistic tensor force and the opep central force , the @xmath8- or @xmath9-dependence of the shell structure is well described with m3y - p6 and p7 , as with the previous set m3y - p5 . the isotope shifts of the pb nuclei have also been argued . future study includes application of the semi - realistic interactions to the excitations in the rpa , as well as to deformed nuclei . moreover , extensive applications to nuclear reactions and to the neutron stars may be within the reach , which give further test of the effective interactions and a step toward unified description of nuclear structure , reactions and neutron stars . this work is financially supported as grant - in - aid for scientific research ( c ) , no . 22540266 , by japan society for the promotion of science , and as grant - in - aid for scientific research on innovative areas , no . 24105008 , by the ministry of education , culture , sports , science and technology , japan . numerical calculations are performed on hitac sr16000s at institute of media and information technology in chiba university , yukawa institute for theoretical physics in kyoto university , research institute for information technology in kyushu university , information technology center in university of tokyo , and information initiative center in hokkaido university . pieper , k. varga and r.b . wiringa , phys . c * 66 * , 044310 ( 2002 ) . p. navratl and b.r . barrett , phys . c * 57 * , 3119 ( 1998 ) ; 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new parameter - sets of the semi - realistic nucleon - nucleon interaction are developed , by modifying the m3y interaction but maintaining the tensor channels and the longest - range central channels . the modification is made so as to reproduce microscopic results of neutron - matter energies , in addition to the measured binding energies of doubly magic nuclei including @xmath0sn and the even - odd mass differences of the @xmath1 and @xmath2 nuclei in the self - consistent mean - field calculations . separation energies of the proton- or neutron - magic nuclei are shown to be in fair agreement with the experimental data . with the new parameter - sets m3y - p6 and p7 , the isotropic spin - saturated symmetric nuclear matter remains stable in the density range as wide as @xmath3 , while keeping desirable results of the previous parameter - set on finite nuclei . isotope shifts of the pb nuclei and tensor - force effects on shell structure are discussed .

You are an expert at summarizing long articles. Proceed to summarize the following text: with the discovery of the standard model ( sm ) higgs in the atlas @xcite and cms @xcite experiments , we have taken one step further toward understanding the electroweak symmetry breaking ( ewsb ) through the spontaneous symmetry breaking ( ssb ) mechanism in the scalar sector . the next mission for the high - luminosity large hadron collider ( lhc ) is to explore not only the detailed properties of the sm higgs , but also the new physics effects . since problems related to the origin of neutrino mass , dark matter ( dm ) , and matter - antimatter asymmetry can not be resolved in the sm , it is believed that the sm of particle physics is an effective theory at the electroweak scale . if new physics exists at the tev scale , the lhc can detect it . some potential events indicating the existence of new effects indeed have been observed in the recent atlas and cms experiments . for instance , diboson excess of @xmath20 with @xmath21 at around 2 tev was shown by atlas @xcite and cms @xcite ; the branching ratio ( br ) for lepton - flavor - violating higgs decay @xmath22 with a @xmath23 significance was presented by cms @xcite ; a resonance at a mass of 750 gev in the diphoton invariant mass spectrum was reported by atlas @xcite and cms @xcite . although the results are not conclusive yet , these experimental measurements have inspired theorists to speculate various effects to interpret the excesses . ever since the sm higgs was observed , the higgs measurements have approached to the precision level . it becomes an important issue to uncover the physics beyond the sm through the higgs portal . precise higgs measurements can also give strict bounds on the new couplings ; for instance , @xmath24 in the two - higgs - doublet model has been limited to be close to the decoupling limit @xcite , and the sm with a fourth generation of chiral fermions has become disfavored @xcite . although the extension of the sm with chiral fermions has been severely limited , the constraint on the vector - like quark ( vlq ) models may not have the same situation due to the use of different representations and coupling structures . unlike chiral fermion models , where the appearance of chiral quarks has to accompany chiral leptons due to gauge anomaly , the gauge anomaly in vlq models is cancelled automatically . therefore , it is not necessary to introduce the exotic heavy leptons into the sm when vlqs are added . due to their interesting properties , the phenomena of some specific vlqs at the lhc have been investigated from a theoretical viewpoint @xcite . in experiments , single vlqs and pairs of vlqs have been produced at atlas @xcite and cms @xcite . based on the sm gauge symmetry @xmath25 , the representations of vlqs can basically be any @xmath26 multiplets . however , in order to consider the vlq decays , the possible representations of vlq couplings to the sm quarks are singlet , doublet , and triplet . to interpret the excesses of dibosons and diphotons indicated by atlas and cms , we proposed a model that contains one higgs singlet and two triplet vlqs with hypercharges of @xmath27 and @xmath28 , respectively @xcite . since the representations of the vlqs are different from those of the sm quarks , the higgs- and @xmath1-mediated flavor - changing neutral currents ( fcncs ) are induced at the tree level and the cabibbo - kobayashi - maskawa ( ckm ) matrix is non - unitary matrix . in our earlier studies , besides the resolutions of the excesses , we focused on the leading effects , which were from the left - handed flavor - mixing matrices , and found that they led to interesting contributions to top fcncs @xmath29 and the sm higgs production and decays . in this study , we systematically discuss the left- and right - handed flavor - mixing effects together . we revisit the constraints and present the bounds from @xmath30 processes in detail . with the values of constrained parameters , it is found that the modified top coupling to the sm higgs , which arises from the right - handed flavor mixing , can diminish the influence of the sm higgs production and the decay to diphotons by around @xmath2 and @xmath31 deviations from the sm results , respectively . we demonstrate how the changes of the sm ckm matrix elements can be smeared so that the severe bounds from the current measurements of the ckm matrix elements are satisfied @xcite . in addition to the phenomena in flavor physics , we also investigate the single and pair production of vlqs in this work . in the proposed model , the new quarks are @xmath32 , @xmath33 , @xmath34 , and @xmath35 , whose associated electric charges are @xmath36 , @xmath37 , @xmath38 , and @xmath39 , respectively . therefore , @xmath32 and @xmath33 can be regarded as top and bottom partners , respectively . since vlqs @xmath34 and @xmath35 carry the unusual electric charges , they do not have fcnc couplings to the sm quarks . as a result , their single production and decays are only through charged weak interactions . since the top and bottom partners involve more complicated fcnc interactions , we concentrate the study on vlqs @xmath34 and @xmath35 . it is found that the single production cross sections of @xmath34 and @xmath35 can be larger than the pair production cross sections , which are dominant from quantum chromodynamics ( qcd ) . in order to understand this phenomenon , we analyze each process @xmath40 for the single production of @xmath34 and @xmath35 . it is found that with @xmath41 tev , the cross sections for @xmath42 and @xmath43 modes can be of the order of 100 fb while the pair production cross sections are smaller by a factor of around 2 . we postpone the study of event simulation to another paper . the rest of this paper is organized as follows . we establish the model , discuss the new flavor mixing effects , and derive the new higgs and gauge couplings in section ii . we present the constraints from low - energy and higgs measurements in section iii . we also study the implications on top - quark fcnc processes @xmath29 . in section iv , we discuss the single and pair production for vlqs @xmath44 and @xmath45 , and thoroughly analyze the production mechanism in @xmath7 collisions . the conclusions are given in section v. we extend the sm by including one real higgs singlet and two vector - like triplet quarks ( vltqs ) , where the representations of vltqs in @xmath46 gauge symmetry are chosen as @xmath47 and @xmath48 @xcite . for suppressing the mixing between higgs singlet and doublet , we impose a @xmath49 discrete symmetry on the scalar potential , where the scalar fields follow the transformations @xmath50 and @xmath51 under the @xmath49 transformation . thus , the scalar potential is expressed as : @xmath52 we adopt the following representation of @xmath53 : h= ( cc g^+ + ( v+ h + ig^0 ) ) , where @xmath54 and @xmath55 are goldstone bosons , @xmath56 is the sm higgs field , and @xmath57 is the vacuum expectation value ( vev ) of @xmath53 . the @xmath58 field can not develop a non - vanishing vev in the scalar potential of eq . ( [ eq : vhs ] ) when @xmath59 . due to the @xmath49 symmetry , @xmath56 and @xmath58 do not mix at the tree level ; thus @xmath60 is the mass of @xmath58 , @xmath61 , and @xmath62 gev is the mass of the sm higgs @xcite . we note that the @xmath49 symmetry is softly broken by some other sector of lagrangian . the gauge - invariant yukawa couplings of vltqs to the sm quarks , the sm higgs doublet , and the new higgs singlet are expressed as : -l^y_vltq & = & tr(|f_2l f_2r ) s + & + & m_f_1 tr(|f_1l f_1r ) + m_f_2 tr(|f_2l f_2r)+ h.c . , [ eq : yukawa ] where @xmath63 is the left - handed sm quark doublet , all flavor indices are hidden , @xmath64 , and @xmath65 is the @xmath66 vltq with hypercharge @xmath67 , whose representations of @xmath68 are : f_1 = ( cc u_1/ & x + d_1 & -u_1/ ) , f_2 = ( cc d_2/ & u_2 + y & -d_2/ ) . under @xmath49 transformation , @xmath69 . the electric charges of @xmath70 , @xmath71 , @xmath34 , and @xmath35 are @xmath72 , @xmath37 , @xmath73 , and @xmath74 , respectively . therefore , @xmath75 can mix with up ( down)-type sm quarks . the masses of vltqs do not originate from the electroweak symmetry breaking . due to the gauge symmetry , vltqs in a given multiplet state are degenerate , and denoted by @xmath76 . since the mass terms of vltqs do not involve the @xmath58 field and the associated operators are dimension-3 , the discrete @xmath49 symmetry is softly broken by @xmath77 terms . it is worth mentioning that @xmath78 results in the mixture of the sm quarks and vltqs ; consequently , the @xmath56 coupling to the top quark is modified and the @xmath56 couplings to vltqs are induced . as a result , the sm higgs production cross section via gluon - gluon fusion ( ggf ) and its decays will be modified . in the next subsection , we discuss the modifications in detail . we note that the @xmath49 breaking effects will induce @xmath79 term through one - loop diagrams in the scalar potential . however , in addition to the suppression factor @xmath80 , the loop effects are suppressed by the small yukawa couplings @xmath78 ( see the detailed discussions later ) . as a result , the induced vev of @xmath58 field and the induced mixing between @xmath56 and @xmath58 are small , and the br for @xmath81 decay is a factor of two smaller than that for @xmath82 decay @xcite . next , we discuss the weak interactions of vltqs . as usual , we write the covariant derivative of @xmath25 as : d_= _ + i ( t^+ w^+_+t^- w^- _ ) + i ( t^3 - s^2_w q ) z_+ i e q a _ , where @xmath83 , @xmath84 , and @xmath85 are the gauge bosons in the sm , @xmath86 is the gauge coupling of @xmath87 , @xmath88 , @xmath89 is the weinberg angle , @xmath90 , and the charge operator @xmath91 , where @xmath35 is the hypercharge of the particle . thus , the gauge interactions of vltqs are written as @xcite : _ vff & = & -g + & - & , [ eq : vff ] where we express the triplet vlqs as @xmath92 and @xmath93 , diag@xmath94 , diag@xmath95 , and diag@xmath96 . to further understand the weak interactions in terms of physical states , we have to investigate the structures of flavor mixings when @xmath78 effects in eq . ( [ eq : yukawa ] ) are involved . the introduced two @xmath87 triplet vlqs contain the quarks with electric charges of @xmath97 and @xmath37 . from the new higgs yukawa couplings to the sm higgs and vltqs , the mixture between the sm quarks and vltqs is generated after ewsb . in order to get the physical mass eigenstates of quarks and the new flavor mixings , we have to diagonalize the mass matrices of the sm quarks and vltqs . with the yukawa couplings @xmath98 and @xmath99 ( i=1 - 3 ) in eq . ( [ eq : yukawa ] ) , the quark mass terms are given by : -l_mass = |*q*_l * m*_q * q*_r + |*q*_l * y*^q v * f*_r + |*f*_l * m*_f * f*_r + h.c . , where @xmath100 or @xmath101 denotes the sm up- or down - type quarks . we have chosen the basis such that @xmath102 is a @xmath103 diagonal matrix , @xmath104 or @xmath105 is the vltq with charge @xmath97 or @xmath37 , diag@xmath106 ) , and =( cc y_11/2 & y_21/ + y_12/2 & y_22/ + y_13/2 & y_23/ ) , * y^d*= ( cc y_11/ & -y_21/2 + y_12/ & -y_22/2 + y_13/ & -y_23/2 ) . [ eq : yud ] we do not have @xmath107 terms due to gauge invariance . the quark mass matrices for electric charges @xmath97 and @xmath37 now become @xmath108 matrices . one can introduce the @xmath109 unitary matrices @xmath110 and @xmath111 to diagonalize the mass matrices , namely @xmath112 . in order to obtain the information of @xmath113 , we consider the multiplications of mass matrices to be @xmath114 and @xmath115 , where @xmath116 and @xmath117 are expressed as : @xmath118 with @xmath119 , and @xmath120 . it is clear that the off - diagonal matrix elements in @xmath116 are related to @xmath121 while those in @xmath117 are @xmath122 . due to @xmath123 , the unitary matrices @xmath124 can be expanded with respect to @xmath125 and @xmath126 ; at the leading order approximation , they can be formulated as : v^q _ ( cc _ 33 & -(^q_)_32 + ( ^q _ ) _ 23 & _ 22 ) , [ eq : vqlr ] where @xmath127 , @xmath128 , and @xmath129 . we find that the effects of @xmath130 are suppressed by @xmath131 while those of @xmath132 are associated with @xmath133 . since the top and bottom quarks are much heavier than other sm quarks , in this study we keep the contributions from @xmath134 and @xmath135 , and ignore other @xmath136 that involve the light quark masses . we use the flavor mixing matrices of eq . ( [ eq : vqlr ] ) to investigate the new flavor couplings of the higgs and the weak gauge bosons below . from the yukawa couplings in eq . ( [ eq : yukawa ] ) , the sm higgs couplings to the quarks in the flavor space are written as : -l_hqq & = |q_l v^q q_r h + h.c . , [ eq : hqq ] where @xmath137 or @xmath138 , @xmath139 , and @xmath140 are the physical states of vltqs and carry the electric charges of @xmath72 and @xmath37 , respectively , and @xmath141 is the mixing matrix for the @xmath142-type quark and is given by : @xmath143 the small effects , such as @xmath144 , @xmath145 , and @xmath146 , have been dropped . according to eq . ( [ eq : vq ] ) , the @xmath56-mediated fcncs for the sm quarks ( e.g. , @xmath56-@xmath142-@xmath147 ) are proportional to @xmath148 . if the mass effects of the first two generations of quarks are neglected , we have the flavor - changing higgs interactions : @xmath149 where @xmath150 quark , @xmath151 quark , the definition of @xmath152 in eq . ( [ eq : yud ] ) is applied , and @xmath153 . the @xmath154 and @xmath155 oscillations can be induced via the tree - level higgs mediation in the vltq model . additionally , the brs for the flavor - changing processes @xmath156 , which are highly suppressed in the sm , become sizable . in addition to the new fcnc couplings , the flavor - conserving couplings are also modified : -l_hqq = |t_l t_r h + quark will affect the sm higgs production and decays in the @xmath7 collisions at the lhc . if we take @xmath157 tev , @xmath158 , and @xmath159 gev , the @xmath56 production cross section by the top - quark loop will be reduced by @xmath160 of the sm prediction . that is , the influence of new effects can not be ignored arbitrarily . from the flavor mixing matrix in eq . ( [ eq : vq ] ) , we can also obtain the sm higgs interactions with the vltqs as : @xmath161 with @xmath162 . since vltqs are color triplet states in @xmath163 and carry the same color charges as those of the sm quarks , the new couplings @xmath164 also contribute to the @xmath56 production cross section via the ggf channel . we will study their influence on the process @xmath165 in the numerical analysis . by combining the charged weak interactions of the sm quarks with those of vltqs in eq . ( [ eq : vff ] ) , the charged current interactions of quarks can be formulated by : _ w & = & - |*u*_l ^v^l_ckm * d*_l w^+ _ - |*u*_r ^v^r_ckm * d*_r w^+_+h.c . , [ eq : wud ] where @xmath166 and @xmath167 are respectively the physical up- and down - type quarks , and @xmath168 is the @xmath109 ckm matrix for left ( right)-handed quarks , defined by : v^l_ckm & = & v^u_l ( cc ( v_ckm ) _ 33 & 0_32 + 0_23 & _ 22 ) v^d_l , v^r_ckm = v^u_r ( cc 0_33 & 0_32 + 0_23 & _ 22 ) v^d_r . the @xmath103 matrix @xmath169 is associated with the sm ckm matrix . since the weak isospin of a triplet quark differs from that of a doublet quark , the new @xmath109 ckm matrices @xmath170 are non - unitary . by using the results of eq . ( [ eq : vqlr ] ) , the ckm matrix elements for the three - generation sm quarks are modified to be : ( v^sm_ckm)_i j ( v_ckm)_i j + ( _ 1i _ 1j - _ 2i _ 2j ) . [ eq : mckm ] with @xmath171 and @xmath172 tev , the changes of the sm ckm matrix elements are roughly estimated as @xmath173 . as indicated by experiments @xcite , the value of @xmath174 has the same order of magnitude as @xmath175 and is larger than @xmath176 . to satisfy the constraints of @xmath176 , the possible schemes are : ( a ) @xmath177 is less than @xmath178 , the smallest ckm matrix element , ( b ) @xmath179 so that @xmath180 and ( c ) @xmath181 which leads to @xmath180 . moreover , if we adopt @xmath182 @xmath1831 - 3 ) , all ckm matrix elements return to the sm ones . with the leading - order approximation for @xmath184 , the @xmath185-boson interactions with the sm quarks and vltqs are given by @xcite : @xmath186w^+_\mu \non \\ & - \frac{g}{2 } \left [ \zeta_{2i } \bar d_{il } \gamma^\mu y_l + \frac{m_b \zeta_{23}}{m_{f_2 } } \bar b_r \gamma^\mu y_r - \zeta_{1i } \bar x_l \gamma^\mu u_{il } - \frac{m_t \zeta_{13}}{m_{f_1 } } \bar x_r \gamma^\mu t_r \right ] w^+_\mu + h.c . \label{eq : wfq } \end{aligned}\ ] ] the charged weak interactions of vltqs can be directly read from eq . ( [ eq : vff ] ) . we next discuss the neutral weak interactions . it is known that the left - handed and right - handed quarks in the sm are @xmath187 doublets and singlets , respectively ; however , the vltqs are @xmath87 triplets . since the isospin of a triplet is different from those of doublets and singlets , in order to combine the vltqs with the sm quarks into the same representation in the flavor space , we need to rewrite the vertex structure of the @xmath1-boson , @xmath188 , in eq . ( [ eq : vff ] ) to fit the cases of doublets and singlets , such as @xmath189 , where @xmath190 for doublets and @xmath191 for singlets . due to the isospin difference , @xmath1-mediated fcncs are induced at the tree level . since vltqs @xmath34 and @xmath35 carry the electric charges of @xmath73 and @xmath192 , respectively , they can not mix up with other quarks in the neutral current interactions . in terms of weak eigenstates , we write the weak neutral current interactions in eq . ( [ eq : vff ] ) as : @xmath193\ , , \label{eq : zff } \end{aligned}\ ] ] where @xmath194 and @xmath195 are composed of vltqs with electric charges of @xmath72 and @xmath37 , respectively , @xmath196 , @xmath190 for @xmath197 , @xmath198 , and @xmath199 . we succeed in expressing the @xmath1 couplings to vltqs by using the sm @xmath1 couplings . it is clear that the first two terms in eq . ( [ eq : zff ] ) lead to the flavor - conserving couplings when the sm quarks and vltqs form a representation in the dimension-5 flavor space . since the sm quarks do not have the interaction structures , as shown in the last three terms of eq . ( [ eq : zff ] ) , as a result , fcncs via @xmath1 mediation are generated . hence , the @xmath1-boson interactions with quarks , which carry electric charges of @xmath72 and @xmath37 , can be formulated as : _ zqq & = & - c^q_l_ij |q_il ^q_j l z_- c^q_r_ij |q_i r ^q_jr z _ , + [ eq : zqq ] c^q_l_ij & = & ( i^q_3 -s^2_w q_q ) _ ij + ( -v^q_li4 v^q*_lj4 + v^q_li5 v^q*_lj5 ) , + c^q_r_ij & = & -s^2_w q_q _ ij + _ q ( v^q_r)_i _ q ( v^q*_r)__q j where @xmath200 or @xmath138 , @xmath201 are defined in eq . ( [ eq : vqlr ] ) , @xmath202 , and @xmath203 . using eq . ( [ eq : vqlr ] ) and the leading - order approximation , the new gauge couplings of the @xmath1-boson to the sm quarks are given by : _ zq_iq_j = - ( a_q _ 1i _ 1j - b_q _ 2i _ 2j ) |q_il ^q_jl z _ , [ eq : zqiqj ] where @xmath204 denote the up- or down - type sm quarks , @xmath205 , and @xmath206 . it can be seen that the fcnc effects can contribute to @xmath30 neutral meson mixings . a comparison with the results in eq . ( [ eq : mckm ] ) indicates that the induced new coupling structures in charged and neutral currents are different . it is interesting to investigate the possible schemes that can simultaneously satisfy the constraints from the ckm matrix elements and the data of neutral meson oscillations . the interactions of the @xmath1-boson couple to one vltq and one sm quark are shown as : @xmath207 one can get the @xmath1 couplings to vltqs from eq . ( [ eq : vff ] ) . in this section , we discuss the constraints from low - energy @xmath30 processes and from the data of the sm higgs production and decay into diphotons . from eqs . ( [ eq : hqq ] ) and ( [ eq : zqiqj ] ) , we know that the @xmath56- and @xmath1-mediated fcncs appear and contribute to the @xmath30 processes , such as @xmath209 and @xmath210 mixings , where the current experimental data can give strict constraints on the free parameters . since the fcnc couplings in the up - type quarks are the same as those in the down - type quarks and the hadronic effects in the @xmath211-meson system are dominated by unclear non - perturbative effects , we focus on @xmath212 and @xmath213 processes . following the notations in previous studies @xcite , the transition matrix elements for @xmath209 and @xmath210 mixings are given by : @xmath214\ , , \\ m^{k}_{12}(z ) & = \frac{(\delta^{sd}_l(z))^2}{2 m^2_z } c^{\rm vll}_1 ( \mu_z ) \bar p^{\rm vll}_1(k,\mu_z)\ , , \\ m^{b_q}_{12}(z ) & = \frac{(\delta^{bd}_l(z))^2}{2 m^2_z } c^{\rm vll}_1 ( \mu_z ) \bar p^{\rm vll}_1(b_q , \mu_z)\ , . \end{aligned}\ ] ] @xmath215 is the wilson coefficient with @xmath216 qcd corrections , and @xmath217 denotes the hadronic effects that include the renormalization group ( rg ) evolution from high energy to low energy , whose expressions are @xcite : @xmath218_{\rm sll } r_{b_q } b^{\rm sll}_{1}(\mu_b ) -\frac{3}{2 } [ \eta_{21}(\mu_b)]_{\rm sll } r_{b_q } b^{\rm sll}_{2}(\mu_b ) \ , , \non \\ p^{\rm sll}_2(b_q,\mu_b ) & = - \frac{5}{8 } [ \eta_{12}(\mu_b ) ] _ { \rm sll } r_{b_q } b^{\rm sll}_{1}(\mu_b ) -\frac{3}{2 } [ \eta_{22}(\mu_b)]_{\rm sll } r_{b_q } b^{\rm sll}_{2}(\mu_b ) \ , , \end{aligned}\ ] ] @xmath219 , where @xmath220 and @xmath221 are the mass and decay constant of the @xmath222-meson , respectively , @xmath223 gev for the @xmath224-meson , @xmath225 , and the values of other hadronic effects and rg evolution effects are given in table [ tab : values ] . @xmath226 are from the short - distance interactions of eqs . ( [ eq : hqq ] ) and ( [ eq : zqiqj ] ) and are written as : @xmath227 since we have ignored the effects of light quark masses , the @xmath56-mediated fcnc has no contribution to @xmath209 mixing . to constrain the parameters , we assume that the obtained @xmath228 in the model should be less than the experimental measurements . to understand the individual influences of @xmath56 mediation and @xmath1 mediation , we show their constraints separately . with @xmath229 , @xmath230 , and the inputs of table [ tab : values ] , we obtain the constraints as : @xmath209 mixing : @xmath231 @xmath154 mixing : @xmath232 @xmath155 mixing : @xmath233 from these results , we find that the constraint from @xmath234 is only a factor of 2 stronger than that from @xmath235 . since the ratio @xmath236 in experiments is very close to the wolfenstein s parameter @xmath237 @xcite , the difference of a factor of @xmath238 between eq . ( [ eq : bdc ] ) and eq . @xmath239 is reasonable . . [ cols="^,^,^,^,^",options="header " , ] finally , we briefly discuss the new physics in connection to the flavor physics . in this study , we do not introduce new couplings to the lepton sector , therefore , the contributions to the lepton flavor - changing processes are similar to the sm predictions . however , the introduced vlqs lead to fcncs at the tree level in the quark sector , where the strict constraints from @xmath30 processes have been considered in section iii . besides the rare decays @xmath240 and @xmath241 that were discussed earlier , it is also interesting to investigate the fcnc effects in the low energy physics . for instance , the coupling @xmath242 can contribute to the @xmath243 and @xmath244 decays , where the sm predicted brs are of @xmath245 , both are sensitive to the new physics effects , and the theoretical uncertainties are well - controlled @xcite . furthermore , the na62 experiment at cern can achieve the @xmath246 to be a precision of @xmath2 @xcite ; and the koto experiment at j - parc for @xmath247 decays can reach the sm sensitivity . thus , it is important to search for new physics in rare @xmath248 decays . in @xmath249-meson physics , the tree - level couplings @xmath250 with @xmath251 can contribute to @xmath252 decays . although the measured @xmath253 is consistent with the sm prediction @xcite , a @xmath254 deviation from the sm prediction in the angular analysis of @xmath255 is observed @xcite . it is worthy to explore the excess in our model . since the detailed analysis of flavor physics is beyond the scope of this paper , a complete analysis will be studied elsewhere @xcite . we have studied the phenomenology of which two triplet vlqs with @xmath0 and a higgs singlet are embedded in the sm . because the isospin of vlqs is different from that of the sm quarks , higgs- and @xmath1-mediated fcncs are generated at the tree level and the new ckm matrix becomes a non - unitary matrix . we find that the modifications of the ckm matrix elements coupled to the sm quarks can be smeared out if two triplet vlqs are introduced and the scheme @xmath256 is adopted , where @xmath257 are the parameters from flavor mixings . although the tree - level fcncs can not be removed , it was found that when the constraints from @xmath30 processes are applied , the upper limits of brs for @xmath258 decays are @xmath259 , which is two orders of magnitude smaller than the current experimental bounds . with the values of constrained parameters , we examined the influence of the model on the sm higgs production and its diphoton decay ; we found that @xmath260 and @xmath261 can have @xmath262 and @xmath31 deviations from the sm results , respectively . as a result , the signal strength for @xmath165 is thus changed by @xmath2 . the main purpose of this work was to explore the single production of exotic vlqs @xmath34 and @xmath35 in the @xmath7 collisions at @xmath8 tev . we gave a detailed analysis for each possible @xmath6 scattering , where @xmath204 and @xmath263 are the possible initial quarks . it was found that the contributions of @xmath264-channel annihilations are much smaller than those of @xmath265-channel annihilations . from this study , we comprehend the contribution of each subprocess to the production cross section of a specific vlq . the interesting production channels are @xmath266 , @xmath267 , and @xmath268 , where the corresponding production cross sections for @xmath12 tev are @xmath13 , @xmath14 , and @xmath15 fb , respectively . from our analysis , it is clear to see that the single production cross sections of vlqs are much larger than the pair production cross sections , which are through qcd processes . the dominant decay modes of the vlqs are @xmath16 and @xmath17 . each br can be 1/2 in our chosen scheme . for illustration , we estimate the significances for the channels proposed in eq . 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two triplet vector - like quarks ( vlqs ) with hypercharges of @xmath0 and one singlet scalar boson are embedded in the standard model ( sm ) to resolve the 750 gev diphoton excess . the constraints on the tree - level higgs- and @xmath1-mediated flavor - changing neutral currents are discussed in detail . besides the resolution of excess , it is found that the signal strength of diphoton higgs decay can have a @xmath2 deviation from the sm prediction and that the upper limits of the branching ratios for rare top - quark decays are @xmath3 . we find that the production cross section of a single vlq by electroweak processes is larger than that of vlq - pair by qcd processes . to explore the signals of the heavy vlqs at the lhc , we throughly analyze the production of single @xmath4 and @xmath5 via @xmath6 annihilations in @xmath7 collisions at @xmath8 tev . it is found that the electroweak production cross sections for @xmath9 , @xmath10 , and @xmath11 channels with @xmath12 tev can be @xmath13 , @xmath14 , and @xmath15 fb , respectively ; and the dominant decay modes are @xmath16 and @xmath17 . with adopting kinematic cuts , the significance for @xmath18 channel can be over @xmath19 . kias - p16026

You are an expert at summarizing long articles. Proceed to summarize the following text: there are two approaches to characterizing spacetime singularities in a cosmological context . the first approach may be called _ geometric _ and consists of finding sufficient and/or necessary conditions for singularity formation , or absence , _ independently _ of any specific solution of the field equations under general conditions on the matter fields . methods of this sort include those based on an analysis of geodesic congruences in spacetime and lead to the well known singularity theorems , cf . @xcite , as well as those which are depend on an analysis of the geodesic equations themselves and lead to completeness theorems such as those expounded in cbc02 , and the classification of singularities in @xcite . the second approach to the singularity problem can be termed _ dynamical _ and refers to characterizing cosmological singularities in a geometric theory of gravity by analysing the dynamical field equations of the theory _ _ _ _ it uses methods from the theory of dynamical systems and can be _ global _ , referring to the asymptotic behaviour of the system of field equations for large times , or _ local _ , giving the behaviour of the field components in a small neighborhood of the finite - time singularity . in this latter spirit , we present here a local method for the characterization of the asymptotic properties of solutions to the field equations of a given theory of gravity in the neighborhood of the spacetime singularity . we are interested in providing an asymptotic form for the solution near singularities of the gravitational field and understanding all possible dominant features of the field as we approach the singularity . we call this approach the _ method of asymptotic splittings . _ in the following sections , we give an outline of the method of asymptotic splittings with a view to its eventual application to cosmological spacetimes in different theories of gravity . for the sake of illustration , in the last section we analyze the asymptotic behaviour of a friedmann - robertson - walker ( frw ) universe filled with perfect fluid in einstein s general relativity , which provides the simplest , nontrivial cosmological system . it is advantageous to work on any differentiable manifold @xmath0 , although for specific applications we restrict attention to open subsets or @xmath1 we shall use interchangeably the terms vector field @xmath2 and dynamical system defined by @xmath3 on @xmath0 , @xmath4 , with @xmath5 . also , we will use the terms integral curve @xmath6 of the vector field @xmath3 with initial condition @xmath7 , and solution of the associated dynamical system @xmath8 passing through the point @xmath9 , with identical meanings . given a vector field @xmath3 on the @xmath10-dimensional manifold @xmath11 , we define the notion of a _ general _ solution of the associated dynamical system as a solution that depends on @xmath10 arbitrary constants of integration , @xmath12 . these constants are uniquely determined by the initial conditions in the sense that to each @xmath13 we can always find a @xmath14 such that the solution @xmath15 is the unique solution passing through the point @xmath13 . therefore , a property holds _ independently _ of the initial conditions if and only if it is a property of a general solution of the system . a _ particular _ solution of the dynamical system is any solution obtained from the general solution by assigning specific values to at least one of the arbitrary constants . the particular solutions containing @xmath16 arbitrary constants can be viewed as describing the evolution in time of sets of initial conditions of dimension @xmath16 strictly smaller than @xmath10 . a particular solution is called an _ exact _ solution of the dynamical system when @xmath17 . thus , in our terminology , a particular solution is a more general object than any exact solution , the latter having the property that all arbitrary constants have been given specific values . the hierarchy : exact ( no arbitrary constants ) to particular ( strictly less than maximum number of arbitrary constants ) to general solutions , will play an important role in what follows . general , particular , or exact solutions of dynamical systems can develop _ finite - time singularities _ ; that is , instances where a solution @xmath18 , misbehaves at a finite value @xmath19 of the time @xmath20 . this is made precise as follows . we say that the system @xmath21 ( equivalently , the vector field @xmath3 ) has a _ finite - time singularity _ if there exists a @xmath22 and a @xmath23 such that for all @xmath24 there exists an @xmath25 such that @xmath26for @xmath27 . here @xmath28 , @xmath29 for some @xmath30 , and @xmath31 ( resp . @xmath32 ) . note also , that @xmath33 is an arbitrary point in the domain @xmath34 and may be taken to mean ` now ' . alternatively , we may set @xmath35 , @xmath36 , and consider the solution in terms of the new time variable @xmath37 , @xmath38 , with a finite - time singularity at @xmath39 . we see that for a vector field to have a finite - time singularity there must be at least one integral curve passing through the point @xmath13 of @xmath40 such that at least one of its @xmath41 norms diverges at @xmath42 . we write @xmath43to denote a finite - time singularity at @xmath44 . one of the most interesting problems in the theory of singularities of vector fields is to find the structure of the set of points @xmath13 in @xmath45 such that , when evolved through the dynamical system defined by the vector field , the integral curve of @xmath3 passing through a point in that set satisfies property ( [ sing2 ] ) . another important question , of special interest in relativistic cosmology , is to discover the precise relation between the finite - time singularities of vector fields that arise as reductions of the field equations and those that emerge in the form of geodesic incompleteness . the difficulty here is that the finite - time singularities of vector fields appear to be unconnected to geodesic incompleteness and conversely , singularities which arise through the formation of conjugate points do not seem to demand , or require , any dynamical description . finite - time singularities of ( general or particular ) solutions of linear dynamical systems are located at the singularities of their coefficients and are _ fixed _ because they are known from the singularities of the coefficients of the system . the fixed singularities in a solution are therefore independent of the choice of initial conditions . in contrast , solutions of nonlinear systems can develop finite - time singularities that are either fixed or movable . a singularity is _ fixed _ if it is a singularity of @xmath46 for all @xmath47 ; otherwise , we say it is a _ movable singularity_. note that any fixed finite - time singularity of a particular solution ( @xmath48 ) can not be a fixed singularity of a general solution since at least one of the constants appearing in the general solution has been set to zero , and so this singularity is not one of a @xmath46 for _ all _ @xmath49 , that is independent of the initial conditions . hence , fixed finite - time singularities _ in a general solution _ can not be understood by studying fixed singularities in particular solutions . however , movable singularities of a particular solution , if they _ are _ singularities ( in the sense of the definition above ) of the general solution , will always be movable ones . therefore , movable finite - time singularities in particular solutions make a nonzero contribution to the singularity pattern of the vector field and must be taken into consideration in the general study of its singularities . it may also happen that a dynamical system has no movable singularities in the general solution but still has a singular or particular solution with a movable singularity . hence , a fixed ( or movable ) singularity in a general solution can be a fixed ( or movable ) singularity of some particular solution but not vice versa . in general , movable singularities are more interesting , since the issue of choosing initial conditions plays an important role for them ; consequently , we shall restrict our attention to them almost exclusively in what follows . how should we tackle the geometric problem of describing the behaviour of vector fields and their integral curves solutions of the associated dynamical system in the neighborhood of a finite - time movable singularity ? assume that we are given a vector field , and we know that at some point , @xmath44 , a system of integral curves , corresponding to a particular or a general solution , has a ( future or past ) finite - time singularity in the sense of definition ( [ sing1 ] ) . the approach we take in this paper is an asymptotic one . the vector field ( or its integral curves ) can basically do two things sufficiently close to the finite - time singularity , namely , it can either show some dominant feature or not . in the latter case , the integral curves can ` spiral ' in some way around the singularity _ ad infinitum _ so that ( [ sing1 ] ) is satisfied and the dynamics are totally controlled by the subdominant ( lower - order in terms of weight - see below ) terms , whereas in the former case solutions share a distinctly dominant behaviour on approach to the singularity at @xmath44 determined by the most nonlinear terms . to describe both cases invariably , we decompose the vector field into simpler , component vector fields and examine whether the most nonlinear one of these shows a dominant behaviour while the rest become subdominant in some exact sense . using this picture , we then built a system of integral curves corresponding , where feasible , to the general solution , and sharing exactly its characteristics in a sufficiently small neighborhood of the finite - time singularity . the construction of these solutions is given as a formal asymptotic series expansion around the singularity and is done term - by - term . we begin by introducing some useful notation and terminology . we write @xmath50 to denote the function @xmath51 where @xmath52 and @xmath53 a function of the form ( [ scale function ] ) is _ scale invariant _ in the sense that a change in the time scale , @xmath54 , @xmath55 , reveals that we must also have @xmath56 , and conversely . now we demand that a scale invariant function @xmath57 of the form ( [ scale function ] ) is an integral curve of the vector field @xmath3 passing through @xmath13 , or equivalently , the associated dynamical system @xmath21 has a particular solution that is scale invariant , valid in a neighborhood of the assumed finite - time singularity . this means that @xmath58the notation @xmath59 stands for the monomial @xmath60 and is valid for each @xmath61 ( range convention , no summation ) . we say that a vector field @xmath3 ( or the associated dynamical system ) is _ scale invariant _ if it satisfies @xmath62more generally , a vector field is called _ weight - homogeneous _ with _ weighted degree _ @xmath63 if there is a vector @xmath64 , called _ the weight _ , and a vector @xmath65 such that @xmath66we denote the degree by @xmath67 . when @xmath68 the vector field is called _ homogeneous _ of degree @xmath63 . note that any scale invariant vector field is a weight - homogeneous field @xmath69 such that each component @xmath70 has degree @xmath71 , from which it follows that @xmath72 has weight @xmath73 . if @xmath3 has degree @xmath63 , then @xmath74 has degree @xmath75 , with @xmath76 the @xmath77-th unit vector . there are weight - homogeneous vector fields that are not scale invariant ; for instance , all linear dynamical systems with constant coefficients which are homogeneous vector fields . using eqs . ( [ basic1 ] ) , ( [ basic2 ] ) we see that given any nonzero vector @xmath78 , a scale - invariant vector field @xmath3 admits a scale - invariant integral curve provided that the inhomogeneous linear system @xmath79has nontrivial solutions for @xmath80 . note that there may be nontrivial solutions for @xmath80 with some components zero . when at least one of the components of @xmath80 is nonzero , the corresponding solution of the form ( [ scale function ] ) is a particular solution of the dynamical system . therefore , in this case , we know the exact asymptotic behaviour of the vector field @xmath3 in the neighborhood of a finite - time singularity , given by the solution ( [ scale function ] ) with suitable @xmath81 s and @xmath78 s . unfortunately , most vector fields are neither scale invariant nor weight - homogeneous . however , since any _ analytic _ vector field @xmath82 can be expanded in a power series in some domain @xmath83 , by taking any @xmath84 for any @xmath85 distinct from @xmath86 any such vector field can be decomposed into weight - homogeneous _ components _ by taking for instance the first @xmath87 terms in its taylor expansion around @xmath9 . a simpler example of a vector field admitting a weight - homogeneous splitting is to take @xmath3 to be any polynomial vector field ; then , a possible decomposition is to split it such that each component is a suitable combination of monomials . we say that the nonlinear vector field @xmath3 on @xmath0 admits a _ weight - homogeneous decomposition _ with respect to a given vector @xmath78 if it splits as a combination of the form @xmath88where the _ components _ @xmath89 are weight - homogeneous vector fields , namely , @xmath90for some non - negative numbers @xmath91 and all @xmath80 in some domain @xmath92 of @xmath93 . in terms of individual components , condition ( [ dec2 ] ) reads @xmath94 in a slightly vague but suggestive manner we can say that a weight - homogeneous decomposition splits the original vector field in parts starting by collecting together the most nonlinear part and then proceeding down to the ` weakest ' component such that each term in the splitting is ` less nonlinear ' than the previous one in a precise sense . there are two important features of such a vector - field decomposition ( dec1 ) , ( [ dec2 ] ) . firstly , it is not unique . in general , for the given vector field @xmath3 , many different vectors @xmath78 can be found which each lead to different weight - homogeneous decompositions of the vector field in the same domain @xmath95 . secondly , since the _ subdominant exponents _ @xmath96 can be ordered , @xmath97the degrees of the component vector fields in the decomposition ( [ dec2 ] ) are also ordered and ( only ) the first vector field @xmath98 appearing in the decomposition is scale invariant . therefore , a weight - homogeneous decomposition of a vector field with respect to a vector @xmath78 is a splitting into @xmath87 weight - homogeneous components each with degree @xmath99 such that the lowest - order vector field in the decomposition is scale invariant . the lowest - order vector field , @xmath98 , in a splitting is sometimes called _ the dominant part _ of @xmath3 and includes the most nonlinear terms in @xmath3 , whereas the remaining sum of parts , @xmath100 , is called _ the subdominant part_. given a vector field @xmath101 it is very important to have the complete list of all possible weight - homogeneous decompositions it admits ; in other words , to know all the possible dominant and subdominant ways it can be split . the asymptotic method we employ to trace the behaviour of vector fields and their integral curves in a neighborhood of a movable finite - time singularity ( of a particular or general solution ) , begins by finding all weight - homogeneous decompositions of the vector field valid in that neighborhood . suppose that there exists a decomposition @xmath102into ( @xmath87 ) weight - homogeneous components , such that the dominant part @xmath103 is scale invariant , and each subdominant component @xmath104 is weight - homogeneous . the scale invariant solution @xmath105is a solution only , as complex ones do not describe the behaviour near finite time singularities in the sense of eq . ( [ sing2 ] ) . ] of the dominant part @xmath106 of the vector field provided that @xmath107 . thus , some of the components of the vector @xmath80 may vanish . we call the components of the vector @xmath108 the _ dominant exponents . _ in this case , we sometimes say that ( [ dec2 ] ) is an _ asymptotic solution _ of the original system @xmath109 . on the other hand , the @xmath16 subdominant components @xmath110 satisfy @xmath111with the subdominant exponents @xmath112 which are ordered and strictly positive . dividing both sides by @xmath113 and taking the limit as @xmath114 , the true meaning of the subdominant exponents is revealed , and we have _ 0=0 , which proves that the subdominant part of the vector field , @xmath115 , is less dominant than the dominant part , @xmath116 , which of course asymptotes as @xmath117 . we say that the pair @xmath118 is a _ ( dominant ) balance _ for the vector field @xmath3 if the latter admits a decomposition satisfying eqs . ( [ dec1])-([dec3 ] ) . there may be several different balances for any particular decomposition of @xmath3 and a way to classify them is by means of their order . the _ order _ of a balance @xmath119 is the number of the nonzero components of the vector @xmath80 . for a vector field on @xmath0 , the highest order of any possible balance is @xmath10 and in this case the scale - invariant solution ( [ dec2 ] ) corresponds to a possible dominant behaviour of a _ general _ solution of the original system @xmath120 near the singularity . on the other hand , balances of a lower order than @xmath10 describe possible asymptotics of _ particular _ solutions . there is an elegant convex - geometric explanation of the dominant balances of a vector field @xmath3 . this requires us first to express the vector field in the so - called quasi - monomial form . then , a dominant balance of order @xmath121 corresponds precisely to a @xmath121-dimensional face of the newton - puiseux - bruno polyhedron associated to @xmath3 ( cf . @xcite , @xcite ) . we can now study an important square matrix , the so - called _ ( k-)ovalevskaya matrix _ , associated with a given vector field @xmath3 . consider the dominant part , @xmath116 , of @xmath3 which admits an exact solution of the form ( [ dec2 ] ) , described by the dominant balance @xmath118 ( of any nonzero order ) . the _ k - matrix of the vector field @xmath3 at the balance _ @xmath118 is the square matrix @xmath122the _ ( k-)ovalevskaya exponents _ associated with the balance @xmath123 are the @xmath10 eigenvalues @xmath124 of @xmath125 . when the order of the balance is @xmath10 ( @xmath126 @xmath61 ) , the _ k_-exponents are called the _ resonances _ of @xmath127 . setting @xmath128 and differentiating with respect to @xmath129 we have @xmath130while by the chain rule @xmath131where the last equality is most easily understood if we expand the derivative to obtain @xmath132 , for the @xmath133-th component . thus , from the last two equations , we find @xmath134that is , the k - matrix always has the vector @xmath135 as an eigenvector with eigenvalue equal to @xmath136 . we say that a balance is _ hyperbolic _ if the remaining @xmath137 k - exponents have positive real parts . suppose now that we know all the eigenvectors @xmath138 and eigenvalues @xmath139 of the k - matrix . by simply inspecting the form of the two sides in the variational equation for the dominant part of the vector field , namely , the equation @xmath140we can write the set of fundamental solutions @xmath141 of this linear equation for @xmath142 in the form @xmath143where , depending on whether or not @xmath127 is semi - simple , the @xmath144 s are the eigenvectors @xmath141 or , in general , polynomials in @xmath145 . hence , the solutions of the variational equation will be appropriate sums of terms by the form ( [ var ] ) . therefore , any solution of the original system will be well approximated ( cf . @xcite , p.299 , for the precise conditions ) by a solution of the form ( this is a taylor estimate ) : @xmath146where @xmath147where the @xmath148 s are , in general , polynomials in @xmath145 . furthermore , we arrive at the interesting conclusion that in any particular or general solution , the arbitrary constants characterizing it will first appear in those terms whose coefficients have indices equal to a k - exponent . in a general solution , the arbitrary constants normally appear at different places in an expansion , and consequently , a solution in which a k - exponent ( different from the @xmath149 value which as we have shown always exists ) is either negative , or has nontrivial multiplicity , may or may not be a general solution . we shall see in the next section how we can use eq . ( [ main1 ] ) to obtain an important series representation of the solutions to the original dynamical system near a finite - time singularity . apart from the principal use made here in unravelling the nature of finite - time singularities , there are various important connections between the k - exponents , first integrals , integrability properties of hamiltonian systems and complex algebraic geometry , cf . @xcite , @xcite . in the previous subsection , we derived the formal expansion ( [ main1 ] ) for the solution of the dynamical system @xmath150 near a finite - time singularity by including only the first terms in a power - series expansion of @xmath151 . that solution corresponds to a given dominant balance @xmath152 and depends on the k - exponents @xmath139 associated with the dominant part , @xmath116 , of the original vector field @xmath153 . writing eq . ( [ main1 ] ) in full , that is including terms of all orders , can lead to very complicated expansions in general , and the essential feature is the appearance of logarithmic terms . we note that there are a number of general theorems guaranteeing the existence and convergence of such expansions see , for instance , @xcite for a review . let @xmath154 be the set of all k - exponents with positive real parts . for simplicity we assume that all k - exponents in @xmath154 are rational , and define the number @xmath155 to be the least common multiple of the denominators of the numbers in the set @xmath156 . in the case where any log terms are absent ( for instance , when k is semi - simple ) , we can write , by ( [ main1 ] ) , the full expansion of the general solution around the finite - time singularity in the form of a _ puiseux series _ , @xmath157where , as we know already from the previous section , each of the @xmath10 arbitrary constants in ( [ main2 ] ) will first appear in the term with coefficient @xmath158 and @xmath159 . hence , finding the final form of the solution ( general or particular , depending on the number of arbitrary constants appearing in the series expansion ) can be now reduced to knowing the coefficients @xmath160 in the expansion . these coefficients are computed by inserting the puiseux series ( [ main2 ] ) into the original system . this leads to a set of _ recursion relations _ , a linear system for the coefficients @xmath160 . for the @xmath77th - order coefficient we find @xmath161 where the forms @xmath162 are polynomial in its variable , read off from the original equation . there is an important consistency condition to be satisfied for the above analysis to be valid . multiplying both sides of ( [ main3 ] ) by @xmath163 , an eigenvector of the k - matrix , we see that when @xmath164 , an eigenvalue of the k - matrix , we must have the following _ compatibility condition _ ( @xmath165 denotes the adjoint eigenvector of @xmath166 ) : @xmath167therefore , if the above compatibility condition is _ violated _ at some eigenvalue , then we conclude that no solution in the form of a puiseux series can exist and we have to search for more general solutions which may contain logarithmic terms . such a more general series will be of the form of a _ @xmath168-series _ ( cf . @xcite for this terminology ) : a direct generalization of the form ( [ main2 ] ) [ main4 ] = ^(+_i=1^_j=1^ _ ij^i / s(^)^j / s ) , where @xmath169 is the first k - exponent for which the compatibility condition is not satisfied and @xmath170 as defined above . the procedure for the calculation of the coefficients in this more general case is the same as before , leading again to the form of the general solution in a suitable neighborhood of the finite - time singularity as a @xmath168-series . suppose that ( [ dec2 ] ) is an asymptotic solution of the vector field @xmath171 . we set @xmath172and imagine that an equation of the form @xmath173regards the coefficient @xmath80 of a given balance @xmath174 of the vector field @xmath3 as an equilibrium point of the new system given , in terms of the new variables @xmath175 , by @xmath176where @xmath177 and @xmath178 is the least common multiple of the denominators of the subdominant exponents in the original system . we call the dynamical system ( [ comp ] ) the _ companion system _ of the original vector field @xmath21 . thus , the transformation ( [ comp ] ) to the companion system associates a different system of this sort to each one of the balances @xmath118 of the original system provided that any given pair of balances has different dominant exponents @xmath179 . consider now the _ linearized system _ @xmath180which is the variational equation of the companion system around the equilibrium point @xmath181 . this is a constant - coefficient , linear system . from the fundamental theorem of such systems , it follows that the general solution passing through the initial condition @xmath182 is given by @xmath183now we know ( cf . @xcite ) that for any @xmath184 matrix @xmath185 : @xmath186where @xmath187 are the eigenvalues of @xmath185 and @xmath188 are @xmath189-valued polynomials , the latter being constant if and only if @xmath190 is semi - simple ( diagonalizable ) . so @xmath191 in terms of the @xmath37-time . hence , using this form of the local solution to the companion system , we can express the local solution to the original system around its finite - time singularity in the form @xmath192which is precisely the taylor estimate leading eventually to the @xmath193-series representation we arrived at in previous sections . we therefore reach the interesting conclusion that around the equilibrium point @xmath194 , origin of the companion system ( [ comp ] ) , the eigenvalues of that system are simply @xmath195 ; that is , the dominant exponents of the asymptotic solution of the original system together with the number @xmath178 characterizing the subdominant part of the vector field . moreover , around the equilibrium point @xmath196 of the companion system ( [ comp ] ) , the eigenvalues @xmath197 of the companion system are @xmath198 ; that is , they are precisely the k - exponents of the original system together with the subdominant number @xmath199 . we have an interpretation of the results of the previous sections concerning the local behaviour of the original system around its finite - time singularities from a dynamical systems perspective . using the companion transformation , a local analysis of the companion system @xmath200 _ around its equilibrium points _ ( with @xmath201 necessarily ) will provide the local analysis of the solutions of all possible balances of the original system _ around its singularities _ ( @xmath202 since @xmath203 always , for an acceptable decomposition ) . note that , since we need @xmath204 , we are only interested in the unstable manifold ( eigenvalues with positive real parts ) of the equilibrium points of the companion system . therefore , the negative k - exponents ( corresponding to the stable manifold of the companion system ) are not connected to the behaviour of solutions of the original system at the finite - time singularity , but are associated with its behaviour as @xmath205 . for a discussion of the companion transformation in connection with integrability and complex dynamics , see @xcite and refs . therein . the results in the previous subsections suggest a general procedure to uncover the nature of singularities by constructing series expansion representations of particular or general solutions of dynamical systems in suitable neighborhoods of their finite - time singularities . this method consists of building splittings of vector fields that are valid asymptotically and trace the dominant behaviour of the vector field near the singularity . a resulting series expansion connected to a particular dominant balance helps to decide whether or not the arrived solution is a general one and to spot the exact positions of the arbitrary constants as well as their role in deciding about the nature of the time singularity . the method we suggest below is analogous to the so - called ars procedure connected with the painlev and integrability properties of dynamical systems @xcite , but here the whole approach and viewpoint are completely different . we are not concerned with notions of integrability but solely with the problem of the nature of finite - time singularities . also , our systems are real - valued with a real time variable . to apply this _ method of asymptotic splittings _ to a particular dynamical system in an effort to discover the nature of its time singularities , we must follow this recipe : 1 . write the system of equations in the form of a dynamical system @xmath206 with @xmath207 , and identify the vector field @xmath208 . 2 . find all the different weight - homogeneous decompositions of the system ; that is , the splittings of the form @xmath209and choose one of these splittings to start the procedure . 3 . substitute the scale - invariant solution @xmath210into the equation @xmath211 . study the resulting algebraic systems , and find all dominant balances @xmath118 together with their orders . 4 . identify the non - dominant exponents , that is the positive numbers @xmath212 , such that @xmath213 5 . construct the k - matrix @xmath127 : @xmath214 6 . compute the spectrum of @xmath127 , @xmath215is @xmath127 semi - simple ? are the balances hyperbolic ? 7 . find the eigenvectors @xmath138 of @xmath127 . identify @xmath170 as the multiplicative inverse of the least common multiple of all the subdominant exponents and positive k - exponents . [ puiseux ] substitute the puiseux series @xmath216into the original system . identify the polynomials @xmath217 and solve for the final recursion relations which give the unknown coefficients @xmath218 . check the compatibility conditions at the k - exponents , @xmath219 12 . if the puiseux series is valid , then the method is concluded for this particular splitting . otherwise , if compatibility conditions are violated at the eigenvalue @xmath220 , restart from step [ puiseux ] by substituting the logarithmic series ( [ main4 ] ) . 13 . get coefficient at order @xmath220 . write down the final expansion with terms up to order @xmath220 verify that compatibility at @xmath221 is now satisfied . repeat whole procedure for each of the other possible decompositions . note that although the whole spectrum of possible behaviours of the system near a time singularity is concluded once we find valid series expansions corresponding to each balance in each particular decomposition , an additional analysis of the phase space of the companion systems corresponding to each one of the balances of the original system may lead to valuable insights as to the geometric structure of the phase space _ how the orbits behave _ in the neighborhood of the finite - time singularity . following the above steps even up to that of calculating a dominant balance in one particular decomposition , can be very useful since it gives you one particular possible asymptotic behaviour of the system near the time singularity . in this respect , the whole method expounded here is truly generic since it helps to decide the generality of any behaviour found in an exact solution that is , how many arbitrary constants there are in the final solution that shares that behaviour ( particular or general solution ) . it is rare that a puiseux series is inadequate to describe the dynamics ( semi - simplicity of k ) , but in such uncommon cases one must resort to the more complex logarithmic solutions . as a concrete example of the above analysis , consider the homogeneous and isotropic frw cosmological equations in general relativity for a perfect fluid of pressure @xmath222 , density @xmath223 and equation of state @xmath224 , where @xmath225 is a constant . they read @xmath226where @xmath227 is the scale factor , @xmath228 the hubble expansion rate and @xmath16 the constant spatial curvature . setting @xmath229 , this system reads @xmath230with @xmath231and is subject to the integral constraint ( the friedmann equation ) @xmath232this system is weight - homogeneous . the unique balance is determined by @xmath233and the parameters @xmath234 depend on @xmath225 via eq . ( [ 7 ] ) . since the field is weight - homogeneous ( i.e. , @xmath235 ) , this balance corresponds to an exact , scale - invariant solution of the original system . note that there is one arbitrary coefficient , and so we expect that one of the k - exponents will be zero . for the vector field @xmath236 the associated k - matrix , @xmath237 has characteristic equation @xmath238hence the k - exponents are @xmath239we can use the integral constraint ( [ 8 ] ) to get a value for the coefficient @xmath240 . balancing the terms in ( [ 8 ] ) leads to @xmath241 in the case when all components of the vector @xmath80 are real , the solution @xmath242 of the dynamical system experiences a finite - time singularity . when @xmath243 , that is when either @xmath244 or @xmath245 , solutions are general ; while , in the range @xmath246 we have only behaviours corresponding to particular solutions of the system . note that when @xmath247 , we have @xmath248 and we find a behaviour similar to that of dust , but when @xmath249 , @xmath250 the behaviour is that of the standard radiation models . further , calculating the recursion relations to compute the coefficients of the series expansion term by term , we find the following asymptotic solution for radiation , @xmath251while the dust - dominated expansion is found to be @xmath252 the method that we have described in detail here provides a toolkit for the investigation of the general form of a range of finite - time singularities in general relativistic cosmologies . in particular , the sudden singularities introduced by one of us @xcite , in which @xmath253 and @xmath169 remain finite but @xmath254 and @xmath255 at finite time in situations where no functional relationship is assumed between @xmath256 and @xmath257 , have been widely studied @xcite . such situations , allow sudden singularities to develop at finite time without violating the strong - energy condition of general relativity and are require less severe conditions than future big rip singularities @xcite with @xmath258 . elsewhere , we will report on the results of applying these methods to determine the general behaviour on approach to these singularities in general relativity and in higher - order gravity theories . we thank yvonne choquet - bruhat , alain goriely and antonis tsokaros for useful discussions . the work of s.c . was partly supported by the ministry of education and religious affairs ( 25% ) and by e.u.(75% ) under the grant `` pythagoras '' . 99 s. w. hawking and g. f. r. ellis , _ the large - scale structure of space - time _ , ( cup , 1973 ) . y. choquet - bruhat and s. cotsakis , j. geom . 43 ( 2002 ) 345 - 350 ( arxiv : gr - qc/0201057 ) ; see also , y . choquet - bruhat , s. cotsakis , in _ recent developments in gravity _ , proceedings of the 10th hellenic relativity conference , k. d. kokkotas and n. stergioulas eds . , ( world scientific 2003 ) , pp . 145 - 149 . taylor , _ partial differential equations , basic theory _ , ( springer , 1996 ) , p. 19 . barrow , class . quantum gravity * 21 * ( 2004 ) l79 ; j.d . barrow , class . quantum grav . * 21 * ( 2004 ) 5619 ; j.d . barrow and c.g . tsagas class . quantum grav . * 22 * ( 2005 ) 1563 . s. cotsakis and i. klaoudatou , j. geom . @xmath259 ( 2005 ) 306 _ _ ; _ _ p.f . _ _ _ _ gonzlez - daz , phys . rev__. _ _ @xmath260 ( 2003 ) 021303(r ) ; g. calcagni , phys . rev . d @xmath261 ( 2004 ) 103508 ; v. gorini , a. kamenshchik , u. moschella and v. pasquier , phys . _ _ _ _ d @xmath261 ( 2004 ) 123512 ; g. kofinas , r. maartens and e. papantonopoulos , jhep @xmath262 ( 2003 ) 066 ; v. sahni and y. shtanov , class . quantum grav__. _ _ @xmath263 ( 2002 ) l101 ; s. nojiri and s. d. odintsov , phys . lett__. _ _ b @xmath264 ( 2004)1 ; m. dabrowski , t. stachowiak and m. sydlowski , phys . d @xmath260 ( 2003 ) 067301 ; p. elizalde and j. quiroga , mod . lett . a @xmath263 ( 2004 ) 29 ; p.f . gonzlez - daz , phys . b @xmath265 ( 2004 ) 1 ; a. feinstein and s. jhingan , _ _ _ _ mod . phys . _ _ _ _ a _ _ @xmath263 _ _ ( _ _ 2004 ) 457 ; l.p . chimento and r. lazkoz , _ _ _ _ phys . lett . _ _ @xmath266 ( 2003 ) 211301 ; e. elizalde , s. nojiri and s.d . odintsov , phys . d @xmath267 ( 2004 ) 043539 ; l.p . chimento and r. lazkoz , mod . _ _ _ _ a @xmath261 ( 2004 ) 123512 .

we define the notion of a finite - time singularity of a vector field and then discuss a technique suitable for the asymptotic analysis of vector fields and their integral curves in the neighborhood of such a singularity . having in mind the application of this method to cosmology , we also provide an analysis of the time singularities of an isotropic universe filled with a perfect fluid in general relativity .

You are an expert at summarizing long articles. Proceed to summarize the following text: star and planet formation are connected through disks . disk formation , long thought to be a trivial consequence of angular momentum conservation during core collapse and star formation ( e.g. , bodenheimer1995 ) , turned out to be much more complicated than originally envisioned . the complication comes from magnetic fields , which are observed in dense , star - forming , cores of molecular clouds ( see crutcher2012 for a recent review ) . the field can strongly affect the angular momentum evolution of core collapse and disk formation through magnetic braking . there have been a number of studies aiming at quantifying the effects of magnetic field on disk formation . in the ideal mhd limit , both analytic considerations and numerical simulations have shown that the formation of a rotationally supported disk ( rsd hereafter ) is suppressed by a realistic magnetic field ( corresponding to a dimensionless mass - to - flux ratio of @xmath4 a few ; trolandcrutcher2008 ) during the protostellar mass accretion phase in the simplest case of a non - turbulent core with the magnetic field aligned with the rotation axis ( allen+2003 ; galli+2006 ; pricebate2007 ; mellonli2008 ; hennebellefromang2008 ; dappbasu2010 ; seifried+2011 ; santos - lima+2012 ) . the suppression of rsds by excessive magnetic braking is termed `` magnetic braking catastrophe '' in star formation . rotationally supported disks are routinely observed , however , around evolved class ii young stellar objects ( see williamscieza2011 for a review ) , and increasingly around class i ( e.g. , jorgensen+2009 ; lee2011 ; takakuwa+2012 ) and even one class 0 source ( tobin+2012 ) . when and how such disks form in view of the magnetic braking catastrophe is unclear . the current attempts to overcome the catastrophic braking fall into three categories : ( 1 ) non - ideal mhd effects , including ambipolar diffusion , ohmic dissipation and hall effect , ( 2 ) misalignment between magnetic and rotation axes , and ( 3 ) turbulence . ambipolar diffusion does not appear to weaken the braking enough to enable large - scale rsd formation under realistic conditions ( krasnopolskykonigl2002 ; mellonli2009 ; duffinpudritz2009 ; li+2011 ) . ohmic dissipation can produce small , au - scale , rsd in the early protostellar accretion phase ( machida+2010 ; dappbasu2010 ; dapp+2012 ; tomida+2013 ) . larger , @xmath5-scale rsds can be produced if the resistivity or the hall coefficient of the dense core is much larger than the classical ( microscopic ) value ( krasnopolsky+2010,krasnopolsky+2011 ; see also braidingwardle2012a , braidingwardle2012b ) . @xcite explored the effects of tilting the magnetic field away from the rotation axis on disk formation ( see also machida+2006 ; pricebate2007 ; hennebelleciardi2009 ) . they concluded that keplerian disks can form for a mass - to - flux ratio @xmath6 as low as @xmath7 , as long as the tilt angle is close to @xmath2 ( see their fig . the effects of turbulence were explored by @xcite , who concluded that a strong enough turbulence can induce enough magnetic diffusion to enable the formation of a @xmath5-scale rsd . @xcite and @xcite considered supersonically turbulent massive cores . they found rotationally dominated disks around low - mass stars , although in both cases the turbulence - induced rotation is misaligned with the initial magnetic field by a large angle , which may have contributed to the disk formation ( see also joos+2013 ) . the goal of this paper is to revisit the role of magnetic field - rotation misalignment in disk formation . the misalignment is expected if the angular momenta of dense cores are generated through turbulent motions ( e.g. , burkertbodenheimer2000 ; myers+2012 ) . it is also inferred from the misalignment between the field direction traced by polarized dust emission and the outflow axis , which is taken as a proxy for the direction of rotation ( hull+2012 ) . indeed , in the carma sample of @xcite , the distribution of the angle @xmath8 between the magnetic field and jet / rotation axis is consistent with being random . if true , it would indicate that in half of the sources the two axes are misaligned by a large angle of @xmath9 ( see however chapman+2013 and discussion in [ disk ] ) . such a large misalignment would be enough to allow disk formation in dense cores magnetized to a realistic level ( with @xmath6 of a few ; trolandcrutcher2008 ) according to @xcite . if the alignment angle @xmath8 is indeed random and @xcite s conclusions are generally true , the magnetic braking catastrophe would be largely solved . given their far - reaching implications , it is prudent to check @xcite s conclusions , using a different numerical code . it is the task of this paper . we carry out numerical experiments of disk formation in dense cores with misaligned magnetic and rotation axes using non - ideal mhd code zeus - tw that includes self - gravity . we find that a large misalignment angle does indeed enable the formation of rsds in weakly magnetized dense cores with dimensionless mass - to - flux ratios @xmath1 , but not in dense cores magnetized to higher , more typical levels . our conclusion is that while the misalignment helps with disk formation , especially in relatively weakly magnetized cores , it may not provide a complete resolution to the magnetic braking catastrophe by itself . the rest of the paper is organized as follows . in [ setup ] , we describe the model setup . the numerical results are described in [ misalignment ] and [ strong ] . we compare our results to those of @xcite and discuss their implications in [ discussion ] and conclude with a short summary in [ summary ] . we follow @xcite and @xcite and start our simulations from a uniform , spherical core of @xmath10 and radius @xmath11 in a spherical coordinate system @xmath12 . the initial density @xmath13 corresponds to a molecular hydrogen number density of @xmath14 . we adopt an isothermal equation of state with a sound speed @xmath15 below a critical density @xmath16 , and a polytropic equation of state @xmath17 above it . at the beginning of the simulation , we impose a solid - body rotation of angular speed @xmath18 on the core , with axis along the north pole ( @xmath19 ) . it corresponds to a ratio of rotational to gravitational binding energy of 0.025 , which is typical of the values inferred for nh@xmath20 cores ( goodman+1993 ) . the initial magnetic field is uniform , tilting away from the rotation axis by an angle @xmath8 . we consider three values for the initial field : @xmath21 , @xmath22 and @xmath23 , corresponding to dimensionless mass - to - flux ratio , in units of @xmath24 , @xmath25 , @xmath26 and @xmath27 , respectively , for the core as a whole . the mass - to - flux ratio for the central flux tube @xmath28 is higher than the global value @xmath6 by @xmath29 , so that @xmath30 , @xmath31 and @xmath32 for the three cases respectively . the effective mass - to - flux ratio @xmath33 should lie between these two limits . if the star formation efficiency per core is @xmath34 ( e.g. , alves+2007 ) , then one way to estimate @xmath33 is to consider the ( cylindrical ) magnetic flux surface that encloses @xmath35 of the core mass , which yields @xmath36 , corresponding to @xmath37 , @xmath38 , and @xmath39 for the three cases respectively ; the fraction @xmath35 is also not far from the typical fraction of core mass that has accreted onto the central object at the end of our simulations ( see table 1 ) . for the tilt angle , we also consider three values : @xmath40 , @xmath41 and @xmath2 . the @xmath40 corresponds to the aligned case , with the magnetic field and rotation axis both along the @xmath42-axis ( @xmath19 ) . the @xmath43 corresponds to the orthogonal case , with the magnetic field along the @xmath44-axis ( @xmath45 , @xmath46 ) . models with these nine combinations of parameters are listed in table 1 ; additional models are discussed below . lllllll a & 9.72 & 13.7 & 0@xmath47 & 1 & 0.24 & no + b & 4.86 & 6.85 & 0@xmath47 & 1 & 0.22 & no + c & 2.92 & 4.12 & 0@xmath47 & 1 & 0.33 & no + d & 9.72 & 13.7 & 45@xmath47 & 1 & 0.21 & yes / porous + e & 4.86 & 6.85 & 45@xmath47 & 1 & 0.35 & no + f & 2.92 & 4.12 & 45@xmath47 & 1 & 0.27 & no + g & 9.72 & 13.7 & 90@xmath47 & 1 & 0.38 & yes / robust + h & 4.86 & 6.85 & 90@xmath47 & 1 & 0.46 & yes / porous + i & 2.92 & 4.12 & 90@xmath47 & 1 & 0.47 & no + m & 9.72 & 13.7 & 90@xmath47 & 0 & 0.10 & yes / robust + n & 9.72 & 13.7 & 90@xmath47 & 0.1 & 0.26 & yes / robust + p & 4.03 & 5.68 & 90@xmath47 & 1 & 0.25 & yes / porous + q & 3.44 & 4.85 & 90@xmath47 & 1 & 0.14 & no + as in @xcite , we choose a non - uniform grid of @xmath48 . in the radial direction , the inner and outer boundaries are located at @xmath49 and @xmath11 , respectively . the radial cell size is smallest near the inner boundary ( @xmath50 or @xmath51 ) . it increases outward by a constant factor @xmath52 between adjacent cells . in the polar direction , we choose a relatively large cell size ( @xmath53 ) near the polar axes , to prevent the azimuthal cell size from becoming prohibitively small ; it decreases smoothly to a minimum of @xmath54 near the equator , where rotationally supported disks may form . the grid is uniform in the azimuthal direction . the boundary conditions in the azimuthal direction are periodic . in the radial direction , we impose the standard outflow boundary conditions . material leaving the inner radial boundary is collected as a point mass ( protostar ) at the center . it acts on the matter in the computational domain through gravity . on the polar axes , the boundary condition is chosen to be reflective . although this is not strictly valid , we expect its effect to be limited to a small region near the axis . we initially intended to carry out simulations in the ideal mhd limit , so that they can be compared more directly with other work , especially @xcite . however , ideal mhd simulations tend to produce numerical `` hot zones '' that force the calculation to stop early in the protostellar mass accretion phase , a tendency we noted in our previous 2d ( mellonli2008 ) and 3d simulations ( krasnopolsky+2012 ) . to lengthen the simulation , we include a small , spatially uniform resistivity @xmath55 . we have verified that , in the particular case model g ( @xmath56 and @xmath43 ) , this resistivity changes the flow structure little compared to either the ideal mhd model m ( before the latter stops ) or model n , where the resistivity is reduced by a factor 10 , to @xmath57 . to illustrate the effect of the misalignment between the magnetic field direction and rotation axis , we first consider an extreme case where the magnetic field is rather weak ( with a mass - to - flux ratio @xmath56 for the core as a whole and @xmath58 for the inner @xmath35 of the core mass ) . in this case , a well - formed rotationally supported disk is present in the orthogonal case with @xmath43 ( model g in table 1 ) . such a disk is absent in the aligned case ( with @xmath40 , model a ) . the contrast is illustrated in fig . [ contrast ] , where we plot snapshots of the aligned and orthogonal cases at a representative time @xmath59 , when a central mass of @xmath60 and @xmath61 , respectively , has formed . the flow structures in the two cases are very different in both the equatorial ( panels [ a ] and [ b ] ) and meridian ( panels [ c ] and [ d ] ) plane . in the equatorial plane , the aligned case has a relatively large ( with radius @xmath62 ) over - dense region where material spirals rapidly inward . on the ( smaller ) scale of @xmath63 , the structure is dominated by expanding , low - density lobes ; they are the decoupling enabled magnetic structures ( dems for short ) that have been studied in detail by @xcite and @xcite . no rotationally supported disk is evident . the equatorial structure on the @xmath64 scale in the orthogonal case is dominated by a pair of spirals instead . the spirals merge , on the @xmath63 scale , into a more or less continuous , rapidly rotating structure a rotationally supported disk . clearly , the accretion flow in the orthogonal case was able to retain more angular momentum than in the aligned case . why is this the case ? a clue comes from the meridian view of the two cases ( panels [ c ] and [ d ] of fig . [ contrast ] ) . in the aligned case , there is a strong bipolar outflow extending beyond @xmath65 at the relatively early time shown . the outflow forces most of the infalling material to accrete through a flattened equatorial structure an over - dense pseudodisk ( gallishu1993 ; see panel [ a ] for a face - on view of the pseudodisk , noting the difference in scale between panel [ c ] and [ a ] ) . it is the winding of the magnetic field lines by the rotating material in the pseudodisk that drives the bipolar outflow in the first place . the wound - up field lines act back on the pseudodisk material , braking its rotation . it is the efficient magnetic braking in the pseudodisk that makes it difficult for rotationally supported disks to form in the aligned case . the prominent bipolar outflow indicative of efficient magnetic braking is absent in the orthogonal case , as was emphasized by @xcite . it is replaced by a much smaller , shell - like structure inside which the @xmath63-scale rotationally supported disk is encased ( panel [ d ] ) . to understand this difference in flow structure pictorially , we plot in fig . [ 3d ] the three - dimensional structure of the magnetic field lines on the scale of @xmath66 ( or @xmath67 ) , which is @xmath29 larger than the size of panels ( a ) and ( b ) of fig . [ contrast ] , but half of that of panels ( c ) and ( d ) . clearly , in the aligned case , the relatively weak initial magnetic field ( corresponding to @xmath68 ) has been wound up many turns by the material in the equatorial pseudodisk , building up a magnetic pressure in the equatorial region that is released along the polar directions ( see the first panel of fig . the magnetic pressure gradient drives a bipolar outflow , which is evident in panel ( c ) and in many previous simulations of magnetized core collapse , including the early ones such as @xcite and @xcite . in contrast , in the orthogonal case , the equatorial region is no longer the region of the magnetically induced pseudodisk . in the absence of rotation ( along the @xmath42-axis ) , the dense core material would preferentially contract along the field lines ( that are initially along the @xmath44-axis ) to form a dense sheet in the @xmath69-@xmath42 plane that passes through the origin . the twisting of this sheet by rotation along the @xmath42-axis produces two curved `` curtains '' that spiral smoothly into the disk at small distances , somewhat analogous in shape to two snail - shells ( see the second panel of fig . [ 3d ] ) . the snail - shaped dense curtain in the orthogonal case naturally explains the morphology of the density maps shown in panels ( b ) and ( d ) of fig . [ contrast ] . first , the two prominent spiral arms in the panel ( a ) are simply the equatorial ( @xmath44-@xmath69 ) cut of the curved curtains . an interesting feature of the spirals ( and the snail - shaped dense curtain as a whole ) is that they are the region where the magnetic field lines change directions sharply . this is illustrated in fig . [ pinch ] , which is similar to panel ( b ) of fig . [ contrast ] , except that the magnetic vectors ( rather than velocity vectors ) are plotted on top of the density map . clearly , the spirals separate the field lines rotating counter clock - wise ( lower - right part of the figure ) from those rotating close - wise ( upper - left ) . the sharp kink is analogous to the well - known field line kink across the equatorial pseudodisk in the aligned case , where the radial component of the magnetic field changes direction . it supports our interpretation of the spirals and , by extension , the curtain as a magnetically induced feature , as is the case of pseudodisk . in other words , the spirals are not produced by gravitational instability in a rotationally supported structure ; they are `` pseudospirals '' in the same sense as the `` pseudodisks '' of @xcite . the field line kinks are also evident across the dense curtain in the 3d structure shown in the second panel of fig . [ 3d ] . the 3d topology of the magnetic field and the dense structures that it induces lie at the heart of the difference in the magnetic braking efficiency between the aligned and orthogonal case . in particular , a flattened , rotating , equatorial pseudodisk threaded by an ordered magnetic field with an appreciable vertical component ( along the rotation axis ) is more conducive to driving an outflow than a warped curtain with a magnetic field predominantly tangential to its surface . the outflow plays a key role in angular momentum removal and the suppression of rotationally supported disks , as we demonstrate next . to quantify the outflow effect , we follow @xcite ( @xcite ; see also joos+2012 ) and compare the rates of angular momentum change inside a finite volume @xmath70 through its surface @xmath71 due to infall and outflow to that due to magnetic torque . the total magnetic torque relative to the origin ( from which a radius vector @xmath72 is defined ) is @xmath73\,dv,\ ] ] where the integration is over the volume @xmath70 . typically , the magnetic torque comes mainly from the magnetic tension rather than pressure force . the dominant magnetic tension term can be simplified to a surface integral ( matsumototomisaka2004 ) @xmath74 over the surface @xmath71 of the volume . this volume - integrated magnetic torque is to be compared with the rate of angular momentum advected into the volume through fluid motion , @xmath75 which will be referred to as the advective torque below . since the initial angular momentum of the dense core is along the @xmath42-axis , we will be mainly concerned with the @xmath42-component of the magnetic and advective torque which , for a spherical volume inside radius @xmath76 , are given by @xmath77 and @xmath78 the advective torque consists of two parts : the rates of angular momentum advected into and out of the sphere by infall and outflow respectively : @xmath79 and @xmath80 an example of the magnetic and advective torques is shown in fig . [ torque ] . the torques are evaluated on spherical surfaces of different radii , at the representative time @xmath59 . for the aligned case , the net torque close to the central object is nearly zero up to a radius of @xmath81 , indicating that the angular momentum advected inward is nearly completely removed by magnetic braking there . at larger distances , between @xmath82 and @xmath62 , the net torque @xmath83 is negative , indicating that the angular momentum of the material inside a sphere of radius in this range decreases with time . this is in sharp contrast with the orthogonal case , where the net torque is positive in that radial range , with the angular momentum there increasing ( rather than decreasing ) with time . one may think that the difference is mainly due to a significantly larger magnetic torque @xmath84 in the aligned case than in the orthogonal case . although this is typically the case at early times , the magnetic torques in the two cases become comparable at later times ( see the lowest solid lines in the two panels of fig . [ torque ] ; a movie of the torques is available on request from the authors ) . a bigger difference comes from the total ( or net ) angular momentum @xmath85 advected inward , which is substantially smaller in the aligned case than in the orthogonal case ( see the uppermost solid lines in the two panels ) . the main reason for the difference is that a good fraction of the angular momentum advected inward by infall @xmath86 is advected back out by outflow @xmath87 in the former , but not the latter . this is helped by the fact that @xmath86 is somewhat smaller in the aligned case to begin with ( compare the dotted lines in the two panels ) . the lack of appreciable outward advection of angular momentum by outflow , which is itself a product of field - winding and magnetic braking in the aligned case , appears to be the main reason for the orthogonal case to retain more angular momentum at small radii and form a rotationally supported disk in this particular case of relatively weak magnetic field . the formation of a rotationally supported disk can be seen most clearly in fig . [ rotation ] , where we plot the infall and rotation speed , as well as specific angular momentum as a function of radius along 4 ( @xmath88 and @xmath89 ) directions in the equatorial plane . in the orthogonal case , the infall and rotation speeds display the two tell - tale signs of rotationally supported disks : ( 1 ) a slow , subsonic ( although nonzero ) infall speed much smaller than the free fall value , and ( 2 ) a much faster rotation speed close to the keplerian value inside a radius of @xmath90 . the absence of a rotationally supported disk in the aligned case is just as obvious . it has a rotation speed well below the keplerian value and an infall speed close to the free fall value , especially at small radii up to @xmath90 . this corresponds to the region dominated by the low - density , strongly magnetized , expanding lobes ( i.e. , dems ; see panel [ a ] of fig [ contrast ] ) where the angular momentum is almost completely removed by a combination of magnetic torque and outflow ( see the third panel in fig . [ rotation ] ) . also evident from the panel is that the specific angular momentum of the equatorial inflow drops significantly twice : near @xmath91 and @xmath62 respectively . the former corresponds to the dems - dominated region , and the latter the pseudodisk ( see panel [ a ] of fig . [ contrast ] ) . the relatively slow infall inside the pseudodisk allows more time for magnetic braking to remove angular momentum . it is the pseudodisk ( and its associated outflow ) working in tandem with the dems that suppresses the formation of a rotationally supported disk in the aligned case . interestingly , there is a bump near @xmath62 for the specific angular momentum of the orthogonal case , indicating that the angular momentum in the equatorial plane is transported radially outward along the spiraling field lines from small to large distances ( see fig . [ pinch ] ) . the spiraling equatorial field lines in the orthogonal model g have an interesting property : they consist of two strains of opposite polarity . as the strains get wound up more and more tightly by rotation at smaller and smaller radii , field lines of opposite polarity are pressed closer and closer together , creating a situation that is conducive to reconnection , either of physical or numerical origin ( see the first panel of fig . [ magnetic ] ) . model g contains a small but finite resistivity ( @xmath55 ) . it does not appear to be responsible for the formation and survival of the keplerian disk , because a similar disk is also formed at the same ( relatively early ) time for a smaller resistivity of @xmath92 ( model n ) and even without any explicit resistivity ( model m ) . numerical resistivity may have played a role here , but it is difficult to quantify at the moment . in any case , the magnetic field on the keplerian disk appears to be rather weak , as can be seen from the second panel of fig . [ magnetic ] , where the plasma @xmath93 is plotted along 4 ( @xmath88 and @xmath89 ) directions in the equatorial plane . on the keplerian disk in the orthogonal model g ( inside @xmath90 ) , @xmath93 is of order @xmath94 or more , indicating that there is more matter accumulating in the disk than magnetic field , either because the matter slides along the field lines into the disk ( increasing density but not the field strength ) or because of numerical reconnection that weakens the field , or both . this situation is drastically different from the aligned case , where the inner @xmath95 region is heavily influenced by the magnetically - dominated low - density lobes . we have seen from the preceding section that , in the weakly magnetized case of @xmath56 , the @xmath5-scale inner part of the protostellar accretion flow is dominated by two very different types of structures : a weakly magnetized , dense , rotationally supported disk ( rsd ) in the orthogonal case ( @xmath43 , model g ) and magnetically dominated , low - density lobes or dems in the aligned case ( @xmath40 , model a ) , at least at the relatively early time discussed in [ misalignment ] , when the central mass reaches @xmath96 . this dichotomy persists to later times for these two models and for other models as well , as illustrated by fig . [ all ] , where we plot models a i at a time when the central mass reaches @xmath97 . it is clear that the rsd for model g becomes even more prominent at the later time , although a small magnetically dominated , low - density lobe is evident close to the center of the disk : it is a trapped dems that is too weak to disrupt the disk . in this case , the identification of a robust rsd is secure , at even later times ( up to the end of the simulation , when the central mass reaches @xmath98 or @xmath99 of the initial core mass ) . in the aligned case ( model a ) , the inner accretion flow remains dominated by the highly dynamic dems at late times , with no sign of rsd formation . for the intermediate tilt angle case of @xmath100 ( model d ) , the inner structure of the protostellar accretion flow is shaped by the tussle between rsd and dems . [ all ] shows that , at the plotted time , model d has several spiral arms that appear to merge into a rotating disk . there are , however , at least three low - density `` holes '' near the center of the disk : they are the magnetically dominated dems . movies show that the highly variable dems are generally confined close to the center , although they can occasionally expand to occupy a large fraction of the disk surface . overall , the circumstellar structure in model d is more disk - like than dems - like . we shall call it a `` porous disk , '' to distinguish it from the more filled - in , more robust disk in the orthogonal model g. even though most of the porous disk has a rotation speed dominating the infall speed , the infall is highly variable , and often supersonic . the rotation speed also often deviates greatly from the keplerian value . such an erratic disk is much more dynamic than the quiescent disks envisioned around relatively mature ( e.g. , class ii ) ysos . the intermediate tilt angle case drives a powerful bipolar outflow , unlike the orthogonal case , but similar to the aligned case . this is consistent with the rate of angular momentum removal increasing with decreasing tilt angle ( i.e. , from @xmath2 to @xmath41 ; see also ciardihennebelle2010 ) . as the strength of the initial magnetic field in the core increases , the dems becomes more dominant . this is illustrated in the middle column of fig . [ all ] , where the three cases with an intermediate field strength corresponding to @xmath101 ( and @xmath102 ) are plotted . in model h , where the magnetic and rotation axes are orthogonal , a relatively small ( with radius of @xmath103 ) rotationally dominated disk is clearly present at the time shown . as in the weaker field case of model g , it is fed by prominent `` pseudo - spirals '' which are part of a magnetically - induced curtain in 3d ( see the second panel of fig . [ 3d ] ) . compared to model g , the curtain here is curved to a lesser degree , which is not surprising because the rotation is slower due to a more efficient braking and the stronger magnetic field embedded in the curtain is harder to bend . the disk is also smaller , less dense , and more dynamic . it is more affected by dems , which occasionally disrupt the disk , although it always reforms after disruption . overall , the circumstellar structure in model h is more rsd - like than dems - like . as in model d , we classify it as a `` porous disk . '' as the tilt angle decreases from @xmath2 to @xmath41 ( model e ) and further to @xmath104 ( model b ) , the rotationally dominated circumstellar structure largely disappears ; it is replaced by dems - dominated structures . even though there is still a significant amount of rotation in the accretion flow , a dense coherent disk is absent . we conclude that for a moderately strong magnetic field of @xmath101 the formation of rsd is suppressed if the tilt angle is moderate . in the cases of the strongest magnetic field corresponding to @xmath105 ( and @xmath106 ) , the formation of rsd is suppressed regardless of the tilt angle , as can be seen from the last column of fig . [ all ] . for the orthogonal case ( model i ) , the prominent `` pseudo - spirals '' in the weaker field cases of model g ( @xmath56 ) and h ( @xmath101 ) are replaced by two arms that are only slightly bent . they are part of a well - defined pseudodisk that happens to lie roughly in the @xmath107 ( or @xmath44-@xmath42 ) plane ( see the second panel of fig . [ 3db1e-5 ] ) . in the absence of any initial rotation , one would expect the pseudodisk to form perpendicular to the initial field direction along the @xmath44-axis , i.e. , in the @xmath108 ( or @xmath69-@xmath42 ) plane . over the entire course of core evolution and collapse , the rotation has rotated the expected plane of the pseudodisk by nearly @xmath2 . nevertheless , at the time shown ( when the central mass reaches @xmath97 ) , there is apparently little rotation left inside @xmath109 to warp the pseudodisk significantly . except for the orientation , this pseudodisk looks remarkably similar to the familiar one in the aligned case ( model c ; see the first panel of fig . [ 3db1e-5 ] ) . in particular , there are low - density `` holes '' in the inner part of both pseudodisks which are threaded by intense magnetic fields and surrounded by dense filaments : they are the dems . in the intermediate tilt angle case of @xmath41 ( model f , not shown in the 3d figure ) , the pseudodisk is somewhat more warped than the two other cases , and its inner part is again dominated by dems . it is clear that for a magnetic field of @xmath6 of a few , the inner circumstellar structure is dominated by the magnetic field , with rotation playing a relatively minor role ; the rsd remains suppressed despite the misalignment . to better estimate the boundary between the cores that produce rsds and those that do not , we carried out two additional simulations with @xmath43 ( models p and q in table 1 ) . we found a porous disk in model p ( @xmath110 and @xmath111 ) , as in the weaker field case of model h , but no disk in model q ( @xmath112 and @xmath113 ) , as in the stronger field case of model i. from this , we infer that the boundary lies approximately at @xmath114 ( or @xmath115 ) . our most important qualitative result is that the misalignment between the magnetic field and rotation axis tends to promote the formation of rotationally supported disks , especially in weakly magnetized dense cores . this is in agreement with the conclusion previously reached by @xcite ( @xcite ; jhc12 hereafter ) , using a different numerical code and somewhat different problem setup . their calculations were carried out using an adaptive mesh refinement ( amr ) code in the cartesian coordinate system , with the central object treated using a stiffening of the equation of state , whereas ours were done using a fixed mesh refinement ( fmr ) code in the spherical coordinate system , with an effective sink particle at the origin . despite the differences , these two distinct sets of calculations yield qualitatively similar results . the case for the misalignment promoting disk formation is therefore strengthened . quantitatively , there appears to be a significant discrepancy between our results and theirs . according to their fig . 14 , a keplerian disk is formed in the relatively strongly magnetized case of @xmath116 if the misalignment angle @xmath43 . formally , this case corresponds roughly to our model i ( @xmath105 and @xmath43 ) , for which we can rule out the formation of a rotationally supported disk with confidence ( see fig . [ all ] ) . we believe that the discrepancy comes mostly from the initial density profile adopted , which affects the degree of magnetization near the core center for a given global mass - to - flux ratio @xmath6 . @xcite adopted a centrally condensed initial mass distribution @xmath117 with the characteristic radius @xmath118 set to @xmath35 of the core radius @xmath119 , so that the central - to - edge density contrast is 10 ( see also ciardihennebelle2010 ) . it is easy to show that , for this density profile and a uniform magnetic field , the mass - to - flux ratio for the flux tube passing through the origin is @xmath120 where @xmath121 and @xmath6 is the global mass - to - flux ratio for the core as a whole . for @xmath122 , we have @xmath123 . in other words , the central part of their core is substantially less magnetized relative to mass than the core as a whole , due to the initial mass condensation . for the @xmath116 case under consideration , we have @xmath124 , which makes the material on the central flux tube rather weakly magnetized ( relative to mass ) . the magnetization of the central region is important , because the central part is accreted first and the star formation efficiency in a core may not be 100% efficient ( e.g. , alves+2007 ) . if we define as in [ setup ] an effective mass - to - flux ratio for the ( cylindrical ) magnetic flux surface that encloses @xmath35 of the core mass , then @xmath125 for the density distribution adopted by jhc12 ( equation [ [ profile ] ] ) . it is significantly different from the effective mass - to - flux ratio of @xmath36 for the uniform density that we adopted . the above difference in @xmath33 makes our strongest field case of @xmath105 ( @xmath106 ) more directly comparable to jhc12 s @xmath126 ( @xmath127 ) case . there is agreement that , in both cases , the formation of a rsd is suppressed , even when the tilt angle @xmath43 . these results suggest that rsd formation is suppressed when the effective mass - to - flux ratio @xmath128 , independent of the degree of field - rotation misalignment , consistent with the conclusion we reached toward the end of [ strong ] based on models p and q. similarly , jhc12 s @xmath116 ( @xmath129 ) models may be more directly comparable to our @xmath101 ( @xmath102 ) models . our calculations show that a ( more or less ) rotationally supported disk is formed in the extreme @xmath43 case for @xmath102 ( model h , see fig . [ all ] ) but not in the intermediate tilt angle @xmath100 case ( model e ) . this is consistent with their fig . 14 , where a keplerian disk is formed for @xmath43 for @xmath129 , but not for @xmath100 . in the latter case , jhc12 found a disk - like structure with a flat rather than keplerian rotation profile . they attributed the flat rotation to additional support from the magnetic energy , which dominates the kinetic energy at small radii . we believe that their high magnetic energy comes from strongly magnetized , low - density lobes ( i.e. , dems ) , which are clearly visible in our model e ( see fig . [ all ] ) . although there is still significant rotation on the @xmath5 scale in model e , we find the morphology and kinematics of the circumstellar structure too disorganized to be called a `` disk . '' we conclude that for a moderate field strength corresponding to @xmath130@xmath131 , a rotationally supported disk does not form except when the magnetic field is tilted nearly orthogonal to the rotation axis . our weak - field case of @xmath56 ( @xmath58 ) can be compared with jhc12 s @xmath132 ( @xmath133 ) case . in both cases , a well - defined rotationally supported disk is formed when @xmath43 ( see our fig . [ all ] and their fig . 12 ) . for the intermediate tilt angle @xmath100 , jhc12 obtained a disk - like structure with a flat rotation curve , with the magnetic energy dominating the kinetic energy at small radii . this is broadly consistent with our intermediate tilt angle model d , where a highly variable , `` porous '' disk is formed , with the central part often dominated by strongly magnetized , low - density lobes . there is also agreement that the disk formation is suppressed if @xmath134 even for such a weakly magnetized case . it therefore appears that , for the weak - field case of @xmath135 , a rotationally supported disk can be induced by a relatively moderate tilt angle of @xmath136 . the result that a misalignment between the magnetic field and rotation axis helps disk formation by weakening magnetic braking may be counter - intuitive . @xcite showed analytically that , for a uniform rotating cylinder embedded in a uniform static external medium , magnetic braking is much more efficient in the orthogonal case than in the aligned case . this analytic result may not be directly applicable to a collapsing core , however . as emphasized by jhc12 , the collapse drags the initially uniform , rotation - aligned magnetic field into a configuration that fans out radially . jhc12 estimated analytically that the collapse - induced field fan - out could in principle make the magnetic braking in the aligned case more efficient than in the orthogonal case . the analytical estimate did not , however , take into account of the angular momentum removal by outflow , which , as we have shown in [ torque ] , is a key difference between the weak - field ( @xmath56 , @xmath137 ) aligned case ( model a ) where disk formation is suppressed and its orthogonal counterpart ( model g ) that does produce a rotationally supported disk ( see figs . [ contrast ] and [ torque ] , and also ciardihennebelle2010 ) . the generation of a powerful outflow in the weak - field aligned case ( model a ) is facilitated by the orientation of its pseudodisk , which is perpendicular to the rotation axis ( see fig . [ 3d ] ) . this configuration is conducive to both the pseudodisk winding up the field lines and the wound - up field escaping above and below the pseudodisk , which drives a bipolar outflow . when the magnetic field is tilted by @xmath2 away from the rotation axis , the pseudodisk is warped by rotation into a snail - shaped curtain that is unfavorable to outflow driving ( see fig . the outflow makes it more difficult to form a rotationally supported disk in the aligned case . disk formation is further hindered by magnetically dominated , low - density lobes ( dems ) , which affect the inner part of the accretion flow of the aligned case more than that of the perpendicular case , at least when the field is relatively weak ( see fig . [ contrast ] and [ all ] ) . for more strongly magnetized cases , the dems becomes more dynamically important close to the central object , independent of the tilt angle @xmath8 ( see fig . [ 3db1e-5 ] ) . dems - like structures were also seen in some runs of jhc12 ( e.g. , the case of @xmath126 and @xmath40 ; see their fig . 19 ) but were not commented upon . as stressed previously by @xcite and @xcite and confirmed by our calculations , the dems presents a formidable obstacle to the formation and survival of a rotationally supported disk . while there is agreement between jhc12 and our calculations that misalignment between the magnetic field and rotation axis is beneficial to disk formation , it is unlikely that the misalignment alone can enable disk formation around the majority of young stellar objects . the reason is the following . @xcite obtained a mean dimensionless mass - to - flux ratio of @xmath138 through oh zeeman observations for a sample of dense cores in nearby dark clouds near the center relative to that in the envelope . ] . for such a small @xmath33 , disk formation is completely suppressed , even for the case of maximum misalignment of @xmath43 . @xcite argued , however , that there is a flat distribution of the total field strength in dense cores , from @xmath139 to some maximum value @xmath140 ; the latter corresponds to @xmath141 , so that the mean @xmath142 stays around 2 . if this is the case , some cores could be much more weakly magnetized than others , and disks could form preferentially in these cores . however , to form a rotationally supported disk , the core material must have ( 1 ) an effective mass - to - flux ratio @xmath33 greater than about 5 _ and _ ( 2 ) a rather large tilt angle ( see discussion in the preceding section ) . if one assumes that the core - to - core variation of @xmath33 comes mostly from the field strength rather than the column density ( as done in crutcher+2010 ) , then the probability of a core having @xmath143 ( or @xmath144 ) is @xmath145 . since a large tilt angle of @xmath146 is required to form a rsd for @xmath130@xmath131 , the chance of disk formation is reduced from @xmath145 by at least a factor of 2 ( assuming a random orientation of the magnetic field relative to the rotation axis ) , to @xmath147 or less . the above estimate is necessarily rough , and can easily be off by a factor of two in either direction . it is , however , highly unlikely for the majority of the cores to simultaneously satisfy the conditions on both @xmath33 and @xmath8 for disk formation . the condition @xmath143 is especially difficult to satisfy because , as noted earlier , it implies that the dense cores probed by oh observations must have a total field strength @xmath148 , less than or comparable to the well - defined median field strength inferred by @xcite for the much more diffuse , cold neutral _ atomic _ medium ( cnm ) . it is hard to imagine a reasonable scenario in which the majority of dense cores have magnetic fields weaker than the cnm . we note that @xcite independently estimated a range of @xmath149@xmath29 for the fraction of dense cores that would produce a keplerian disk based on fig . 14 of jhc12 . their lower limit of @xmath147 is in agreement with our estimate ; in both cases , the fraction is dominated by weakly magnetized cores that have mass - to - flux ratios greater than @xmath150 and moderately large tilt angles . their upper limit of @xmath151 is much higher than our estimate , mainly because it includes rather strongly magnetized cores with mass - to - flux ratios as small as @xmath152 . since our calculations show that such strongly magnetized cores do not produce rotationally supported disks even for large tilt angles , we believe that this upper limit may be overly generous . whether dense cores have large tilt angle @xmath8 between the magnetic field and rotation axis that are conducive to disk formation is unclear . @xcite measured the field orientation on the @xmath109-scale for a sample of 16 sources using millimeter interferometer carma . they found that the field orientation is not tightly correlated with the outflow axis ; indeed , the angle between the two is consistent with being random . if the outflow axis is aligned with the core rotation axis and if the field orientation is the same on the core scale as on the smaller , @xmath109-scale , then @xmath8 would be randomly distributed between @xmath153 and @xmath2 , with half of the sources having @xmath154 . however , the outflow axis may not be representative of the core rotation axis . this is because the ( fast ) outflow is thought to be driven magnetocentrifugally from the inner part of the circumstellar disk ( on the au - scale or less ; shu+2000 ; koniglpudritz2000 ) . a parcel of core material would have lost most of its angular momentum on the way to the outflow launching location ; the torque ( most likely magnetic or gravitational ) that removes the angular momentum may also change the direction of the rotation axis . similarly , the field orientation on the @xmath109-scale may not be representative of the initial field orientation on the larger core scale . the magnetic field on the @xmath109 scale is more prone to distortion by collapse and rotation than that on the core scale . indeed , @xcite found that the field orientation on the core scale measured using single - disk telescope cso is within @xmath155 of the outflow axis for 3 of the 4 sources in their sample ; the larger angle measured in the remain source may be due projection effects because its outflow axis is close to the line - of - sight . if the result of @xcite is valid in general and if the outflow axis reflects the core rotation axis , then dense cores with large tilt angle @xmath8 would be rare . in this case , disk formation would be rare according to the calculations presented in this paper and in jhc12 , even in the unlikely event that the majority of dense cores are as weakly magnetized as @xmath143 . rotationally supported disks are observed , however , routinely around evolved class ii ysos ( see williamscieza2011 for a recent review ) , and increasingly around younger class i ( e.g. , jorgensen+2009 ; lee2011 ; takakuwa et al . 2012 ) and even class 0 ( tobin+2012 ) sources . when and how such disks form remain unclear . our calculations indicated that the formation of large observable ( @xmath5-scale ) rotationally supported disks . small disks may be needed to drive fast outflows during the class 0 phase , and may form through non - ideal mhd effects ( machida+2010 ; dappbasu2010 ; dapp+2012 ) . ] is difficult even in the presence of a large tilt angle @xmath8 during the early protostellar accretion ( class 0 ) phase ( e.g. , alves+2007 ) . ] . this may not contradict the available observations yet ( maury+2010 ) , since only one keplerian disk is found around a ( late ) class 0 source so far ( l1527 ; tobin+2012 ) ; it could result from a rare combination of an unusually weak magnetic field and a large tilt angle @xmath8 . if it turns out , through future observations using alma and jvla , that large - scale keplerian disks are prevalent around class 0 sources , then some crucial ingredients must be missing from the current calculations . possible candidates include non - ideal mhd effects and turbulence . the existing calculations indicate that realistic levels of the classical non - ideal mhd effects do not weaken the magnetic braking enough to enable large - scale disk formation ( mellonli2009 ; li+2011 ; krasnopolsky+2012 ; see also krasnopolskykonigl2002 and braidingwardle2012a , braidingwardle2012b ) , although misalignment has yet to be considered in such calculations . supersonic turbulence was found to be conducive to disk formation ( santos - lima+2012,santos - lima+2013 ; seifried+2012 ; myers+2012 ; joos+2013 ) , although dense cores of low - mass star formation typically have subsonic non - thermal line - width and it is unclear whether subsonic turbulence can enable disk formation in dense cores magnetized to a realistic level . if , on the other hand , it turns out that large - scale keplerian disks are rare among class 0 sources , then the question of disk growth becomes paramount : how do the mostly undetectable class 0 disks become detectable in the class i and ii phase ? if the magnetic braking plays a role in keeping the early disk undetectable , then its weakening at later times may promote rapid disk growth . one possibility for the late weakening of magnetic braking is the depletion of the protostellar envelope , either by outflow stripping ( mellonli2008 ) or accretion ( machida+2011 ) . it deserves to be better quantified . we carried out a set of mhd simulations of star formation in dense cores magnetized to different degrees and with different tilt angles between the magnetic field and the rotation axis . we confirmed the qualitative result of @xcite that misalignment between the magnetic field and rotation axis tends to weaken magnetic braking and is thus conducive to disk formation . quantitatively , we found however that the misalignment enables the formation of a rotationally supported disk only in dense cores where the star - forming material is rather weakly magnetized , with a dimensionless mass - to - flux ratio @xmath1 ; 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stars form in dense cores of molecular clouds that are observed to be significantly magnetized . in the simplest case of a laminar ( non - turbulent ) core with the magnetic field aligned with the rotation axis , both analytic considerations and numerical simulations have shown that the formation of a large , @xmath0-scale , rotationally supported protostellar disk is suppressed by magnetic braking in the ideal mhd limit for a realistic level of core magnetization . this theoretical difficulty in forming protostellar disks is termed `` magnetic braking catastrophe . '' a possible resolution to this problem , proposed by @xcite and @xcite , is that misalignment between the magnetic field and rotation axis may weaken the magnetic braking enough to enable disk formation . we evaluate this possibility quantitatively through numerical simulations . we confirm the basic result of @xcite that the misalignment is indeed conducive to disk formation . in relatively weakly magnetized cores with dimensionless mass - to - flux ratio @xmath1 , it enabled the formation of rotationally supported disks that would otherwise be suppressed if the magnetic field and rotation axis are aligned . for more strongly magnetized cores , disk formation remains suppressed , however , even for the maximum tilt angle of @xmath2 . if dense cores are as strongly magnetized as indicated by oh zeeman observations ( with a mean dimensionless mass - to - flux ratio @xmath3 ) , it would be difficult for the misalignment alone to enable disk formation in the majority of them . we conclude that , while beneficial to disk formation , especially for the relatively weak field case , the misalignment does not completely solve the problem of catastrophic magnetic braking in general .

You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years , the problem of anisotropic quantum scattering in two spatial dimensions ( 2d ) attracts increasing interest . it is stimulated by the spectacular proposals for prospects to create exotic and highly correlated quantum systems with dipolar gases @xcite . particularly , there were considered anisotropic superfluidity @xcite , 2d dipolar fermions @xcite , and few - body dipolar complexes @xcite . the recent experimental production of ultracold polar molecules in the confined geometry of optical traps @xcite has opened up ways to realize these phenomena . noteworthy also is a rather long history of research of 2d quantum effects in condensed matter physics . one can note superfluid films @xcite , high - temperature superconductivity @xcite , 2d materials , such as graphene @xcite , and even possibilities for topological quantum computation @xcite . unique opportunities for modeling these 2d effects in a highly controlled environment have recently appeared with the development of experimental techniques for creating quasi-2d bose and fermi ultracold gases @xcite . interest in the processes and effects in 2d - geometry has stimulated the theory of elementary quantum two - body systems and processes in the plane . special consideration should be given to the anisotropy and long - range character of the dipole - dipole interaction . actually , usual partial - wave analysis becomes inefficient for describing the dipole - dipole scattering due to the strong anisotropic coupling of different partial - waves in the asymptotic region @xcite . recently , considerable progress in the analysis of the 2d and quasi-2d ( q2d ) scattering of dipoles has been achieved @xcite . thus , the 2d dipolar scattering in the threshold and semiclassical regimes was studied in the case of the dipole polarization directed orthogonally to the scattering plane @xcite . an arbitrary angle of polarization was considered in @xcite . in this work , we develop a method for quantitative analysis of the 2d quantum scattering on a long - range strongly anisotropic scatterer . particularly , it permits the description of the 2d collisions of unpolarized dipoles . our approach is based on the method suggested in @xcite for the few - dimensional scattering which was successfully applied to the dipole - dipole scattering induced by an elliptically polarized laser field in the 3d free - space @xcite . the key elements of the method are described in section ii . in section iii , we apply the method to the 2d scattering on the cylindrical potential with the elliptical base and the 2d dipole - dipole scattering of unpolarized dipoles . we reproduce the threshold formula @xcite for the scattering amplitude on the cylinder potential with the circular base and the results of @xcite for the 2d scattering of polarized dipoles . high efficiency of the method has been found in all problems being considered . the last section contains the concluding remarks . some important details of the computational scheme and illustration of the convergence are given in appendices . the quantum scattering on the anisotropic potential @xmath0 in the plane is described by the 2d schrdinger equation in polar coordinates @xmath1 @xmath2 with the scattering boundary conditions @xmath3 in the asymptotic region @xmath4 and the hamiltonian of the system @xmath5 the unknown wave function @xmath6 and the scattering amplitude @xmath7 are searched for the fixed momentum @xmath8 defined by the colliding energy @xmath9 ( @xmath10 and the direction @xmath11 of the incident wave ( defined by the angle @xmath12 and for the scattering angle @xmath13 . here @xmath14 is the reduced mass of the system . in the polar coordinates , the angular part of the kinetic energy operator in @xmath15 has a simple form @xmath16 . the interaction potential @xmath17 can be anisotropic in the general case , i.e. to be strongly dependent on @xmath13 . it is clear that varying the direction of the incident wave @xmath11 can be replaced by the rotation @xmath18 of the interaction potential by the angle @xmath19 for the fixed direction of the incident wave , which we choose to be coincident with the x - axis . thus , in the case of anisotropic potential @xmath17 the task is to solve the problem ( [ eq1 ] ) with the interaction potential @xmath20 for all possible @xmath19 and fixed @xmath9 with the scattering boundary conditions @xmath21 if the scattering amplitude @xmath7 is found , one can calculate the differential scattering cross section @xmath22 where @xmath23 , as well as the total cross section @xmath24 by averaging over all possible orientations @xmath19 of the scatterer and integration over the scattering angle @xmath13 . to integrate the problem ( [ eq1]),([eq2 ] ) , we use the method suggested in @xcite to solving a few - dimensional scattering problem and applied in @xcite for the dipole - dipole scattering in the 3d free - space . following the ideas of these works we choose the eigenfunctions @xmath25 of the operator @xmath26 as a fourier basis for the angular - grid representation of the searched wave - function @xmath27 . we introduce the uniform grid @xmath28 ) over the @xmath13 and @xmath19-variables and search the wave function as expansion @xmath29 where @xmath30 is the inverse matrix to the @xmath31 square matrix @xmath32 defined on the angular grid , we use the completeness relation for the fourier basis @xmath33 , which in our grid representation reads @xmath34 . ] . in the representation ( [ eq7 ] ) the unknown coefficients @xmath35 are defined by the values of the searched wave function on the angular grid @xmath36 , any local interaction is diagonal @xmath37 and the angular part @xmath38 of the kinetic energy operator has a simple form @xmath39 note that the presence in the interaction potential of the `` nonlocal '' angular part ( i.e. the integration or differentiation over angular variable ) leads to destroying the diagonal structure in ( [ eq8 ] ) . thus , the 2d schrdinger equation ( [ eq1 ] ) is reduced in the angular - grid representation ( [ eq7 ] ) to the system of coupled ordinary differential equations of the second order : @xmath40 since the wave function @xmath41 must be finite at the origin @xmath42 , the `` left - side '' boundary condition for the functions @xmath35 reads as @xmath43 in the asymptotic region @xmath4 the scattering boundary condition ( [ eq3 ] ) accepts the form @xmath44 by using the fourier expansion for the plane wave @xmath45 and the scattering amplitude @xmath46 are the first kind bessel functions of integer order . their asymptotic behavior @xcite : @xmath47{}\sqrt { \frac{2}{\pi z } } \cos \left ( { z-\frac{m\pi } { 2}-\frac{\pi } { 4 } } \right)+e^{\left| { imz } \right|}{\rm o}\left ( { \left| z \right|^{-1 } } \right),\\ \left ( { \left| { \arg ( z ) } \right|<\pi } \right)\end{aligned}\ ] ] ] @xmath48 @xmath49 we eliminate the angular dependence from the asymptotic equation ( [ eq12 ] ) and represent the `` right - side '' boundary condition for the functions @xmath50 in the form @xmath51 to solve the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , we introduce the grid over the @xmath52 and reduce the system of differential equations ( [ eq10 ] ) by using the finite - difference approximation of the sixth order to the system of @xmath53 algebraic equations @xmath54 with the band - structure of the matrix @xmath55 with the width @xmath56 of the band . by using the asymptotic equations ( [ eq15 ] ) in the last two points @xmath57 and @xmath58 one can eliminate the unknown vector @xmath59 from equation ( [ eq15 ] ) and rewrite the `` right - side '' boundary condition in the form @xmath60 analogously , one can eliminate unknown constant from expression ( [ eq11 ] ) by considering asymptotic equations ( [ eq11 ] ) at the first points @xmath61 and @xmath62 . the acquired `` left - side '' boundary condition reads @xmath63 thus , the scattering problem is reduced to the boundary value problem ( [ my_eq15]-[my_eq17 ] ) @xmath64 which can be efficiently solved with standard computational techniques such as the sweeping method @xcite or the lu - decomposition @xcite . the detailed structure of the matrix of the coefficients @xmath65 is discussed in appendix a. after the solving of eq.([my_eq19 ] ) and finding the wave function @xmath66 the scattering amplitude @xmath67 is constructed according to eqs.([eq15 ] ) and ( [ eq14 ] ) . first , we have analyzed the 2d scattering on the cylindrical potential barrier with the elliptical base @xmath68 the case of the circular base @xmath69 was considered in @xcite , where analytic formula for the scattering amplitude @xmath70+i\frac{\pi } { 2}}\ ] ] was obtained at the zero - energy limit @xmath71 . here and @xmath72 is the euler constant . we have analyzed the scattering on the potential barrier with circular base @xmath73 for arbitrary momentum @xmath74 . the results of calculation presented in figs . [ fig1 ] and [ fig3 ] confirm the convergence of the scattering amplitude @xmath67 to the analytical value ( [ eq17 ] ) at @xmath75 . in this subsection all the calculations were performed in the units @xmath76 . in the limiting case of the infinitely high potential barrier ( [ eq16 ] ) with the circular base @xmath77 the asymptotic formula ( [ eq17 ] ) becomes exact for arbitrary @xmath74 . this is confirmed by investigation presented in table [ tab1 ] which illustrates the convergence of the numerical values @xmath78 with increasing ( @xmath79 ) and narrowing ( @xmath80 ) of the potential barrier to the analytic result ( [ eq17 ] ) . in the limit case @xmath79 and @xmath81 we obtain @xmath82 for the scattering length @xmath83 extracted from the calculated amplitude @xmath84 by the formula ( [ eq17 ] ) , what is in agreement to the estimate given in @xcite . the range of applicability of eq . ( [ eq17 ] ) was investigated recently in @xcite . [ cols="^,^,^,^,^ " , ] the boundary - value problem ( [ eq10]),([eq11 ] ) and ( [ eq15 ] ) , obtained in section ii in the angular - grid representation ( [ eq7 ] ) , reads in a matrix form as : @xmath85\bm { \psi } ( \rho ) \,+\\ \hspace{5 cm } + \,\hat{h}^{(0 ) } \bm { \psi } ( \rho ) = 0 \\ \bm { \psi } \xrightarrow[{\rho \to 0}]{}const\cdot \sqrt \rho \\ \frac{2\pi}{(2 m + 1)\sqrt \rho } \sum\limits_{j=0}^{2 m } { e ^{-im\phi_j } \psi _ j ( \rho ) } = i^mj_m ( q\rho ) \sqrt { 2\pi } + \\ \hspace{5 cm } + \,\frac{f_m(\phi_q)}{\sqrt { -i\rho } } e^{iq\rho } \,,\\ \end{array } } \right.\ ] ] where and @xmath86 . in this representation the angular dependence is built into the matrix @xmath87 and the interaction is included into the diagonal matrix @xmath88 of values of the potential in the angular grid nodes . the constant matrix @xmath87 couples all equations in a system and does not depend on the radial variable . there is no need to compute any matrix elements of the potential , what essentially minimizes the computational costs . for solving boundary value problem ( [ eq23 ] ) the seven - point finite - difference approximation for second derivatives of sixth order @xmath89 is applied in the points @xmath90 of the radial grid , where @xmath91 . as a result , the system ( [ eq23 ] ) reduces to the system of linear algebraic equations ( [ my_eq16 ] ) with the matrix @xmath55 whose band structure reads @xmath92 where the @xmath93 coefficients are the square @xmath94 matrices , and the elements @xmath95 of the right - side of ( [ eq25 ] ) are the @xmath96 dimensional vectors . after employing the `` right - side '' boundary condition in the form ( [ my_eq16 ] ) in the last three grid points @xmath97 and , analogously , the `` left - side '' boundary condition in the form ( [ my_eq17 ] ) in the first two points @xmath98 , the detailed structure of the matrix @xmath55 is represented as @xmath99 the block structure of the system ( [ eq26 ] ) provides several significant advantages . the block matrix can be stored in a packaged form , which allows the use of optimal resource . the system ( [ eq26 ] ) can be efficiently solved by a fast implicit matrix algorithm based on the idea of the block sweep method @xcite . in the table [ tab4 ] we illustrate the convergence of the calculated scattering amplitude @xmath67 over the number of angular grid points @xmath96 for the scatterers with weak and essential anisotropy at @xmath100 in the potential barrier ( [ eq16 ] ) . for the case @xmath101 we reach the accuracy of four significant digits in the scattering amplitude on the angular grids with @xmath102 . for stronger anisotropy @xmath103 the accuracy of two significant digits was reached at @xmath104 . 33 m. a. baranov , phys . rep . * 464 * , 71 ( 2008 ) . i. bloch , j. dalibard and w. zwerger , rev . phys . * 80 * , 885 ( 2008 ) . c. ticknor , r. m. wilson and j. l. bohn , phys . lett . * 106 * , 065301 ( 2011 ) . g. m. brunn and e. taylor , phys . * 101 * , 245301 ( 2008 ) . j. c. cremon , g. m. brunn and s. m. reimann , phys . lett . * 105 * , 255301 ( 2010 ) . al . , science * 322 * , 231 ( 2008 ) ; s.ospelkaus et . al . , science * 327 * , 853 ( 2010 ) . l. d. carr et . new j. phys * 11 * , 055049 ( 2009 ) . m. h. g. de miranda et . al . , nat . phys . * 7 * , 502 ( 2011 ) . p. minnhagen , rev . phys . * 59 * , 1001 ( 1987 ) . a. lee , n. nagaosa and x .- g . wen , rev . mod . phys . * 78 * , 17 ( 2006 ) . k. s. novoselov , rev . * 83 * , 837 ( 2011 ) . c. nayak , s. h. simon , a. stern , m. freedman and s. d. sarma , rev . 80 * , 1083 ( 2008 ) . k. martiyanov , v. makhalov and a. turlapov , phys . * 105 * , 030404 ( 2010 ) ; a. turlapov , jetp letters * 95 * , 96 ( 2012 ) . m. marinescu and l. you , phys . * 81 * , 4596 ( 1998 ) ; b. deb and l. you . a * 64 * , 022717 ( 2001 ) v. s. melezhik and chi - yu hu , phys . lett . * 90 * , 083201 ( 2003 ) . c. ticknor , phys . a * 80 * , 052702 ( 2009 ) . c. ticknor , phys . a * 84 * , 032702 ( 2011 ) . c. ticknor , phys . a * 81 * , 042708 ( 2010 ) . j. p. dincao and c. h. greene , phys . a * 83 * , 030702 ( 2011 ) . z. li , s. v. alyabishev and r. v. krems , phys . * 100 * , 073202 ( 2008 ) . v. s. melezhik , j. comput . phys . * 92 * , 67 - 81 ( 1991 ) . l. d. landau and e. m. lifshitz , in _ quantum mechanics : non - relativistic theory _ , vol . 3 ( pergamon press , 1977 ) 3rd ed . , chap .

[ txt : abstract ] we study the quantum scattering in two spatial dimensions ( 2d ) without the usual partial - wave formalism . the analysis beyond the partial - wave approximation allows a quantitative treatment of the anisotropic scattering with a strong coupling of different angular momenta nonvanishing even at the zero - energy limit . high efficiency of our method is demonstrated for the 2d scattering on the cylindrical potential with the elliptical base and dipole - dipole collisions in the plane . we reproduce the result for the 2d scattering of polarized dipoles in binary collisions obtained recently by ticknor [ phys . rev . a * 84 * , 032702 ( 2011 ) ] and explore the 2d collisions of unpolarized dipoles .

You are an expert at summarizing long articles. Proceed to summarize the following text: x - ray spectra of bhxrbs show evidence for a two - phase structure to the accretion flow , an optically thick , geometrically thin accretion disc @xcite giving rise to a blackbody component in the x - ray spectrum , and a hot optically - thin component , modelled as a power law . the relative strengths of these two components define the appearance of different ` states ' ( e.g. @xcite ) . in the hard state , which we focus on in this paper , the power - law emission dominates the total luminosity . it has been suggested that the power - law is produced by an inner , optically - thin advection dominated accretion flow ( adaf ) ( e.g. @xcite ) , which replaces the inner optically thick disc at low accretion rates , and extends down to the innermost stable circular orbit ( isco ) . this implies that the optically thick disc is truncated at some transition radius . alternatively , the optically thick and optically thin components may co - exist over some range of radii , e.g. if the thin disc is sandwiched by a hot flow or corona @xcite . the corona may in turn evaporate the innermost regions of the disc at low accretion rates ( e.g. @xcite ) , so that the optically thin flow is radially separated from the optically thick disc as for the adaf model . it has also been suggested @xcite that a cold thin accretion disc extends close to the black hole in the hard - state . in this model , most of the accretion power is transported away from the disc to power a strong outflowing corona and jet . since the adaf and corona perform similar roles in that they upscatter soft photons to produce the observed power - law , we shall henceforth refer to both components interchangeably as the corona , without necessarily favouring either picture . in the hard state , the variability of the power - law continuum offers further clues to the structure of the accretion flow . studies of the timing properties of hard state bhxrbs show that the frequencies of broad lorentzian features in their power - spectral density functions ( psds ) correlate with the strength of the reflection features as well as the steepness of the power law continuum @xcite . these correlations can naturally be explained if the lorentzian frequencies correspond to a characteristic timescale at the disc truncation radius , e.g. the viscous time - scale , so that as the truncation radius increases the lorentzian frequency decreases , with disc reflection and compton cooling of the optically thin hot flow by disc photons decreasing accordingly . in this picture , the truncation radius of the thin disc acts to generate the signals of the lowest - frequency lorentzian in the psd , while the highest - frequency lorentzians may be generated at the innermost radius of the hot inner coronal flow , i.e. , at the isco @xcite . regardless of whether the corona is radially or vertically separated from the thin disc , photons upscattered by the corona should interact with the disc . this interaction gives rise to reflection features in the x - ray spectrum , including fluorescent iron line emission and a reflection continuum due to compton scattering off the disc material . an often - neglected consideration is that a significant fraction of the photons interacting with the disc are absorbed and the disc heated in a process known as thermal reprocessing . provided that the disc subtends at least a moderate solid angle as seen by the corona , this effect should be particularly significant in the hard state , where the coronal power law continuum dominates the total luminosity . when the power - law luminosity impinging on the disc is high compared to the disc luminosity due to internal heating , then a significant fraction of the disc blackbody emission should be reprocessed and will therefore track variations of the power law continuum . the anticipated correlated variations of the blackbody and power - law emission can be studied using variability spectra , e.g. the rms spectrum , which show only the variable components of the spectrum @xcite . if the geometry is such that the observed power - law produces reprocessed blackbody emission by x - ray heating the disc , then both power - law and blackbody components should appear together in the variability spectra . furthermore , by selecting different time ranges covered by these variability spectra ( i.e. , analogous to the method of fourier - resolved spectrosopy , @xcite ) , it is possible to determine whether the low - frequency part of the psd has a different origin to the high - frequency part in terms of the contributions of blackbody and power - law components . if the optically thick disc does drive the low - frequency lorentzian , we predict that the blackbody component should be stronger in the corresponding variability spectrum . in this work , we examine the variability spectra of two hard state black hole x - ray binaries , swift j1753.5 - 0127 and gx 339 - 4 , which have good _ xmm - newton _ data for studying simultaneous variations of disc and power - law . previous analyses of time - averaged spectra for these _ xmm - newton _ observations have shown the existence of the expected blackbody components , together with relativistically broadened reflection features , which have been used to argue that the disc is truncated at significantly smaller radii than previously thought , perhaps only a few gravitational radii @xcite . however , see @xcite for arguments in favour of disc truncation at larger radii . in the following section , we describe the observations and data reduction . in section [ anres ] we show the soft and hard - band psds obtained from the data , and present a technique to produce a type of rms spectrum , the ` covariance spectrum ' which we use to identify the variable spectral components for each hard state source . in particular we show that , although both power - law and disc blackbody emission are correlated , as expected from thermal reprocessing , the disc is relatively more variable than the power - law on longer time - scales ( corresponding to the low - frequency lorentzian ) , contrary to what we would expect from simple reprocessing models . in section [ discuss ] we discuss the interpretation of our results and present further analysis to suggest that the low - frequency lorentzian corresponds to fluctuations intrinsic to the disc . a summary of our conclusions is given in section [ conc ] . swift j1753.5 - 0127 was observed during revolution 1152 for 42 ks by _ xmm - newton _ epic - pn on 2006 march 24 in _ pn - timing _ mode using the medium optical filter . the events list was screened using the perl script xmmcleankaa / xselect / xmmclean ] to select only events with flag=0 and pattern @xmath0 4 . examination of light curves showed no evidence for background flaring . using sas version 7.0 , evselect was used to extract the mean spectrum , following the procedure of @xcite , using event positions between 20 and 56 in rawx and using the full rawy range . the sas tasks arfgen and rmfgen were used to generate the ancillary response file ( arf ) and redistribution matrix file ( rmf ) . gx 339 - 4 was observed by _ xmm - newton _ on 2004 march 16 during revolutions 782 and 783 , again in _ pn - timing _ mode using the medium optical filter . the sas command evselect was used to filter the events on time to avoid background flaring , producing a total combined exposure of 127 ks . data reduction was very similar to that for swift j1753.5 - 0127 , using sas version 7.0 , but care was taken to avoid pile - up . successive columns from the centre of the image were excised in rawx and spectra extracted until no discernable difference in spectral shape could be identified between successive selections on rawx , implying that pile - up is no longer significant . the epic - pn data was deemed free of pile - up using extraction regions in rawx from columns 30 to 36 and 40 to 46 ( i.e. , columns 37 to 39 inclusive were excised ) . background spectra were selected in rawx from columns 10 to 18 over the full rawy range . to generate an appropriate arf for the data made in this way , it was necessary to use arfgen to generate an arf for the full region in rawx from columns 30 to 46 and generate a second arf from the spectrum of the excluded region ( rawx columns 37 to 39 ) and then subtract the latter from the former using the command addarf . for both sources , the ftool grppha was used to scale the background spectra and to ensure that a minimum of 20 counts were in each bin for @xmath1 fitting . the epic - pn covers the energy range from 0.2 - 10 kev , but the epic calibration status document recommends restricting the fit to energies greater than 0.5 kev . to be conservative , our epic - pn spectral fits were restricted to the range 0.7 to 10.0 kev . we extracted data from _ rxte _ observations which were contemporaneous with the _ xmm - newton _ observations . we used the high time and spectral - resolution pca event mode data to extract mean spectra and make rms and covariance spectra using the same time binning as the epic - pn data ( see section [ method ] ) . throughout this work we use _ rxte _ pca data in the 3 - 25 kev range , and fit these together with the corresponding epic - pn spectra , tying all fit parameters together but allowing _ rxte _ spectral fits to be offset by a constant factor with respect to the simultaneous epic - pn fits , to allow for the difference in flux calibration between the pca and epic - pn instruments , and also for the fact that slight flux differences may result from the fact that the _ rxte _ observations covered shorter intervals than the _ xmm - newton _ data . a 1 per cent systematic error was assumed in all spectral fits to account for uncertainties in instrumental response and cross - calibration . the signal to noise of _ rxte _ hexte data was not sufficient to perform the covariance analysis we describe in section 3.2 . before describing the spectral analysis method and results , we first examine the timing properties of each source in soft and hard spectral bands using the power - spectral density function ( psd ) . for each source , we used the epic - pn data to generate light curves with 1 ms time binning in two energy bands : 0.5 - 1 kev ( soft ) and 2 - 10 kev ( hard ) . only complete segments of 131 s duration were used to construct the psds , segments with gaps ( which can be common at high count rates in timing mode ) were skipped over . the resulting poisson - noise - subtracted psds are shown in figure [ psds ] . it is clear from the figure that the psds are significantly different in shape between the two bands . in gx 339 - 4 , the hard - band psd shows two broad components , with the higher - frequency component appearing to be shifted to even higher frequencies in the hard band . in swift j1753.5 - 0127 , soft and hard psds appear , within the noise , to overlap more closely at higher frequencies . however , in both sources the soft band psd shows relatively larger low - frequency power compared to the psd components at frequencies above @xmath2 hz . this @xmath3 extra low - frequency power at lower x - ray energies may be associated with the soft excess emission , e.g. if the disc is varying more than the power - law on longer time - scales . alternatively , there could be power - law spectral variability that applies only on longer time - scales and causes the soft band variability amplitude to be enhanced , i.e. , due to steepening of the power - law spectral slope on longer time - scales . to determine the origin of the extra low - frequency power in the soft band , we must carry out a spectral analysis of the variations on different time - scales , the methodology for which we describe in the next section . when fitting models to x - ray spectra , it is typical to use mean x - ray spectra that only describe the time - averaged spectral shape of a source . such fits say nothing about the way the different spectral components such as disc blackbody and power - law vary with respect to each other in time . by looking at the absolute amplitude of variations in count rate as a function of energy we can construct ` variability spectra ' which pick out only the time - varying components . one technique is to construct fourier - frequency resolved spectra , obtaining a psd for each individual energy channel and integrating the psd over a given fourier - frequency range in order to measure the variance in that channel , which is used to obtain the rms and so construct the spectrum due to components which vary over that frequency range ( e.g. @xcite ) . this approach is attractive in that it allows the user to pick out complex patterns of spectral variability where components have different time - scales of variation , as may be implied by the soft and hard band psds shown in figure [ psds ] . here we will consider only two time - scales of variability , corresponding roughly to the high and low - frequency parts of the psd which show significant relative differences between the soft and hard bands . to approximate the more complex fourier - resolved approach we will measure over two time - scale ranges a variant of the ` rms spectrum ' , which measures the absolute root - mean - squared variability ( rms ) as a function of energy ( e.g. see @xcite ) . producing the rms spectrum involves allocating each photon event to a time and energy bin , dividing the light curve into segments consisting of @xmath4 time bins per segment and then working out the variance in each segment for each energy bin according to the following standard formula : @xmath5 where @xmath6 is the count rate in the @xmath7 bin and @xmath8 is the mean count rate in the segment . the expectation of poisson noise variance , given by the average squared - error @xmath9 , is subtracted , leaving the ` excess variance ' , @xmath10 , in each segment . the excess variances can then be averaged over all of the segments of the light curve . the square root of the average excess variance plotted against energy forms the rms spectrum . by selecting the time bin size and the segment size , we can isolate different time - scales of variability , effectively replicating the fourier resolved approach for the two time - scale ranges that we are interested in . for this purpose , we choose two combinations of bin size and segment size . to look at variations on shorter time - scales we choose 0.1 s time bins measured in segments of 4 s ( i.e. , 40 bins long ) , i.e. , covering the frequency range 0.25 - 5 hz where @xmath11 is the time bin size . ] . for longer time - scale regions we use 2.7 s bins in segments of 270 s , i.e. , covering the range 0.0037 - 0.185 hz note that these two frequency ranges do not overlap and also cover the two parts of both source psds which show distinct behaviour in soft and hard bands . there is , however , a problem with the rms spectrum when signal to noise is low , e.g. at higher energies . it is possible for the expectation value of the poisson variance term to be larger than the measured average variance term , producing negative average excess variances . if this is the case , it is not possible to calculate the rms at these energies and this introduces a bias towards the statistically higher - than - average realisations of rms values , which can still be recorded . in order to overcome these problems we have developed a technique called the ` covariance spectrum ' . the covariance is calculated according to the formula : @xmath12 where @xmath13 now refers to the light curve for a ` reference band ' running over some energy range where the variability signal - to - noise is large . in this work , we use reference bands of 14 kev for epic - pn data and 35 kev for pca data . in other words , the covariance spectrum is to the rms spectrum what the cross - correlation function of a time series is to its auto - correlation function . the covariance spectrum therefore does not suffer from the same problems as the rms spectrum , as no poisson error term has to be subtracted , since uncorrelated noise tends to cancel out and any negative residuals do not affect the calculation . to remove the reference band component of the covariance , and produce a spectrum in count - rate units , we obtain the normalised covariance for each channel using : @xmath14 where @xmath15 is the excess variance of the reference band . therefore , the only requirement for there being a valid , unbiased value of covariance at a given energy is that the reference excess variance is not negative . this is usually the case , since the reference band is chosen to include those energies with the largest absolute variability . when the covariance is being calculated for an energy channel inside the reference band , the channel of interest is removed from the reference band . the reasoning behind this is that if the channel of interest is duplicated in the reference band , the poisson error contribution for that channel will not cancel and will contaminate the covariance . one can think of the covariance technique as applying a matched filter to the data , where the variations in the good signal - to - noise reference band pick out much weaker correlated variations in the energy channel of interest that are buried in noise . in this way the covariance spectrum picks out the components of the energy channel of interest that are correlated with those in the reference band . it is important to note that the covariance spectrum only picks out the correlated variability component and is therefore a more appropriate measure than the rms spectrum in constraining the reprocessing of hard photons to soft photons , which will result in correlated variations . when the raw counts rms and covariance spectra are overlaid , as in figure [ fig : covrms ] , they match closely indicating that the reference band is well correlated with all other energies ( the spectral ` coherence ' is high , e.g. see @xcite ) . another advantage of the matched filter aspect of the covariance spectrum is that it leads to smaller statistical errors than the rms spectrum . specifically , the errors are given by : @xmath16=\sqrt{\frac{\sigma_{\rm xs , x}^2 \overline{\sigma_{\rm err , y}^2}+\sigma_{\rm xs , y}^2 \overline{\sigma_{\rm err , x}^2}+\overline{\sigma_{\rm err , x}^2}~\overline{\sigma_{\rm err , y}^2}}{nm\sigma_{\rm xs , y}^2}}\ ] ] where @xmath17 denotes the number of segments and subscripts x and y identify excess variances and poisson variance terms for the channel of interest and reference band respectively . this error equation can be derived simply from the bartlett formula for the error on the zero - lag cross - correlation function , assuming that the source light curves have unity intrinsic coherence @xcite . by comparison with equation b2 of @xcite , using the relation : @xmath18 ( which is true because the reference band has good signal to noise ) it can be shown that the errors on the covariance are smaller than corresponding errors on the rms values . finally , we note that , since the covariance is analogous to the zero - lag cross - correlation function , it could be affected by intrinsic time lags in the data . using measurements of the cross - spectral phase - lags between various energy bands , we have confirmed that the lags between hard and soft band variations are smaller than the time bin sizes used to make the long and short time - scale covariance spectra . thus , intrinsic time lags will have no effect on our results . we first consider the covariance spectra of swift j1753.5 - 0127 , and use xspec v12 @xcite to fit the long and short - time - scale data together with the mean spectrum in order to identify the origin of the additional long - time - scale variability which can be seen in the soft band psd . the covariance spectra do not show sufficient signal - to - noise to detect the rather weak iron line present in the mean spectrum of this source @xcite , so for simplicity we fit only a simple power - law and multicolour disc blackbody diskbb , together with neutral absorption . we fit the short and long - time - scale covariance and mean spectra simultaneously , tying the absorbing column density and renormalising constant for pca data to be the same for the mean and covariance spectra . an f - test showed that the disk blackbody temperature does not change significantly between spectra , so that was also tied to be the same for all spectra . the remaining parameters were allowed to be free between the covariance and mean spectra . for those parameters that were free to vary , epic - pn and pca values were tied together . the @xmath1 of the final fit was 1802 for 1948 degrees of freedom ( d.o.f . ) and full fit parameters are listed in table [ tab : swift ] . the best - fitting unfolded spectra and the corresponding data / model ratios are shown in figure [ fig : swifteeufspec ] . to interpret the fits to the covariance spectra , consider the case where the observed psds have identical shapes in both hard and soft bands ( they may have different normalisations ) . in this case , the ratio of soft to hard - band variability amplitudes measured over the same time - scale range will be identical for any given time - scale range . therefore , since ( for high - coherence variations ) the covariance spectrum quantifies variability amplitude as a function of energy , the shape of the covariance spectrum will be independent of time - scale . on the other hand , if the soft band contains more long - time - scale variability relative to short - time - scale variability than the hard band , the long - time - scale covariance spectrum will appear _ softer _ than the short - time - scale covariance spectrum : lower energies show correspondingly greater variability on long time - scales and therefore larger fluxes in the covariance spectra . just such an effect is seen in the model fits to the covariance spectra : the long - time - scale covariance spectrum is softer than the short - time - scale covariance spectrum , because the disc blackbody normalisation is higher on long time - scales . the power - law slope is remarkably similar on both long and short time - scales however . therefore the additional long - term variability in the soft - band psd seems to result from _ additional _ variability of the disc blackbody , not any extra power - law variability ( e.g. due to spectral pivoting ) . .fit parameters for mean and covariance spectra of swift j1753.5 - 0127 for the model constant*phabs*(diskbb+powerlaw ) [ cols="<,^,^,^",options="header " , ] for the definitions of the parameters listed in the first six rows see table [ tab : swift ] . the additional parameters shown are for the relativistically smeared reflection ( from top to bottom ) : disc innermost radius ( units of @xmath19 ) ; disc inclination ( fixed to be the same in both kdblur and hrefl ) ; gaussian line energy ; gaussian normalisation ( photons @xmath20 s@xmath21 ) ; covering fraction of the reflection ( where 1.0 corresponds to @xmath22 steradians ) . [ tab : gx ] the fits to the covariance spectra for gx 339 - 4 show a similar pattern to that seen for swift j1753.5 - 0127 , in that the long time - scale covariance spectrum is softer than the short time - scale covariance spectrum because of a significantly stronger disc blackbody component , while their power - law indices are very similar . we can confirm this interpretation using a plot of the covariance ratio , which is shown in figure [ fig : gxcovratios ] , and shows a similar rise at low energies to that seen for swift j1753.5 - 0127 , which underlines the interpretation that the disc blackbody component is the main contributor to the additional variability seen at low frequencies in the soft band in gx 339 - 4 . the reader is referred to sections 4.1 and 4.4 for a consistent interpretation of the short time - scale covariance spectrum in gx 339 - 4 and swift j1753.5 - 0127 . note that the apparent emission feature around 2 kev is probably an instrumental effect , possibly related to mild pile - up in timing mode , since it disappears when larger regions of rawx are excised from the data . due to the 1 per cent systematic included in the spectral fitting , this feature has no effect on the fit results . based on the ratio of component normalisations to those in the mean spectrum , the fractional rms values for the disc component are 39 per cent and 30 per cent over the long and short time - scale ranges respectively . the corresponding power - law fractional rms values at 1 kev are 41 per cent and 40 per cent . using the same approach , we can also define a fractional rms for the iron line emission for long and short time - scales , at @xmath23 per cent in each case . interestingly , the @xmath24 per cent increase in iron line rms over that of the power - law emission ( which drives the line variability ) is comparable to the increase in reflection covering fraction from the mean to covariance spectra . these results may imply the presence of an additional constant power - law component which dilutes the power - law fractional variability but does not contribute to reflection . it is also interesting to note that the power - law component in the covariance spectra for gx 339 - 4 is softer than in the mean spectrum , i.e. , the difference is in the opposite sense to that seen in swift j1753.5 - 0127 . we have shown that disc blackbody emission contributes significantly to the x - ray variability spectra in the hard state of the black hole candidates swift j1753.5 - 0127 and gx 339 - 4 , and moreover , that the disc emission is the origin of the additional soft band variability seen on longer time - scales in both sources , which manifests itself as an enhanced low - frequency component in the psd . in this section , we discuss the evidence for a connection between disc and power - law variability through x - ray reprocessing , and then consider two possible explanations for the enhanced disc variability on long time - scales , in terms of geometry changes or fluctuations intrinsic to the accretion disc . finally we will compare our results and interpretation to the wider picture of different accretion states . the power - law emission clearly dominates the x - ray luminosity in both sources , as can be seen in figures [ fig : swifteeufspec ] and [ fig : gx339eeufspec ] . in this situation , if the disc sees a reasonable fraction of the power - law emission , we should expect that x - ray heating of the disc , i.e. , thermal reprocessing of power - law emission , will produce a significant fraction of the observed disc luminosity . in fact , the thermal reprocessed emission is directly related to the disc reflection component in the spectrum : if a fraction of incident power - law luminosity @xmath25 is reflected by the disc ( both through compton reflection and emission line fluorescence ) , then a fraction @xmath26 must be absorbed and will be reprocessed into thermal blackbody radiation . the reflected fraction @xmath25 depends on disc ionisation state but simple exploration of the disc reflection models in xspec shows that it is typically 30 - 40 per cent of the incident luminosity , so that around 60 - 70 per cent of the incident luminosity is reprocessed into disc blackbody emission . for typical iron k@xmath27 line equivalent widths ( around 1 kev with respect to the reflection continuum ) , we then expect the line flux to be of order 1 per cent of the thermally reprocessed flux . we might reasonably assume that the blackbody component in the short time - scale covariance spectrum is produced by thermal reprocessing of the varying power - law , which also drives the line emission in the same spectrum . the unabsorbed disc flux is @xmath28 erg @xmath20 s@xmath21 , and the iron line flux is @xmath29 erg @xmath20 s@xmath21 which is 1.3 per cent of the reprocessed flux , i.e. , consistent with a reprocessing origin for the thermal emission , at least on short time - scales . in swift j1753.5 - 0127 , the disc blackbody emission is considerably weaker than in gx 339 - 4 , and correspondingly we would expect a relatively weak iron line , with a few tens of ev equivalent width , which is only just consistent with the lower - limits on line strengths reported by @xcite and @xcite . it is possible that the disc in swift j1753.5 - 0127 is substantially ionised ( e.g. see @xcite ) which would enhance line emission relative to the absorbed ( and hence thermally reprocessed ) emission . since it is likely that there is substantial x - ray heating of the disc , one must interpret the observed blackbody normalisations in the mean spectra with caution . the disc emissivity may be more centrally concentrated than the theoretically expected @xmath30 law , and so the normalisations indicate better the emitting surface area and can not be simply translated to an inner radius . nonetheless , the inferred emitting areas are still relatively small , implying distance scales of tens to hundreds of km ( assuming distances @xmath31 kpc and 6 - 15 kpc for swift j1753.5 - 0127 and gx 339 - 4 respectively ; @xcite ) . we also note here that although the need for disc blackbody emission to explain the spectrum of swift j1753.5 - 0127 has been questioned by @xcite , the model - independent covariance ratio plots in figure [ fig : swiftcovratios ] show that a distinct soft component must be present in order to explain the difference in the shapes of the covariance spectra . it is interesting to note that the power - law indices of the covariance spectra are different to those of the mean spectra , but in an opposite sense for each of the two sources considered here : compared to the mean spectrum swift j1753.5 - 0127 shows a harder power - law in the covariance spectra , while gx 339 - 4 shows a softer power - law . the difference may be caused by flux - dependent spectral pivoting or steepening , so that as flux increases the spectrum gets softer in gx 339 - 4 , increasing the covariance at soft energies relative to the mean , while the opposite effect occurs in swift j1753.5 - 0127 ( it hardens as it gets brighter ) . the difference in behaviour may be related to the source luminosity : if they lie at similar distances swift j1753.5 - 0127 is at least a factor 10 less luminous than gx 339 - 4 , implying a significantly lower accretion rate . correlations between flux and spectral - hardness have been seen in bhxrb hard states , and interestingly the sign of the correlation appears to switch over from negative to positive at low luminosities , both on short time - scales @xcite and in the long - term global correlation @xcite . the same switch in flux - hardness correlation could be related to the different power - law behaviour of swift j1753.5 - 0127 and gx 339 which we see here . the same pattern appears on both long and short time - scales , which show almost identical power - law indices in their covariance spectra , so that the effect of the power - law spectral variability will be to change the normalisation of the psd , but not the shape . since most of the power - law luminosity will be found at tens of kev ( assuming a thermal cutoff at around 100 kev ) , spectral steepening with flux will cause the observed gx 339 - 4 0.5 - 10 kev variability amplitude to be enhanced compared to the true luminosity variations . conversely , observed variations in swift j1753.5 - 0127 will be smaller than the total luminosity variations . therefore , the fractional luminosity variations for swift j1753.5 - 0127 and gx 339 - 4 may be similar , but gx 339 - 4 shows a significantly greater normalisation in the psds shown in figure [ psds ] . we have seen that it is likely that reprocessing of the power - law drives at least some of the disc variability seen in hard state sources , and possibly all of it on time - scales @xmath32 s. however , model - independent covariance ratio plots and spectral fitting show that the covariance spectra of both sources demonstrate increased disc blackbody variability with respect to the power law on longer timescales . there are several possible explanations for this pattern , but the key thing that any successful model needs to achieve is an increase in disk variability on longer timescales without a concomitant rise in power law variability . also , it is important to note that the additional disc variability must still be correlated with power - law variations , because the enhanced disc variations appear in the covariance spectrum , which is identical to the rms spectrum over the high signal - to - noise energy range covered by the disc ( i.e. , coherence is unity ) . if the disc variations were independent of the power - law they would cancel to some extent , since they would be uncorrelated with the power - law component , and covariance would be smaller than the rms . thus the blackbody variations map on to power - law variations but with larger amplitude . one possibility is to change the geometry of the system on longer time - scales . for example , a variable coronal scale height on longer timescales could lead to changes in the solid angle of disc heated by the power - law , thus increasing the disc blackbody variability amplitude relative to the power - law . weaker correlated power - law variation could then be produced if scale - height correlates with power - law luminosity . alternatively , correlated power - law variations could be due to variable seed photon numbers from the disc due to the enhanced variable heating , but in either case the additional power - law variability must be of smaller amplitude than the observed blackbody variability . regardless of these model - dependent arguments , one can make a simple observational test of the variable - geometry model , by comparing the variability of reflection on long and short time - scales . gx 339 - 4 shows significant reflection features in its covariance spectra , so any variation in coronal geometry should manifest itself as increased variability in the reflection components as the disk sees the varying power law . however , the spectral fit parameters given in table [ tab : gx ] indicate that there is little change in reflection amplitude between long and short time - scales , both in terms of reflection covering fraction and iron line equivalent width ( e.g. ratio of line flux to power - law normalisation , which is meaningful because the power - law indices are so similar ) . to highlight this similarity in the covariance spectra , we show in figure [ fig : gxrefl ] the data / model ratios for the pca spectra with the reflection components taken out of the fit . the spectra demonstrate no significant change in the reflection continuum and associated iron line . this implies that if there is any change in geometry , it is small , and the increase in the thermal component of the variability spectra on longer timescales has a different origin . to place this result on a more rigorous statistical footing , we show in figure [ fig : contour ] a contour plot of the best - fitting short and long - time - scale reflection covering fractions , which are obtained only from fits to the pca data , which are most sensitive to the reflection continuum . if variable geometry is the cause of the enhanced long - term blackbody variability , we would expect the covering fraction to show a similar enhancement in variability on long time - scales . the dashed line in the figure shows the largest ratio of short - to - long - time - scale covering fraction which crosses the 99 per cent confidence contour , with a value of 0.81 , placing a 99 per cent confidence upper limit of 23 per cent on any increase in the covering fraction on long time - scales compared to short time - scales ( the 90 per cent confidence upper limit on any increase is 12 per cent ) . in contrast , the long - time - scale blackbody variability amplitude increases by 30 per cent compared to that on shorter time - scales , which can not be explained by the permitted increase in variable reflection at greater than the 99 per cent confidence level . thus , in gx 339 - 4 at least , we can rule out long - time - scale changes in coronal geometry as a viable explanation for the enhanced blackbody variability amplitude . note also that the same arguments apply to any other non - geometric arguments which seek to produce the extra long - term disc variability by varying the power - law contribution as seen by the disc , e.g through variable beaming of the power - law towards the disc . due to the weakness of the iron line in swift j1753.5 - 0127 it is not possible to place similar constraints on coronal geometry changes in this source , although by analogy we expect the same interpretation to apply . having established that variable coronal geometry can not explain the extra blackbody variability on longer timescales , we are left to consider the possibility that the variations are intrinsic to the disc itself . perhaps the simplest possibility is that the fluctuations are due to accretion rate fluctuations in the disc which undergo viscous damping before they reach the corona . such damping is expected in thin discs ( e.g. see @xcite ) , but will not be as significant in geometrically thick flows , which might correspond to the corona . therefore one can envisage that the long time - scale variability , e.g. corresponding to the low - frequency lorentzian in the psd , is generated in the thin disc , producing relatively large amplitude blackbody variations but being damped before reaching an inner coronal emitting region ( perhaps inside the innermost radius of the thin disc ) . the correlated power - law variability could then arise either from the residual undamped variations in accretion rate which reach the corona , or be driven by seed photon variations from the disc . if the inner radius of the disc were to fluctuate on long time - scales , then the varying disc area would also introduce extra blackbody variability . this model is in some sense analogous to that of changes in coronal scale height , since any change in disc area will vary the solid angle of disc seen by the corona which will cause an increase in reflection as well as correlated power - law variability driven by seed photon variations . one could mitigate these effects if the coronal properties were linked to those of the disc inner - radius , e.g. as inner radius decreases , so does coronal scale - height . this situation might be expected if the corona is formed by evaporation of the disc @xcite , so that condensation of the disc will drain and cool the coronal plasma , leading to a reduction in scale - height . we have established that the hard - state variability on time - scales greater than seconds , corresponding to the low - frequency lorentzian psd component , is very probably produced by variations intrinsic to the accretion disc , perhaps in the form of propagating accretion rate fluctuations , as envisaged by @xcite to explain the broad shapes of observed psds . these variations then manifest as weaker power - law variations , either through propagation of accretion fluctuations to the corona @xcite , or through variations of seed photons from the disc , which are compton upscattered in the corona . on shorter time - scales , the blackbody variations are probably mostly produced by x - ray heating of the disc by the power - law , which is a required outcome of the x - ray reflection directly observed in these systems . the exact split between direct emission from intrinsic disc variations and x - ray heating is difficult to judge , since some residual intrinsic variations may remain on short time - scales , and x - ray heating will also contribute on long time - scales . but in the case of intrinsic disc accretion fluctuations , the variable x - ray heating is itself a product of those fluctuations , and so any intrinsic disc variations on long time - scales must be large , at least comparable to the fractional rms of blackbody emission , i.e. , 40 per cent in both sources . it is interesting to contrast the large intrinsic disc variability in these hard state sources with that in the soft states , where the disc emission dominates the bolometric luminosity . the soft states are well - known for showing very weak , if any , variability @xcite , and the strongest variability which is seen , e.g. in cyg x-1 , is associated with the power - law , with the disc being remarkably constant @xcite . therefore it seems likely that hard state discs are inherently unstable compared to soft state discs . this difference may represent just another observable distinction between hard and soft states , but it is interesting to speculate that it may play a more primary role in creating the other observed differences , such as a strong corona and jet formation in the hard state . certainly , it seems likely that intrinsic disc variability plays an important role in determining the psd shape in the hard state . for instance , the low energy psd of grs 1915 + 105 in the @xmath33-class hard intermediate state @xcite shows a low frequency component which disappears at higher energies , possibly indicating that this source is demonstrating intrinsic disc variability . it is worth noting here that x - ray / optical studies of agn also show evidence for reprocessing on short time - scales and intrinsic disc variations on longer time - scales @xcite . however these agn are relatively luminous and radio - quiet and so are likely analogues of bhxrb soft states , perhaps indicating that disc - stability shows a mass - dependence , e.g. related to the transition between gas- and radiation - pressure dominated discs @xcite . our results strongly suggest that disc variations are responsible for the low - frequency component in the hard state psd . a number of authors have suggested that the low - frequency lorentzian corresponds to the viscous time - scale of the inner , truncation radius of the thin disc ( e.g. @xcite ) , a view which is consistent with our results . the higher - frequency psd components may then be produced in the corona , which is likely to be geometrically thick so will show naturally shorter variability time - scales . the viscous time - scale scales with scale - height ( @xmath34 ) over radius ( @xmath35 ) as @xmath36 . assuming the ratio of disc scale - height to radius @xmath37 ( where @xmath27 is the viscosity parameter ) , the predicted inner disc radius corresponding to the observed low - frequency psd peak around @xmath38 hz is 15 r@xmath39 ( see @xcite ) . however , this radius may be even smaller for smaller @xmath40 , which would make the disc truncation radius consistent with the results from fits to the iron line ( this work , and @xcite ) . however , in the latter case the corona would need to be very compact , or seed photon variations from intrinsic disc variability would modulate the power - law with a similar amplitude to the disc . in either case , assuming the disc is thin and that it varies on the viscous time - scale , its inner radius must be relatively small . such a picture is very different from earlier models for the hard state , where the disc is highly truncated and power - law emission is produced by a very extended corona or adaf ( e.g. @xcite ) . spectral fits to observations of bhxrb sources in the hard state show increasing evidence for both power law and black body components . in this work we have explored the hard - state variability of two sources with known soft excesses , swift j1753.5 - 0127 and gx 339 - 4 . our findings are summarised below . 1 . we have introduced a new spectral analysis technique , the covariance spectrum , which measures the correlated variability in different energy bands . this technique overcomes the problems of low signal - to - noise and bias associated with the rms spectrum and has smaller statistical errors . 2 . psds of the two sources demonstrate larger low - frequency power in the soft band . 3 . the longer time - scale ( 2.7 - 270 s ) covariance spectra of both sources are softer than the short time - scale ( 0.1 - 4 s ) covariance spectra , due to additional disc variability , i.e. , extra disc variability occurs on longer time - scales without a concomitant rise in power law variability on such timescales . however , the coherence of the rms and covariance spectra show clearly that disc variations are not independent of the power law variations the strength of reflection features that are detected in the short time - scale covariance spectra of gx 339 - 4 are consistent with the observed blackbody variations on those time - scales being driven by thermal reprocessing of the power - law emission absorbed by the disc . however , the reflection covering fraction and iron line equivalent width show little change between short and long time - scales , implying that additional reprocessing , due to coronal geometry change , is not responsible for the additional blackbody variability seen on longer time - scales . the extra blackbody variability seen on longer time - scales appears to be intrinsic to the accretion disc itself , giving rise to the extra low - frequency power in the psd . this represents the first clear evidence that the low - frequency lorentzian component in hard state psds is produced by disc variability . models invoking damped mass accretion rate variations or oscillations in the disc truncation radius can satisfactorily explain the observed pattern of variability . 6 . the implication of such variations occurring in a thin disc on viscous timescales , is that the disc truncation radius is @xmath41 r@xmath39 . this work highlights the importance of measuring spectral variability on a range of time - scales . mean spectra , which describe the average properties of a source , provide no information on how different spectral components are related to one another as a function of time . by using the covariance spectra we have been able to disentangle the correlated spectral components in these two sources , identify thermal reprocessing as the mechanism by which variability is correlated in different bands , produce model - independent evidence for additional blackbody variability on longer time - scales and therefore associate intrinsic disc variability with the low frequency lorentzian feature seen in hard - state psds . we would like to thank beike hiemstra and the anonymous referee for useful comments . we are grateful to maria daz trigo for providing the gx 339 - 4 epic - pn events files and helpful advice . tw is supported by an stfc postgraduate studentship grant , and pu is supported by an stfc advanced fellowship . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center , and also made use of nasa s astrophysics data system . 1 arvalo , p. , uttley , p. , kaspi , s. , breedt , e. , lira , p. , m@xmath42hardy , i. m. , 2008 , mnras , 389 , 1479 arnaud , k. a. 1996 , astronomical data analysis software and systems v , 101 , 17 axelsson , m. , hjalmarsdotter , l. , borgonovo , l. , larsson , s. , 2008 , a&a , 490 , 253 bartlett , m.s . , 1955 , an introduction to stochastic processes , cup , cambridge box , g.e.p . , jenkins , g.m . , 1976 , time series analysis : forecasting and control , 2nd edn . holden - 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_ xmm - newton _ x - ray spectra of the hard state black hole x - ray binaries ( bhxrbs ) swift j1753.5 - 0127 and gx 339 - 4 show evidence for accretion disc blackbody emission , in addition to hard power - laws . the soft and hard band power - spectral densities ( psds ) of these sources demonstrate variability over a wide range of time - scales . however , on time - scales of tens of seconds , corresponding to the putative low - frequency lorentzian in the psd , there is additional power in the soft band . to interpret this behaviour , we introduce a new spectral analysis technique , the ` covariance spectrum ' , to disentangle the contribution of the x - ray spectral components to variations on different time - scales . we use this technique to show that the disc blackbody component varies on all time - scales , but varies more , relative to the power - law , on longer time - scales . this behaviour explains the additional long - term variability seen in the soft band . comparison of the blackbody and iron line normalisations seen in the covariance spectra in gx 339 - 4 implies that the short - term blackbody variations are driven by thermal reprocessing of the power - law continuum absorbed by the disc . however , since the amplitude of variable reflection is the same on long and short time - scales , we rule out reprocessing as the cause of the enhanced disc variability on long time - scales . therefore we conclude that the long - time - scale blackbody variations are caused by instabilities in the disc itself , in contrast to the stable discs seen in bhxrb soft states . our results provide the first observational evidence that the low - frequency lorentzian feature present in the psd is produced by the accretion disc . [ firstpage ]

You are an expert at summarizing long articles. Proceed to summarize the following text: for the last three decades , equations of the type @xmath0 have been studied intensively . here , @xmath1 , @xmath2 , @xmath3 , @xmath4 are transition rates on @xmath5 , and @xmath6 is a familiy of independent brownian motions . the following special cases with very different interpretations and different behaviour are quite common in the literature . [ ex1 ] the ( wright - fisher ) stepping stone model from mathematical genetics : @xmath7 . [ ex2 ] the parabolic anderson model ( with brownian potential ) from mathematical physics : @xmath8 . [ ex3 ] the super random walk from pure probability theory : @xmath9 . [ ex4 ] the critical ( spatial ) ornstein - uhlenbeck process : @xmath10 . for the super random walk , @xmath11 is the branching rate which in this case is time - space independent . in @xcite , a two type model based on two super random walks with time - space dependent branching was introduced . the branching rate for one species is proportional to the value of the other species . more precisely , the authors considered @xmath12 where now @xmath13 and @xmath14 are families of independent brownian motions . solutions are called mutually catalytic branching processes . in the following years , properties of this model were well studied ( see for instance @xcite , @xcite ) . + in this paper , we are interested in a variant of mutually catalytic branching , namely the symbiotic branching model introduced in @xcite for continuous space . the same equations as for the mutually catalytic branching model are considered but additionally the driving noises are correlated in the following way : @xmath15 where @xmath16 $ ] is a correlation parameter . for @xmath17 and as well for general @xmath18 there are basically two approaches to formalize the equations . in @xcite under quite restrictive assumptions on the transition kernel @xmath4 , existence of solutions was obtained in the space of tempered sequences . since their assumptions in particular assume symmetry and exponential decay of @xmath4 , already existence of solutions in cases we are interested in is not clear . this is why we stick to the setup of @xcite , which as well is more popular for interacting diffusions . for the transition kernel @xmath4 we assume @xmath19 two main examples of interest are the following : [ e1 ] the discrete laplacian is given by @xmath20 obviously , @xmath21 , @xmath22 , @xmath23 are fulfilled . further , the one - dimensional riemann walk ( see for instance @xcite ) has transition rates @xmath24 with @xmath25 normalising the total rate to @xmath26 . here @xmath21 , @xmath22 , @xmath23 are also fulfilled but in contrast to the discrete laplacian the assumptions of @xcite are not satisfied . to specify the state space we fix a positive , summable function @xmath27 on @xmath5 satisfying @xmath28 for some finite constant @xmath29 . in @xcite a possible choice ( see their equation ( 4.13 ) ) is given by @xmath30 where @xmath31 is positive and summable , @xmath32 denote the @xmath33-step transition probabilities , and @xmath34 . this is needed to verify the generalized mytnik self - duality which was introduced for the continuous space analogue model in proposition 5 of @xcite . the duality for the discrete space is similar . for the duality in @xmath35 in the special case @xmath17 see lemma 4.1 of @xcite the state space is now defined by pairs of functions of the following liggett - spitzer space @xmath36 the choice of @xmath27 does not influence the results . [ o ] for @xmath37 and @xmath16 $ ] there is a ( weak ) solution of the symbiotic branching model with almost surely continuous paths and state space @xmath35 . for @xmath38 solutions are unique in law . the proof of proposition [ o ] is standard . existence can be proven by finite dimensional approximations as in @xcite . for @xmath39 uniqueness follows from the generalized mytnik self - duality as in @xcite . for @xmath40 uniqueness is true since moments increase slowly enough , and for @xmath41 uniqueness of solutions is not known . + for this work we restrict ourselves to homogeneous initial conditions @xmath42 this is not necessary but simplifies the notation a lot . + the interesting feature of the symbiotic branching model is that it connects examples [ ex1]-[ex3 ] above . being first established as a time - space inhomogeneous version of a pair of super random walks , the examples from above appear as special cases : @xmath17 obviously corresponds to the mutually catalytic branching model . the case @xmath40 with the additional assumption @xmath43 corresponds to the stepping stone model as can be seen as follows : since in the perfectly negatively correlated case @xmath44 , the sum @xmath45 solves a discrete heat equation and with the further assumption @xmath43 stays constant for all time . hence , for all @xmath2 , @xmath46 which shows that @xmath47 is a solution of the stepping stone model with initial condition @xmath48 and @xmath49 is a solution with initial condition @xmath50 . finally , suppose @xmath51 is a solution of the parabolic anderson model , then , for @xmath41 , the pair @xmath52 is a solution of the symbiotic branching model with initial conditions @xmath53 . + the purpuse of this and the accompanying paper @xcite is to understand the nature of the symbiotic branching model better . how does the model depend on the correlation @xmath18 ? are properties of the extremal cases @xmath54 inherited by some regions of the parameters ? since the longtime behaviour of super random walk , stepping stone model , mutually catalytic branching model , and parabolic anderson model are very different , one might guess that for varying @xmath18 different regimes correspond to the different models . + the focus of @xcite lies on the longtime behaviour in law ( unifying the classical results for the stepping stone model , mutually catalytic branching model , and parabolic anderson model ) if @xmath4 generates a recurrent markov process , on ( un)boundedness of higher moments @xmath55 $ ] as @xmath56 , and the wave speed for the continuous space analogue . it was shown that in the recurrent case @xmath55 $ ] is bounded in @xmath57 if and only if @xmath58 for the transient case the behaviour in @xmath18 is open to a large extent . + in contrast to @xcite the present paper focuses on second moments . note that ( [ e ] ) implies that in the recurrent case second moments are bounded if and only if @xmath59 . this can also be seen and analysed in more detail using a moment - duality which will be explained in section [ sm ] . using this duality , we show how to reduce second moments of symbiotic branching processes to moment generating functions and laplace transforms of local times of discrete space markov processes , i.e. @xmath60 , \end{aligned}\ ] ] where @xmath61 and @xmath62 denotes the local time @xmath63 in @xmath64 . for simple random walks the behaviour as @xmath65 was partially analysed in @xcite by analytic methods . here , we present a simple new proof based on a renewal - type equation and tauberian theorems . the simplicity of the proof has the advantage that no further assumptions on the markov process ( in particular no symmetry and no finite range assumptions ) are needed . for any @xmath3 and @xmath66 the technique yields precise growth rates including all constants . + as an application , intermittency and aging for symbiotic branching processes are established . + the main results on intermittency and aging are collected in section [ sec : results ] . in section [ section2 ] we first establish representations for second moments of symbiotic branching processes ( section [ sm ] ) , then prove the results for exponential moments of local times ( section [ sectionlocal ] ) , and , finally , proofs of the main results ( section [ sectionmain ] ) are given . the first property we address is intermittency ( see for instance @xcite for a discussion of the ideas ) . the @xmath67-th lyapunov exponent is defined by @xmath68\label{ly}\end{aligned}\ ] ] if the limit exists ( @xmath69 analogously ) . since in this work we only deal with second moments we further define @xmath70.\end{aligned}\ ] ] in lemma [ duality ] we will see that @xmath71 and hence we abbreviate @xmath72 . one says the system is intermittent ( or weakly intermittent as recently in @xcite ) if @xmath73 . intermittency for the parabolic anderson model ( @xmath41 ) is a well - studied property ( see @xcite and @xcite ) . the existing proofs heavily depend on the linear structure of the system since they employ explicit solutions given as feynman - kac type representations . such explicit solutions are not known to exist for the symbiotic branching model . hence , one might ask whether or not the results obtained for @xmath41 can be transferred to some larger regime of correlation values . indeed , this can be done . + let us first fix some notation . in the following @xmath74 denotes a continuous time markov process with transition rates @xmath4 and @xmath75 $ ] . due to the moment - duality for symbiotic branching processes ( see lemma [ lem : moment - duality ] ) the notation of symmetrization is needed . for two independent markov processes @xmath76 , @xmath77 with transition rates @xmath4 the symmetrization is defined as @xmath78 the transition rates of the symmetrization are given by @xmath79 its transition probabilities are denoted @xmath80 . note that in the symmetric case @xmath81 . the green function of @xmath74 is denoted @xmath82 and we abbreviate @xmath83 . further , we set @xmath84 and abbreviate @xmath85 . analogously , we use @xmath86 for the symmetrization @xmath87 . [ t3]let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions . then @xmath88 is intermittent if and only if @xmath89 in particular , there is no intermittency for non - positive @xmath18 . the previous theorem suggests @xmath64 deviding symbiotic branching into two regimes in which the @xmath90 regime behaves like the parabolic anderson model with respect to intermittency . + although after understanding the two - types particle moment - dual ( lemma [ mom ] ) one sees that the problem can be treated as for @xmath41 , we present a new proof . in @xcite results of @xcite for higher moments of the parabolic anderson model were generalized to more general symmetric transitions than the discrete laplacian . here , in particular , complete results for the asymptotic behaviour ( exponential and subexponential ) of second moments of the parabolic anderson model with arbitrary transitions @xmath4 are proven . + in the course of the proofs we obtain the expression @xmath91 for the second lyapunov exponent , where @xmath92 is the inverse of the laplace transform of the return probabilities ( see proposition [ prop1 ] ) . this expression , by tauberian theorems , gives us the explicit asymptotic behaviour for the lyapunov exponents as function of @xmath11 . in the following , @xmath93 denotes strong asymptotic equivalence , i.e. @xmath94 means @xmath95 . [ cor]let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions . then for @xmath96 , the map @xmath97 has the following properties : * @xmath72 is strictly convex , * @xmath98 for all @xmath11 , and @xmath99 for @xmath100 , * if @xmath101 as @xmath102 , @xmath103 , we have , as @xmath104 , @xmath105 * if @xmath101 , as @xmath102 , @xmath106 , we have , as @xmath107 , @xmath108 here , @xmath109 denotes the gamma function . our approach has the further advantage that the growth rates in the critical and subcritical regimes follow directly : [ pro ] let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions . if @xmath101 , as @xmath56 , then the following hold : * @xmath90 and @xmath106 * * if @xmath110 , then @xmath111\sim \frac{1}{\varrho(1-\kappa\varrho \bar g_{\infty})},\quad\text { as } t\to \infty . \end{aligned}\ ] ] * * if @xmath112 then , as @xmath56 , @xmath113 \sim \begin{cases } \frac{\bar g_{\infty}(\alpha-1)}{c \gamma(2-\alpha)\gamma(\alpha ) } \ , t^{\alpha-1 } & : 1<\alpha<2,\\ \frac{\bar g_{\infty}}{c}\ , \frac{t}{\log t } & : \alpha=2,\\ \frac{\bar g_{\infty}}{\bar h_{\infty}}\ , t & : \alpha>2 . \end{cases } \end{aligned}\ ] ] * @xmath17 @xmath113 \sim \begin{cases } \frac{\kappa c}{1-\alpha}t^{1-\alpha}&:\alpha<1,\\ \kappa c \log(t)&:\alpha=1,\\ 1+\kappa \bar g_{\infty}&:\alpha>1 , \end{cases}\quad\text { as } t\to \infty . \end{aligned}\ ] ] * @xmath59 @xmath114 \sim \begin{cases } 1-\frac{1}{\varrho } & : \alpha\leq 1,\\ 1-\frac{1}{\varrho}+\frac{1}{\varrho(1-\varrho\kappa \bar g_{\infty})}&:\alpha>1 , \end{cases}\quad\text { as } t\to \infty . \end{aligned}\ ] ] for @xmath41 ( parabolic anderson model ) , the subexponential growth was partially analysed for finite range transitions in @xcite ( see their page 15 ) . an example which was not included is for instance the riemann walk defined in example [ e1 ] . since in this case @xmath115 it serves as a convenient example for the above results which exhibits a precise recurrence / transience transition at @xmath116 . further , the simple random walk on @xmath5 is contained with @xmath117 combining the intermittency result with the extension of the results of @xcite given in @xcite , we support the unstable behaviour of symbiotic branching for @xmath90 . it is quite standard ( see for instance @xcite ) that spatial processes being intermittent have a very local property : for large times the mass of the process is concentrated on few sites ( `` islands '' ) . since for symbiotic branching , on each finite box solutions approach each constant configuration infinitely often , the islands do not stabilize . since the diffusion function has the form @xmath118 , we see that @xmath47 will not produce high peaks if @xmath49 is very small and vice versa . hence , we suspect that @xmath119 are concentrated on the same islands . understanding the pathwise behaviour better is an ambitious task for the future . recently in @xcite the concept of aging was discussed for certain classes of interacting diffusions . they say that aging ( for linear test - functions ) appears if the limit @xmath120\end{aligned}\ ] ] depends on the choices of @xmath121 and @xmath57 . aging does not appear if this is not the case . the main results of @xcite were formulated with more general test - functions , though , restricted to finite range transitions . differently , the present technique is restricted to linear test - functions but not to finite range transition . our results suggest that neither finite range nor the linearity of test - functions is crucial . symmetry of the transitions is assumed as in @xcite . + in @xcite it is shown that no aging appears in the parabolic anderson model ( in our model @xmath41 ) in any dimensions for the discrete laplacian . further , for the super random walk ( in our model related to @xmath17 ) it was shown that aging appears exactly in dimensions @xmath122 . this leads to the question if there are different phases for the symbiotic branching model . we show that the model exhibits three different regimes ; an anderson model like behaviour for @xmath90 , a super random walk like behaviour for @xmath17 , and a stepping stone model like behaviour for @xmath59 . the new case @xmath59 and corollary [ coro ] suggest that there are three regimes in which the most prominent examples fall . [ t5 ] let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions and @xmath123 . then , if @xmath124 , as @xmath56 , the following is true . * if @xmath90 , then no aging occurs for any @xmath125 . * if @xmath17 , then * * no aging occurs , for any @xmath106 , * * @xmath126=(1-a)_+$ ] , for @xmath127 , * * @xmath128=\frac{(1+\frac{a}{2})^{1-\alpha}-(\frac{a}{2})^{1-\alpha}}{(1+a)^{\frac{1-\alpha}{2}}}$ ] , for any @xmath129 . * if @xmath59 , then * * no aging occurs , for any @xmath106 , * * @xmath126=(1-a)_+$ ] , for @xmath127 , * * @xmath128= \frac{\int_0 ^ 1 ( 2 r+ a)^{-\alpha } ( 1-r)^{\alpha-1 } d r } { 2^{-\alpha } \gamma(\alpha)\gamma(1-\alpha)}$ ] , for any @xmath129 . we emphasise that our proof of theorem [ t5 ] can be applied to more general interacting diffusions . in particular , the examples from the introduction are included . for finite range transitions , i ) , ii ) , iii ) of the following proposition were proven in @xcite . [ coro]consider solutions of ( [ 111 ] ) with homogeneous initial conditions . then for * @xmath130 , aging appears as in theorem [ t5 ] i ) , * @xmath131 , aging appears as in theorem [ t5 ] ii ) , * @xmath132 , aging appears as in theorem [ t5 ] ii ) , * @xmath133 , aging appears as in theorem [ t5 ] iii ) . in the cases in which aging occurs , the upper and lower limits are bounded by the stated values up to constants depending on @xmath134 . we start with a discussion on how second moments of symbiotic branching processes can be reduced to exponential moments of local times . let us first recall the two - types particle moment - dual introduced in section 3.1 of @xcite . this will be used to calculate second moments explicitly . since the dual markov process is described formally in @xcite we only sketch the pathwise behaviour . to find a suitable description of the mixed moment @xmath135 $ ] , @xmath136 particles are located at position @xmath137 . each particle moves independently as a continuous time markov process on @xmath5 with transition rates given by @xmath4 . at time @xmath64 , @xmath33 particles have type @xmath26 , @xmath138 particles have type @xmath139 . one particle of each pair changes its type when the time the two particles have spent at same sites with same type exceeds an independent exponential time with parameter @xmath11 . let @xmath140 note that since there are only @xmath136 many particles the infinite product is actually a finite product and hence well - defined . [ mom ] let @xmath88 be a solution of the symbiotic branching model with initial conditions @xmath37 and @xmath16 $ ] . then , for any @xmath141 , @xmath2 , @xmath142&=&{\mathbb{e}}\big[(u_0,v_0)^{l_t}e^{\kappa(l_t^=+\varrho l_t^{\neq})}\big ] . \end{aligned}\ ] ] though the proof for the moment - duality was given in @xcite ( see the proof of their proposition 9 ) only for the discrete laplacian we skip a proof . for general transitions @xmath4 the proof follows along the same lines . + note that for homogeneous initial conditions @xmath143 the first factor in the expectation of the right - hand side equals @xmath26 . lemma [ mom ] in the special case @xmath41 , @xmath143 was already stated in @xcite , reproven in @xcite , and used to analyse the lyapunov exponents of the parabolic anderson model . + for @xmath144 , the difficulty of the dual process is based on the two stochastic effects : on the one hand , one has to deal with collision times of random walks which were analysed in @xcite . additionally , particles have types either @xmath26 or @xmath139 which change dynamically . + second moments are special since particles of different types do not change types anymore . hence , when starting with two particles of same type there is precisely one event of changing types . this is used to obtain the following representation of second moments . [ duality ] let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions . then , for any @xmath141 , @xmath2 , @xmath145&={\mathbb{e}}[e^{\kappa\varrho l_t}],\\ { \mathbb{e}}[u(t , k)^2]&={\mathbb{e}}[v(t , k)^2]=\begin{cases } 1+\kappa{\mathbb{e}}[l_t]&:\varrho=0,\\ 1-\frac{1}{\varrho}+\frac{1}{\varrho}{\mathbb{e}}[e^{\kappa\varrho l_t}]&:\varrho\neq 0 , \end{cases } \end{aligned}\ ] ] where @xmath62 denotes the local time in @xmath64 of the symmetrization @xmath87 defined in ( [ sy ] ) started in @xmath64 . the first expression for the mixed second moment follows directly from lemma [ mom ] : there are two particles which start with different types . since pairs of particles of different types are never forced to change their types , they stay of different type for all time . hence , @xmath146 for all @xmath2 and the assertion follows . + for the second expression note that there is only one possible change of types . starting with two particles of same types one of the types may change and the particles can not change their types again . using independence of the particles and the exponential time we can make this explicit . let @xmath147 be an exponential variable with parameter @xmath11 , denote by @xmath148 the law of the two independent markov processes , and @xmath62 their collision local time . integrating out the exponential variable leads to @xmath114 & = { \mathbb{e}}^{x\times y}[e^{\kappa(l_t^=+\varrho l_t^{\neq})}]\\ & = { \mathbb{e}}^{x\times y}[e^{\kappa(l_t^=+\varrho l_t^{\neq})}1_{y < l_t}]+{\mathbb{e}}^{x\times y}[e^{\kappa(l_t^=+\varrho l_t^{\neq})}1_{y\geq l_t}]\\ & = { \mathbb{e}}^{x}\big[\int_0^{l_t}\kappa e^{-\kappa x}e^{\kappa x+\kappa \varrho ( l_t - x)}\,dx\big]+{\mathbb{e}}^x\big[e^{\kappa l_t}{\mathbb{e}}^y[1_{y\geq l_t}]\big]\\ & = \begin{cases } \kappa{\mathbb{e}}[l_t]+{\mathbb{e}}[e^{\kappa l_t}e^{-\kappa l_t}]&:\varrho=0,\\ { \mathbb{e}}\big[e^{\kappa\varrho l_t}\int_0^{l_t}\kappa e^{-\kappa \varrho x}\,dx\big]+{\mathbb{e}}[e^{\kappa l_t}e^{-\kappa l_t}]&:\varrho\neq 0 . \end{cases } \end{aligned}\ ] ] this proves the assertion . now we prepare for the proof of the aging result . [ appli ] let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions and symmetric transitions @xmath4 . then , for any @xmath141 , @xmath2 , @xmath149=1+\kappa\int_0^tp_{2r+s}(k , k){\mathbb{e}}[e^{\kappa \varrho l_{t - r}}]\,dr \end{aligned}\ ] ] and similarly for @xmath49 . the proof is only given for @xmath47 since due to symmetry the same proof works for @xmath49 . we first employ the standard pointwise representation of solutions @xmath150 yielding @xmath151\\ & = 1+{\mathbb{e}}\big[\sum_{i\in{\mathbb{z}}^d}\int_0^tp_{t - r}(i , k)\sqrt{\kappa u(r , i)v(r , i)}\,dw^1_r(i ) \sum_{j\in{\mathbb{z}}^d}\int_0^{t+s}p_{t+s - l}(j , k)\sqrt{\kappa u(l , j)v(l , j)}\,dw^1_l(j)\big ] . \end{aligned}\ ] ] further , since martingale increments are orthogonal this equals @xmath152 . \end{aligned}\ ] ] now using independence of @xmath153 for @xmath154 and it s isometry we continue the chain of equalities as @xmath155\\ & = 1+\int_0^t\sum_{i\in{\mathbb{z}}^d } p_{t - r}(i , k)p_{t+s - r}(i , k)\kappa{\mathbb{e}}[u(r , i)v(r , i)]\,dr , \end{aligned}\ ] ] where we were allowed to change the order of integration since all terms are non - negative . using lemma [ duality ] , which in particular shows for homogeneous initial conditions that second moments do not depend on the spatial variable , symmetry of the transitions , and the chapman - kolmogorov equality , we finish with @xmath156\,dr\\ & = 1+\kappa\int_0^tp_{2t+s-2r}(k , k ) { \mathbb{e}}[e^{\kappa \varrho l_{r}}]\,dr = 1+\kappa\int_0^tp_{2r+s}(k , k ) { \mathbb{e}}[e^{\kappa \varrho l_{t - r}}]\,dr . \end{aligned}\ ] ] since we are going to examine the second lyapunov exponent @xmath72 of solutions we give a simple argument which ensures existence of the exponent . [ existencegamma ] let @xmath88 be a solution of the symbiotic branching model with homogeneous initial conditions . then the lyapunov exponent @xmath72 exists . note that to ensure existence of the limits @xmath157 \end{aligned}\ ] ] it suffices to show subadditivity of @xmath158 $ ] . using lemma [ duality ] this is reduced to showing subadditivity of @xmath159 $ ] , where @xmath62 is the local time in @xmath64 of @xmath87 started in @xmath64 . thus , by conditioning on @xmath160 , we get @xmath161 = \log { \mathbb{e}}^0[e^{\kappa\varrho l_{s}}{\mathbb{e}}^{\bar x_s}[e^{\kappa\varrho l_t } ] ] \leq\log { \mathbb{e}}^0[e^{\kappa\varrho l_{s}}{\mathbb{e}}^{0}[e^{\kappa \varrho l_t } ] ] = \log { \mathbb{e}}^0[e^{\kappa\varrho l_{s}}]+\log{\mathbb{e}}^{0}[e^{\kappa \varrho l_t } ] . \end{aligned}\ ] ] in lemma [ duality ] , we observed that in order to study second moments of symbiotic branching processes it suffices to study exponential moments of local times of the symmetrization @xmath87 . we now take up this issue and discuss exponential moments of @xmath62 in greater generality than needed for the symbiotic branching model . for the following let @xmath74 be a time - homogeneous markov process with countable state space @xmath162 and transition kernel @xmath163 . in particular , the transition rates are not assumed to be symmetric . + we start with a renewal - type equation for exponential moments of local times . [ l3]let @xmath62 be the local time of @xmath74 in @xmath164 for the process started in @xmath165 . then for @xmath61 the following equation holds : @xmath60=1+\kappa \int_0^tp_r(i , i){\mathbb{e}}[e^{\kappa l_{t - r}}]\,dr,\quad t\geq 0 \label{eqn : maineqn}. \end{aligned}\ ] ] we use the exponential series to get @xmath60 & = { \mathbb{e}}\big[e^{\kappa \int_0^t \delta_i(x_s)\,ds}\big ] = { \mathbb{e}}\left[\sum_{n=0}^{\infty}\frac{\kappa^n}{n!}\left(\int_0^t \delta_i(x_{s})\,ds\right)^n\right]\\ & = 1+{\mathbb{e}}\left[\sum_{n=1}^{\infty}\frac{\kappa^n}{n!}\int_0^t \cdots \int_0^t \delta_i(x_{s_1})\ldots \delta_i(x_{s_n})\,ds_n\ldots ds_1\right]\\ & = 1+{\mathbb{e}}\left[\sum_{n=1}^{\infty}\kappa^n\int_0^t \int_{s_1}^{t } \cdots \int_{s_{n-1}}^{t } \delta_i(x_{s_1})\ldots \delta_i(x_{s_n})\,ds_n\ldots d s_2 ds_1\right ] . \end{aligned}\ ] ] the last step is justified by the fact that the function that is integrated is symmetric in all arguments and , thus , it suffices to integrate over a simplex . we can exchange sum and expectation and obtain that the last expression equals @xmath166\,ds_n\ldots d s_2 ds_1 . \end{aligned}\ ] ] due to the markov property , the last expression equals @xmath167 \,ds_n\ldots d s_2 ds_1 \end{aligned}\ ] ] and can be rewritten as @xmath168 \,ds_n\ldots d s_2 \right ) ds_1 . \end{aligned}\ ] ] using the same line of arguments backwards for the term in parenthesis , the assertion follows . a similar renewal - type equation as ( [ eqn : maineqn ] ) can be shown with essentially the same proof for a discrete - time markov process . it reads @xmath169 = 1 + \kappa \sum_{n=0}^{m } p_n(i , i ) { \mathbb{e}}[e^{\kappa l_{m - n}}],\quad m\geq 1 , \end{aligned}\ ] ] where @xmath170 is the return probability after @xmath33 steps and @xmath171 is the number of visits after @xmath33 steps . + similar equations were obtained for symmetric markov chains on @xmath5 in @xcite using a completely different technique . note that neither symmetry nor any structure of the set @xmath162 is needed . the information on the geometry of @xmath162 is completely encoded in @xmath172 . for the rest of this section we fix the markov process @xmath74 , @xmath164 , and abbreviate @xmath173 . \end{aligned}\ ] ] the return probabilities @xmath172 are always assumed to be strongly asymptotically equivalent to @xmath174 , as @xmath175 , for @xmath125 , as for instance for simple random walks on @xmath5 and the riemann walk on @xmath176 . further , @xmath134 is monotone , decreasing , positive with @xmath177 , and @xmath178 is monotone , increasing , positive with @xmath179 . the laplace transform for a function @xmath180 on @xmath181 is denoted by @xmath182 and the convolution of two functions @xmath183 is denoted @xmath184 . in this notation equation [ eqn : maineqn ] reads @xmath185 taking the laplace transform of equation ( [ db ] ) leads to @xmath186 obviously , since @xmath134 is bounded by @xmath26 , @xmath187 is always finite for all @xmath188 . a priori this is not true for @xmath178 but if so , we obtain a useful representation from ( [ eqn : kappa0prime ] ) . [ le ] if @xmath189 , then @xmath190 in the following we proceed in two steps . first , we use ( [ db ] ) to understand in which cases @xmath191 grows exponentially in @xmath57 and discuss properties of the exponential growth rate . the following corresponence between exponential growth and finiteness of laplace transforms holds ( existence of the limit was proven in lemma [ existencegamma ] ) : @xmath192 this observation is particularly important for the second step in which we discuss the behaviour of @xmath191 as @xmath65 . in the cases in which @xmath191 grows subexponentially ( [ a ] ) implies that @xmath189 for all @xmath193 . hence , lemma [ le ] can be used for all @xmath193 . the strategy in this case is the following : by assumption , the asymptotic behaviour of @xmath194 as @xmath57 tends to infinity is known , namely @xmath174 . using tauberian theorems the asymptotic behaviour of @xmath187 as @xmath195 tends to zero can be deduced . by lemma [ le ] this determines the asymptotic behaviour of @xmath196 as @xmath195 tends to zero . using tauberian theorems in the opposite direction , the asymptotic behaviour of @xmath191 as @xmath57 tends to infinity is obtained . + to manage the transfer from the behaviour of @xmath134 to @xmath197 and back from @xmath198 to @xmath178 the following tauberian theorems are used . they are taken from @xcite ( see theorem 1.7.6 , theorem 1.7.1 , corollary 8.1.7 , and the considerations at the beginning of section 8.1 , [ lem : moment - duality ] let @xmath180 be a monotone function on @xmath199 with @xmath200 , then the following hold : 1 . if @xmath129 and @xmath201 , then @xmath202 if and only if @xmath203 as @xmath204 . 2 . if @xmath205 , then @xmath206 as @xmath204 . 3 . if @xmath106 and @xmath207 , then @xmath208 and @xmath209 as @xmath204 . the main point of the analysis is the following representation of the exponential growth rate which follows directly from lemma [ l3 ] . [ prop1 ] let @xmath3 , then @xmath210=\hat{f}^{-1}\big(\frac{1}{\kappa}\big).\ ] ] first , ( [ a ] ) implies that @xmath211 moreover , @xmath212 we are done if we can show @xmath213 first we show `` @xmath214 '' . due to lemma [ l3 ] we obtain @xmath215 which implies @xmath216 . since @xmath217 this shows that @xmath218 . + now we show `` @xmath219 '' . first , iterating ( [ db ] ) yields for fixed @xmath33 @xmath220 using @xmath221 and @xmath222\leq e^{\kappa t}$ ] yields @xmath223 as @xmath224 . hence , for fixed @xmath2 @xmath225 taking laplace transforms we note that @xmath196 is finite if and only if the laplace transform of the right - hand side is finite . however , using fubini s theorem ( note that only @xmath3 needs to be considered ) we obtain @xmath226 which is finite since we assumed @xmath227 . in particular , the previous result shows that understanding @xmath228 suffices to understand the exponential growth rates of @xmath229 $ ] . this is not difficult due to the following observation : @xmath197 is a strictly decreasing , convex function with @xmath230 . hence , @xmath228 is a strictly decreasing , convex function with @xmath231 and @xmath232 if and only if @xmath233 . this implies that @xmath234 precisely for @xmath235 . this and more properties of the exponential growth rate are collected in the following corollary . [ cor1 ] let @xmath3 and @xmath236 $ ] . then with @xmath237 the following hold : * @xmath238 and @xmath239 if and only if @xmath240 , * the function @xmath241 is strictly convex for @xmath240 , * @xmath242 for all @xmath11 , and @xmath243 , as @xmath100 , * if @xmath103 , then @xmath244 and , as @xmath104 , @xmath245 * if @xmath106 , then @xmath246 and , as @xmath247 , @xmath248 parts i ) and ii ) are proven as argued above the corollary . + since @xmath249 , the first part of iii ) follows from @xmath250 continuity of @xmath134 and @xmath177 imply that for @xmath251 there is @xmath252 such that @xmath253 for @xmath254 . hence , @xmath255 since @xmath256 for @xmath257 we obtain @xmath258 the second part of iii ) now follows since as well @xmath259 for all @xmath3 . + finally , for iv ) and v ) note that the asymptotics of @xmath197 for @xmath260 are known from lemma [ lem : moment - duality ] . this translates to @xmath261 and hence to @xmath262 . so far , we have understood the behaviour of @xmath229=g(t)$ ] as @xmath65 for @xmath263 . in this case @xmath191 grows exponentially and the behaviour of the exponential rates in @xmath11 could be analysed . we now come to the case @xmath264 . first , if @xmath265 , there is nothing to be done since the only appearing case is @xmath266 which yields @xmath267 for all @xmath2 . hence , we can stick to @xmath268 . [ prop10 ] let @xmath3 and @xmath269 . then , as @xmath102 , @xmath270 \sim \frac{1}{1-\kappa g_{\infty}}. \end{aligned}\ ] ] since @xmath268 we can apply part iii ) of lemma [ lem : moment - duality ] . hence , @xmath271 , as @xmath204 . as discussed above , since @xmath191 does not grow exponentially , @xmath272 for all @xmath193 , and we can use lemma [ le ] . this implies @xmath273 as @xmath204 . going backwards with lemma [ lem : moment - duality ] , part i ) , @xmath274 , the asymptotic of @xmath178 follows . [ prop11 ] let @xmath3 and @xmath275 . then , as @xmath65 , @xmath270 \sim \begin{cases } t^{\alpha-1}\frac{\alpha-1}{\kappa c \gamma(2-\alpha)\gamma(\alpha ) } & : 1<\alpha<2,\\ \frac{t } { \log t } \frac{1}{\kappa c}&:\alpha=2,\\ t\frac{1}{\kappa h_{\infty } } & : \alpha>2 . \end{cases } \end{aligned}\ ] ] since @xmath268 we can apply lemma [ lem : moment - duality ] , part iii ) . hence , @xmath276 , as @xmath204 . as discussed above , since @xmath191 does not grow exponentially , @xmath272 for all @xmath193 , and we can use lemma [ le ] . since @xmath277 , the denominator @xmath278 appearing in lemma [ le ] does not behave like a constant and we can not apply part i ) of lemma [ lem : moment - duality ] with @xmath274 . instead we use lemma [ lem : moment - duality ] , part iii ) , to obtain @xmath279 as @xmath204 . this , by lemma [ lem : moment - duality ] , part i ) , implies the assertion . we now investigate equation ( [ eqn : maineqn ] ) for @xmath66 . [ prop12 ] if @xmath66 , then , as @xmath102 , @xmath270 \sim \begin{cases } \frac{1}{t^{1-\alpha}}\frac{1}{-\kappa c \gamma(1-\alpha ) \gamma(\alpha ) } & : 0<\alpha<1,\\ \frac{1}{\log t}\frac{1}{-\kappa c } & : \alpha=1,\\ \frac{1}{-\kappa g_{\infty}+1}&:\alpha>1 . \end{cases } \end{aligned}\ ] ] first note that for @xmath66 , @xmath222<1 $ ] and hence for all @xmath193 , @xmath189 which validates the use of lemma [ le ] . this implies @xmath280 as @xmath204 . using lemma [ lem : moment - duality ] in both directions returns the assertion . these follow from lemma [ duality ] and corollary [ cor1 ] . this follows from lemma [ duality ] and propositions [ prop10 ] and [ prop11 ] . for this proof we always denote by @xmath93 the strong asymptotics at infinity and we abbreviate @xmath281 . lemma [ appli ] implies @xmath282=\frac{\int_0^t p_{2r+s } { \mathbb{e}}[e^{\varrho \kappa l_{t - r } } ] \ , d r}{\sqrt{\int_0^t p_{2r } { \mathbb{e}}[e^{\varrho \kappa l_{t - r } } ] \ , d r \ , \int_0^{t+s } p_{2r } { \mathbb{e}}[e^{\varrho \kappa l_{t+s - r } } ] \ , d r}}. \end{aligned}\ ] ] _ step 1 , @xmath17 : _ first , assume @xmath106 which implies @xmath283 . since @xmath284 we obtain , independently of the choice of @xmath57 and @xmath121 , @xmath282= \frac{\int_0^t p_{2r+s } \ , d r}{\sqrt{\int_0^t p_{2r } \ , d r \ , \int_0^{t+s } p_{2r}\ , d r } } \stackrel{s , t\to\infty}{\to } 0 . \end{aligned}\ ] ] here , we used @xmath285 if @xmath286 . we now come to the case @xmath127 , where we get @xmath287 therefore , we have @xmath282\sim\frac{\log\big(\frac{2 t + s + 1}{s+1}\big)}{\sqrt{\log ( 2 t + 1 ) \ , \log ( 2(t+s)+1)}}. \label{eqn : age1 } \end{aligned}\ ] ] for @xmath288 with @xmath289 this expression behaves asymptotically as @xmath290 on the other hand , for @xmath288 with @xmath291 the term in ( [ eqn : age1 ] ) behaves asymptotically as @xmath292 hence , for @xmath293 , we obtain @xmath294 \sim ( 1-a)_+.$ ] + now suppose @xmath129 . then @xmath295 therefore , we have @xmath282\sim\frac{(2 t+s+1)^{1-\alpha } -(s+1)^{1-\alpha}}{\sqrt{((2 t+1)^{1-\alpha } -1 ) \ , ( ( 2 ( t+s)+1)^{1-\alpha } -1)}}. \end{aligned}\ ] ] for @xmath296 this behaves asymptotically as @xmath297 _ step 2 , @xmath59 : _ let us first consider @xmath106 . since @xmath298 this case is exactly the same as @xmath17 , @xmath106 . now suppose @xmath127 . in this case we have by proposition [ prop12 ] @xmath299 we use the scaling @xmath288 with @xmath300 . let @xmath301 . the integral above can be split from @xmath302 to @xmath303 and @xmath303 to @xmath304 . we treat the first integral and show that its order is less than @xmath305 . first note that in the range of integration @xmath306 therefore , @xmath307 on the other hand , the second integral can be treated as follows . in its range of integration we have @xmath308 therefore , @xmath309 thus , @xmath310 \sim \frac{\frac{1}{\log ( t ) } \ , \frac{1}{2 } \log \left(\frac{2(1 - \theta ) t+s}{2+s}\right)}{\sqrt{\frac{1}{\log ( t ) } \ , \frac{1}{2 } \log ( ( 1 - \theta ) t)\ , \frac{1}{\log ( t ) } \ , \frac{1}{2 } \log ( ( 1 - \theta)(t+s ) ) } } \sim \frac { \log \left(\frac { t}{t^a}\right)}{\sqrt{\log ( t)\log ( t ) } } = 1-a.\end{aligned}\ ] ] analogously , for case @xmath291 . therefore , we get @xmath294\sim ( 1-a)_+,$ ] whenever @xmath311 . + for @xmath59 only @xmath129 is left : here , we have @xmath312 we set @xmath313 . the integral can be rewritten as @xmath314 the same way one can see that @xmath315 d r \sim \frac{c}{-\kappa \varrho c \gamma(\alpha)\gamma(1-\alpha ) } \int_0 ^ 1 ( 2 r)^{-\alpha } ( 1-r)^{\alpha-1 } d r.\end{aligned}\ ] ] thus , @xmath310 & \sim \frac{\int_0 ^ 1 ( 2 r+ a)^{-\alpha } ( 1-r)^{\alpha-1 } d r}{\int_0 ^ 1 ( 2 r)^{-\alpha } ( 1-r)^{\alpha-1 } d r}\\ & = \frac{\int_0 ^ 1 ( 2 r+ a)^{-\alpha } ( 1-r)^{\alpha-1 } d r } { 2^{-\alpha } b(\alpha,1-\alpha)}= \frac{\int_0 ^ 1 ( 2 r+ a)^{-\alpha } ( 1-r)^{\alpha-1 } d r } { 2^{-\alpha } \gamma(\alpha)\gamma(1-\alpha)},\end{aligned}\ ] ] when @xmath316 . here , @xmath317 denotes the beta function . + _ step 3 , @xmath90 : _ the transient case ( @xmath106 ) with @xmath110 has already appeared in the case @xmath17 for @xmath106 . @xmath318 $ ] is bounded due to proposition [ prop10 ] . hence , no aging occurs . + for @xmath96 , and for @xmath103 , we proved in corollary [ cor1 ] i ) that @xmath319 $ ] grows exponentially . this implies that there is a @xmath320 depending on the growth rate , i.e. on @xmath18 and @xmath11 , ( for example @xmath321 does the job ) such that for all @xmath2 and @xmath322 we have @xmath323 \leq e^{-\grw r } { \mathbb{e}}[e^ { \varrho \kappa l_t } ] . \end{aligned}\ ] ] therefore , @xmath324 \ , d r \leq p_{s } { \mathbb{e}}[e^ { \varrho \kappa l_{t } } ] \int_0^t e^{-\grw r } \ , d r \leq \frac{1}{\grw}\ , p_{s } { \mathbb{e}}[e^ { \varrho \kappa l_{t } } ] . \end{aligned}\ ] ] on the other hand , by the assumption @xmath325 , the renewal - type equation of lemma [ l3 ] , and since @xmath90 , @xmath326 \ , d r \geq c \int_0^t p_{r } { \mathbb{e}}[e^ { \varrho \kappa l_{t - r } } ] \ , d r = \frac{c}{\varrho \kappa } \ , \left ( { \mathbb{e}}[e^ { \varrho \kappa l_{t } } ] -1\right ) \geq c ' { \mathbb{e}}[e^ { \varrho \kappa l_{t } } ] . \end{aligned}\ ] ] putting these pieces together we obtain that @xmath327 \leq c''\ , \frac{p_{s } { \mathbb{e } } [ e^ { \varrho \kappa l_{t } } ] } { \sqrt{{\mathbb{e } } [ e^ { \varrho \kappa l_{t } } ] { \mathbb{e } } [ e^ { \varrho \kappa l_{t+s } } ] } } = c''p_{s}\ , \sqrt { \frac { { \mathbb{e } } [ e^ { \varrho \kappa l_{t } } ] } { { \mathbb{e } } [ e^ { \varrho \kappa l_{t+s } } ] } } . \end{aligned}\ ] ] note that , since @xmath90 , the term with the square root is bounded by @xmath26 . since clearly @xmath328 tends to zero , the whole expression must tend to zero independently of how @xmath329 . + the only case left is @xmath106 and @xmath112 . first we consider @xmath330 . @xmath331 \ , d r \approx \int_0^t ( r+1)^{-\alpha } ( t - r)^{\alpha-1 } \ , d r = \int_0 ^ 1 ( r+1/t)^{-\alpha } ( 1-r)^{\alpha-1 } \ , d r.\end{aligned}\ ] ] this expression tends to infinity for @xmath65 . the rate is @xmath332 on the other hand , @xmath333 \ , d r \approx \int_0^t ( r+s)^{-\alpha } ( t - r)^{\alpha-1 } \ , d r = \int_0 ^ 1 ( r+s / t)^{-\alpha } ( 1-r)^{\alpha-1 } \ , d r.\end{aligned}\ ] ] this expression is bounded or tends to infinity ( depening on how @xmath329 ) . it is bounded by @xmath334 putting these pieces together we obtain that @xmath335 \ , d r } { \sqrt{\int_0^t p_{2 r } { \mathbb{e } } [ e^ { \varrho \kappa l_{t - r } } ] \ , d r \int_0^{t+s } p_{2 r } { \mathbb{e } } [ e^ { \varrho \kappa l_{t+s - r } } ] \ , d r } } \leq c ' \,\frac { c_1 + c_2 ( s / t)^{1-\alpha}}{\sqrt{t^{\alpha-1 } ( t+s)^{\alpha-1 } } } \to 0.\end{aligned}\ ] ] the calculation is completely analogous in the cases @xmath336 . the pointwise representation of ( [ gr ] ) is not restricted to the symbiotic branching model but also holds for the interacting diffusions of this proposition . hence , the same derivation as for the symbiotic branching model yields @xmath337 = \frac{\int_0^tp_{2r+s}(k , k){\mathbb{e}}[f(u(t - r , k))]\,dr}{\sqrt{\int_0^tp_{2r}(k , k){\mathbb{e}}[f(u(t - r , k))]\,dr\int_0^{t+s}p_{2r}(k , k){\mathbb{e}}[f(u(t+s - r , k))]\,dr}}. \end{aligned}\ ] ] part i ) is contained in theorem [ t3 ] since the parabolic anderson model appears as special case @xmath41 . for part ii ) we can estimate the expectations from above and below to obtain the same result as in theorem [ t5 ] ii ) except constants . the same is true for part iii ) since the pointwise representation ( [ gr ] ) implies @xmath338=1 $ ] . finally , we could interprete part iv ) as a submodel of the symbiotic branching model . instead , we give a direct proof using use the coalescing particles dual of @xcite . the dual process consists of two independent particles started in @xmath339 , performing transitions @xmath4 in continuous time . after spending an exponential time @xmath147 with parameter @xmath11 , independent of the particles , at same sites , the particles coalesce . we denote @xmath148 the law of the particles and suppose @xmath340 . then @xmath114&={\mathbb{e}}^{y\times x}[w^{\text{number of non - coalesced particles}}]\\ & = { \mathbb{e}}^{y\times x}[w1_{y\leq l_t}]+{\mathbb{e}}^{y\times x}[w^21_{y > l_t } ] = w(1-{\mathbb{e}}^x[e^{-\kappa l_t}])+w^2{\mathbb{e}}^x[e^{-\kappa l_t } ] , \end{aligned}\ ] ] where @xmath62 denotes the collision time of the particles . using @xmath338=w$ ] this yields @xmath341={\mathbb{e}}[u(t , k)]-{\mathbb{e}}[u(t , k)^2]=(w - w^2){\mathbb{e}}[e^{-\kappa l_t}]$ ] and we can proceed as for the symbiotic branching model with @xmath59 . we are grateful to j. grtner from tu berlin for simplifying our original proof of lemma [ l3 ] . m. b. marcus and j. rosen . moment generating functions for local times of symmetric markov processes and random walks . in _ probability in banach spaces , 8 ( brunswick , me , 1991 ) _ , volume 30 of _ progr . _ , pages 364376 . birkhuser boston , boston , ma , 1992 . t. shiga . stepping stone models in population genetics and population dynamics . in _ stochastic processes in physics and engineering ( bielefeld , 1986 ) _ , volume 42 of _ math . _ , pages 345355 . reidel , dordrecht , 1988 .

for the symbiotic branching model introduced in @xcite , it is shown that aging and intermittency exhibit different behaviour for negative , zero , and positive correlations . our approach also provides an alternative , elementary proof and refinements of classical results concerning second moments of the parabolic anderson model with brownian potential . some refinements to more general ( also infinite range ) kernels of recent aging results of @xcite for interacting diffusions are given . ,

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Proceed to summarize the following text: the inner region of disk accretion onto neutron stars may be characterized by two unique radii : ( i ) the marginally stable orbit due to general gravity ( gr ) . for nonrotating neutron stars this is located at r_gr=6gmc^2=12.4m_1.4 km , where @xmath11 is the neutron star mass , and @xmath12 . for finite rotation rates , @xmath13 is somewhat smaller . the flow behavior near @xmath13 has been subjected to numerous studies , especially in the context of black hole accretion disks ( e.g. , muchotrzeb & paczyski 1982 ; matsumoto et al . 1984 ; abramowicz et al . 1988 ; narayan et al . 1997 ; chen et al . 1997 ) : close to @xmath13 the inward radial velocity of the accreting gas increases steeply with decreasing radius and becomes supersonic . the existence of such marginally stable orbit for neutron star is predicated on the fact that neutron star models constructed using different nuclear equations of state generally give a stellar radius less than @xmath13 ( arnett & bowers 1977 ; kluniak & wagoner 1985 ) . ( ii ) the magnetospheric radius , @xmath14 , below which magnetic stress dominates disk plasma stress . while the precise value of @xmath14 depends on the ( rather uncertain ) details of the magnetic field disk interactions , it is estimated to close to or slightly less than ( by a factor of a few ) the spherical alfven radius , i.e. , r_mr_ar(b_0 ^ 2r^3m)^2/7 = 18 r_10 ^ 12/7m_1.4 ^ -1/7b_8 ^ 4/7m_17 ^ -2/7 ( km ) , [ alfven]with @xmath15 ( e.g. , pringle & rees 1972 ; lamb , pethick & pines 1973 ; ghosh & lamb 1979 ; arons 1987 ) , where we have scaled various quantities to values appropriate for neutron stars in low - mass x - ray binaries ( lmxbs ) : @xmath16 km is the neutron star radius , @xmath17 g is the dipolar surface field strength , and @xmath18 is the mass accretion rate ( the eddington accretion rate is about @xmath19 g s@xmath20 ) . for highly magnetized neutron stars ( such as x - ray pulsars , typically having @xmath21 g ) , @xmath14 is much greater than @xmath13 and the stellar radius , the disk is therefore truncated near @xmath14 , within which the disk plasma becomes tied to the closed field lines and is funneled onto the magnetic poles of the star , although some plasma may continue to fall in the equatorial plane as a result of interchange instabilities ( spruit & taam 1990 ; see also arons & lea 1980 ) . for weakly magnetized neutron stars , such as those expected in lmxbs , @xmath14 and @xmath13 are comparable , and the plasma may not climb onto the field lines before reaching the stellar surface . a question therefore arises as to how the magnetic field affects the the dynamics of the inner disk and changes the sonic point . in this paper , we present an unified ( albeit phenomenological ) treatment of neutron star accretion disks under the combined influences of magnetic fields and strong gravity . our study is motivated by the recent observations using the rossi x - ray timing explorer ( rxte ) ( bradt , rothschild & swank 1993 ) which revealed kilo - hertz quasi - periodic oscillations ( qpos ) in the x - ray fluxes of at least thirteen lmxbs ( see van der klis 1997 for a review ) . these khz qpos are characterized by their high levels of coherence ( with @xmath22 up to @xmath23 ) , large rms amplitudes ( up to @xmath24 ) , and wide span of frequencies ( @xmath25 hz ) which , in most cases , are strongly correlated with the x - ray fluxes . in several sources , the x - ray power spectra show twin khz peaks moving up and down in frequency together , with the separation frequency roughly constant . moreover , in five atoll sources single qpos ( with a much higher level of coherence ) have been seen during one or more x - ray bursts , with frequencies equal to the frequency differences between the two peaks or twice that . this is a strong indication of beat phenomena ( strohmayer et al . while the origin of these qpos is uncertain , it is clear that the action must take place close to the neutron star , either in the accreting atmosphere ( klein et al . 1996 ) or in the inner disk ( strohmayer et al . 1996 ; miller , lamb and psaltis 1996 ) . a generic beat - frequency model assumes that the qpo with the higher frequency is associated with the kepler motion at some preferred orbital radius around the neutron star , while the lower - frequency qpo results from the beat between the kepler frequency and the neutron star spin frequency . it has been suggested that this preferred radius is the magnetosphere radius ( strohmayer et al . 1996 ) or the sonic radius of the disk accretion flow ( miller et al . 1996 ) . in this paper , we are not concerned with the actual mechanisms by which khz qpos in the x - ray fluxes of lmxbs may be produced ( see miller et al . 1996 and van der klis 1997 for extensive discussion on various possibilities ) . rather , our main purpose is to understand what physical effects determine the characteristics of the inner accretion disks in lmxbs . in the sonic - point model , miller et al . ( 1996 ) suggest that some accreting gas can penetrate inside the magnetosphere , whose boundary is located at a larger radius than the sonic radius . for unknown reasons , they assume that these gases are unaffected by the magnetic field once they are inside the magnetosphere and remains in a keplerian disk . they further suggest that the variation of qpo frequency results from the change in radiative forces on the accretion disk . we note , however , that the effect of radiative forces on the disk fluid ( rather than test particle orbiting the central star ) , is far from clear . calculating particle trajectories without solving for the global disk structure ( m. c. miller 1997 , private communication ) is inadequate for determining the magnitude of the radiative forces . while the radiative forces may be important for high - luminosity z - sources , their effects on the disk dynamics are expected to be be small for low - luminosity systems ( @xmath26 less than @xmath27 of @xmath28 ) . on the other hand , it is well known that millisecond pulsars have magnetic fields in the range of @xmath29 g , and one expect that neutron stars in lmxbs to have the similar range of field strengths . while the magnetic field may not be strong enough to induce a corotating magnetosphere outside the neutron star , it can nevertheless influence the dynamics of the inner disk flow by transporting away angular momentum from the disk . ideally , to properly assess the dynamical effect of magnetic fields on the accretion disks , one needs to solve for both the fluids and the fields self - consistently . this is a difficult task if not impossible . despite many decades of theoretical studies ( e.g. , pringle & rees 1972 ; lamb et al . 1973 ; ghosh & lamb 1979 ; aly 1985 , 1991 ; arons 1987 ; spruit & taam 1990 ; sturrock 1991 ; shu et al . 1994 ; lovelace et al . 1987 , 1995 ; stone & norman 1994 ; miller & stone 1997 ) , there remain considerable uncertainties on the nature of the stellar magnetic field disk interactions . particularly outstanding are the issues related to the efficiency of magnetic field dissipation in and outside the disk and whether the stellar field threads the disk in a closed configuration or it becomes open due to differential shearing between the star and the disk . it seems unlikely that some of these issues can be resolved on purely theoretical grounds . in this paper , we shall not attempt a self - consistent magnetohydrodynamics ( mhd ) calculations . rather , _ we shall adopt a phenomenological approach _ and consider rather general field configurations . we believe that such an approach is useful in bridging the gap between full mhd theories and observations . indeed , as we show in this paper , if the observed khz qpos are associated with the sonic point kepler frequency , then various systematics of khz qpos should provide useful constraints on the nature of magnetic field disk interactions as well as on the magnetic field structure in lmxbs . in 2 we introduce a model of magnetic slim accretion disk . numerical solutions are presented in 3 . because of various uncertainties in disk parameters , we shall focus on the simplest models , leaving more complete exploration to future studies . however , in 4 we introduce the notion of `` generalized marginally stable orbit '' including both the gr and magnetic effects . we derive an analytical expression for the sonic radius , which shows that the sonic point depends mainly on two parameters characterizing the disk magnetic field ( in addition to the neutron star mass ) . we show how different modes of magnetic field disk interactions can lead to different sonic - point orbital frequencies and their scalings with the field strength and mass accretion rate . section 5 concerns the equilibrium spin periods of neutron stars in our slim magnetic disk model . some applications to khz qpos in lmxbs are discussed in 6 , where we show that our phenomenological approach can be used to learn about the physics of magnetic field disk interactions . unless otherwise noted , we use geometrized units in which the speed of light and newton s gravitation constant are unity . we now consider geometrically thin axisymmetric accretion disk in steady state , taking into account of the transonic nature of the flow , and the deviation from keplerian motion in the inner region of the disk . our models generalize the usual `` slim disks '' around black holes ( e.g. , muchotrzeb & paczyski 1982 ; matsumoto et al . 1984 ; abramowicz et al . 1988 ; narayan et al . 1997 ; chen et al . 1997 ) by including the effect of magnetic fields . gr effects are included in our purely newtonian treatment by using the pseudo - newtonian potential introduced by paczyski & wiita ( 1980 ) = -mr-2 m . this potential correctly reproduces the marginally stable orbit ( where @xmath30 is the spin angular momentum ) , we have @xmath31 , and @xmath32 . for a spin frequency of @xmath33 hz ( strohmayer et al . 1996 ) , this amounts to a correction of @xmath34 to @xmath13 and @xmath27 to @xmath35 . these corrections are neglected in this paper . ] located at @xmath36 , and is adequate for this initial exploration , considering the much greater uncertainties in the magnetic field disk interactions . the self - gravity of the disk is neglected . we assume that the accreting material is confined to a thin disk , and we do not formally introduce a magnetosphere in our model . as discussed in 1 , when the field strength is sufficiently high ( as in x - ray pulsars , which typically have @xmath21 g ) , there is no question that a corotating magnetosphere exists outside the neutron star surface , located near @xmath37 ( see eq . [ [ alfven ] ] ; note that , theoretically , the magnetosphere radius is not known to within a factor of two , nor is it clear what the width of the transition zone is ) . in such a high - field regime , we shall find that the sonic point as obtained from our model is approximately equal to the usual alfven radius . although in our model the flow continues to be confined in the disk plane even inside the magnetosphere , in reality it may well behave differently ( e.g. , the plasma may `` jump '' onto the field lines and get funneled onto the magnetic poles ) . thus for high magnetic systems , the supersonic portion of our flow ( inside the sonic point ) may not be realistic . however , for low magnetic systems ( such as lmxbs ) , which is the main focus of this paper , there needs not be a genuine magnetosphere to truncate the disk flow , but the magnetic forces can still shift the sonic point to a radius larger than @xmath38 . in such low - field regimes , we expect our global flow solutions to have a wider validity . the mass continuity equation takes the form m=-2ru , [ mdot]where @xmath39 is the radial velocity of the flow ( @xmath40 for accretion ) , and @xmath41 is the surface density of the disk . the disk half - thickness since only height - integrated equations are used . ] is given by @xmath42 , where @xmath43 is the isothermal sound speed , and @xmath35 is the keplerian angular velocity ( for the pseudo - newtonian potential ) : _ k=(mr^3)^1/2rr-2 m . [ omegak]the radial momentum equation reads ududr=-1dpdr+(^2-_k^2)r + b_zb_r2|_z = h , [ uup ] where @xmath44 is the integrated disk pressure , @xmath45 is the angular velocity . the last term in eq . ( [ uup ] ) represents the dominant radial magnetic force , obtained by integrating over height the force per unit volume @xmath46 and dividing by @xmath47 . note that in eq . ( [ uup ] ) , @xmath48 is evaluated at the upper disk plane , and @xmath49 . in deriving the magnetic force , we have also neglected the @xmath50-component of the magnetic stress . the angular momentum equation reads u dldr = r , [ dl]where the second term on the right - hand - side is the magnetic torque per unit mass , obtained by integrating over height the torque per unit volume @xmath51 and dividing by @xmath47 . equation ( [ dl ] ) can be integrated in @xmath52 to give the conservation equation for angular momentum : m l_0=m l+2r^3 + m n_b(r ) , [ l0 ] where @xmath53 is the integration constant , and m n_b(r)=-_r^drr^2 b_zb_|_z = h . [ n_b]the three terms on the right - hand - side of eq . ( [ l0 ] ) correspond to advective angular momentum transport , viscous torque , and magnetic torque due the threading field lines from @xmath52 to @xmath54 , respectively . the constant @xmath53 is the eigenvalue of the problem ; it should be determined by requiring the flow to be regular at the sonic point . we shall adopt the standard @xmath55prescription for the disk kinematic viscosity , i.e. , @xmath56 ( shakura & sunyaev 1973 ) . the energy equation describing the thermal state of the flow can be written in the form : t u d sdr = e_visc+e_joule -2f_z . here @xmath57 is the specific entropy ( per unit mass ) , @xmath58 is the viscous heating rate per unit area . with the @xmath59-prescription for the disk viscosity , we have e_visc=2h1(rddr)^2 = ( rddr)^2 . the vertical ( optically thick ) radiative transport flux is f_z =- c3ddz(at^4 ) , where @xmath60 is the opacity and @xmath61 is the radiation energy density . the joule heating rate @xmath62 depends on the field dissipation in the disk , and its specific form depends on our ansatz for the magnetic field ( 2.2 ) . finally we need equations of state . for the inner disk region of interest in this paper , radiation pressure dominates over gas pressure . thus we have @xmath63 and @xmath64 . also , the opacity is dominated by thomson scattering , @xmath65 @xmath66 g . the equations above can be applied to general axisymmetric magnetic field disk configurations , as long as mass loss from possible disk wind is negligible , and the accreting material is confined to a thin disk plane . we now specify our ansatz for the magnetic fields . the vertical field component is assumed to take the form b_z = b_0(rr)^n . [ bz]we shall mostly focus on the @xmath67 case , corresponding to a central stellar dipole field threading the disk ( e.g. , ghosh & lamb 1979 ; knigl 1991 ; yi 1995 ; wang 1995 ) , although we will also consider more general values of @xmath68 , as in the cases when high - order multipoles are important ( arons 1993 ) or when field lines become open due differential shearing between the disk and the star ( aly 1985 , 1991 ; sturrock 1991 ; newman et al . 1992 ; lynden - bell & boily 1994 ; lovelace et al . in reality , the power - law relation in eq . ( [ bz ] ) is most likely to be valid only for a small range of @xmath52 , but as we shall see in 4 , the sonic point is mainly determined by the local behavior of the magnetic field . for the azimuthal component of the magnetic field , we consider two possibilities : \(i ) if the stellar magnetic field threads the accretion disk in a closed configurations ( e.g. , ghosh & lamb 1979 ) , then @xmath69 is governed by @xmath70 ( where @xmath71 is the field dissipation time ) . in steady - state , this gives @xmath72 , where @xmath73 is the rotation frequency of the star . we define a dimensionless parameter @xmath74 such that b_|_z = h=(_s-_k)b_z , [ bphi]where @xmath35 is the kepler frequency and @xmath75 is the disk orbital frequency . various ( uncertain ) estimates for the field dissipation timescale in the magnetically threaded disk configurations have been summarized in wang ( 1995 ) . \(ii ) if the magnetic field becomes open ( e.g. , lovelace et al . 1995 ) , we assume b_|_z = h =- b_z , where @xmath76 specifies the maximum twist angle of any field line connecting the star and the disk . clearly , eq . ( [ bphi ] ) encompasses the second possibility if we set @xmath77 . however , we note that the physical meaning of @xmath53 in these two cases are rather different : for closed field configurations ( i ) , @xmath78 measures the total torque on the star , while for the ( partially ) open field configurations ( ii ) , @xmath78 also include the angular momentum carried away from the disk by the magnetic fields of disk outflow . in both cases , @xmath78 is the total angular momentum transported away from the disk per unit time . similar to eq . ( [ bphi ] ) , our ansatz for the radial component @xmath79 of the disk magnetic field is b_r|_z = h=_r(-u_k r)b_z . we expect @xmath80 to be of the same order of magnitude as @xmath74 . with the particular ansatz for the magnetic fields given by eq . ( [ bphi ] ) , the joule heating rate @xmath62 can be calculated as e_joule = dz14b_zb_(r ) = 12r b_z^2(_s-)^2_k . the dissipation due to the current associated with @xmath79 is much smaller and has been neglected . it is useful to define dimensionless field strengths @xmath81 and @xmath82 via @xmath83 where @xmath84 . roughly speaking , @xmath6 is the ratio of the total magnetic torque and the characteristic accretion torque on the neutron star . comparing with eq . ( [ alfven ] ) we see that for dipole magnetic fields , @xmath85 . in all our calculations , we choose @xmath86 ; the flow structure and the sonic radius are rather insensitive to the value of @xmath82 . the radial force equation and the continuity equation can be cast in the form which reveals the existence of a sonic point : ( dudr)=a_s^2r+l^2r^3-_k^2 r + b_r^2u^2l_rr^3_k(rr)^2n-3 , [ up]where @xmath87 . the sonic point ( where @xmath88 ) is a critical point of the differential equation . the other equations can also be rewritten in the forms convenient for numerical integration . the angular momentum equation is l_0=l - r^2 c_s^2u_kddr + n_b , [ l0_2]with = -b^2l_rr(rr)^2n-3 -_s_k . [ dnb]the energy equation is = -c_s^2r 2u_k(ddr)^2 + 12b^2l_rr^2_k ( rr)^3(_s-)^2 -3c c_s_km . [ energy_2 ] the eigenvalue @xmath53 is adjusted so that the solution is regular at the sonic point . equations ( [ up])-([energy_2 ] ) are integrated inward from an outer radius ( far from the sonic point ) where the disk is approximately keplerian . note that in this outer keplerian region , eq . ( [ dnb ] ) can be integrated to give n_b(r)=b^2l_r2n-3(rr)^2n-3\ { 1-(4n-6)_s(4n-9)(m / r^3)^1/2 } , [ nbb]and the angular momentum equation yields u(r)=-3r c_s^22(l_k - l_0+n_b ) ( 1 - 2m/3r1 - 2m / r ) , where @xmath89 . the sound speed @xmath90 can be obtained from the energy equation ( [ energy_2 ] ) . since the radial velocity is small at large radius , the entropy advection term can be neglected . substituting @xmath91 from eq . ( [ l0_2 ] ) into eq . ( [ energy_2 ] ) , we find c_s(r)=m_k4c . [ cs ] equations ( [ up])-([energy_2 ] ) turn out to be a rather stiff set of equations . we have opted to adopt a further simplification by assuming the disk is isothermal . we estimate the range of @xmath90 using the thin disk expression ( [ cs ] ) evaluated near the sonic radius . the physical rationale behind this approximation is that the transition from keplerian disk to supersonic flow is very sharp , and we do not expect the sound speed to change significantly in this transition region ( near the sonic point ) . as we shall see in 4 , the sonic radius is insensitive to the thermal state of the disk when the sound speed is small . we have not studied the general dependence of the sonic radius on the thermal state of the disk . however , considering the very large uncertainties in the disk magnetic fields , the isothermal approximation should be adequate for use in the first step in our investigation . figure 1 depicts two examples of transonic accretion flows , with closed dipolar stellar fields threading the disks ( @xmath67 ) . we choose a standard set of parameters : @xmath92 , @xmath93 , @xmath94 ( a typical value for a @xmath95 , @xmath96 km neutron star ) , and @xmath97 ( corresponding to stellar spin frequency of @xmath98 hz ) . as expected , the magnetic fields slow down the tangential flow velocity , and , together with the strong relativistic gravity , make the radial velocity supersonic at small radii . for small @xmath6 , the deviation of @xmath45 from @xmath35 is small ; for larger @xmath6 , the sonic radius @xmath99 larger , and @xmath45 gradually approaches @xmath73 inside the sonic point . in these examples , the sonic points are located at @xmath100 ( for @xmath101 ) and @xmath102 ( for @xmath103 ) , the corresponding eigenvalues ( @xmath53 ) are @xmath104 and @xmath105 , respectively . in figure 2 we show the sonic radius @xmath99 and the corresponding specific angular momentum @xmath53 as a function of @xmath6 for several different values of @xmath59 and @xmath90 . we assume @xmath94 and @xmath106 for these models . the following trends have been found : for a given @xmath6 , a larger @xmath59 tends to make @xmath99 larger ( i.e. , viscosity tends to `` destablize '' the disk ) , while a larger @xmath90 tends to make @xmath99 smaller ( i.e. , pressure `` stablizes '' the disk ) . however , we emphasize that dependences of @xmath99 on these disk parameters ( @xmath107 ) are rather weak . moreover , as @xmath90 decreases , the sonic @xmath99 converges to a value independent of @xmath59 and @xmath90 the reason for this convergence will become clear in 4 . as our numerical results in 3 indicate , in the limit of small disk pressure and viscosity , the sonic radius approaches a value independent of the disk parameters ( @xmath108 and the equation of state ) . asymptotic sonic radius _ can be derived analytically using a simple mechanical model : consider the equation of motion of a test mass around a neutron star = l^2r^3-_k^2 r , l_0=l+n_b(r ) . [ testeq]imagine that the test mass is `` attached '' to a magnetic field line so that its orbital angular momentum @xmath109 is not conserved by itself . the conserved angular momentum @xmath53 includes the contributions from both the orbital motion and the angular momentum @xmath110 [ cf . ( [ n_b ] ) ] carried by the threading magnetic field . equation ( [ testeq ] ) can be obtained by setting the pressure and viscosity to zero in our general slim disk equations ( 2.1 ) . the radial component of the magnetic force has been neglected . an equilibrium orbit is determined by the condition l_0-n_b=_k^2 r = l_k(r ) . deviation @xmath111 from the equilibrium is governed by the perturbation equation of the form + _ eff^2r=0 , where the the effective epicyclic frequency @xmath112 is given by _ eff^2=2_krddr(l_k+n_b)= m(r-6m)r(r-2m)^3 - 2b^2l_r_kr^2 ( rr)^2n-3(1-_s_k ) , [ kappa2](recall that @xmath113 and @xmath114 is the neutron star radius ) . setting @xmath115 , we obtain a critical orbit , which we dub the _ `` generalized marginal stable orbit '' _ , located at @xmath116 . clearly , @xmath117 depends only on the gravitational potential and the local magnetic torque @xmath118 . the magnetic field enters only through the dimensionless ratio @xmath81 . the corresponding constant eigenvalue @xmath119 , however , depends on the global field structure . in figure 2 we plot @xmath117 and @xmath53 against @xmath6 for @xmath106 ( corresponding to spin frequency of @xmath98 hz ) and @xmath94 . we see that @xmath117 is the upper limit to the numerically determined @xmath99 , i.e. , _ @xmath117 is the asymptotic sonic radius as the disk viscosity and pressure diminish_. it is of interest to consider two limiting cases : ( i ) in the newtonian limit ( neglecting the gr effect ) , or equivalently when @xmath6 is large ( so that @xmath120 ) , we have = ^2/(4n-5 ) . [ largeb]for @xmath67 , this is the standard result for the inner radius of the keplerian disk , as determined by @xmath121 ( e.g. , arons 1993 ; wang 1995 ) ; ( ii ) in the limit of small @xmath6 ( so that @xmath117 is close to @xmath38 ) , we find = 6 + 16b^23(rm)^(4n-5)/2 . [ smallb]this gives the correction to the standard general - relativity - induced mso located at @xmath36 . the consideration of the limiting cases clearly indicates the sonic point ( or the generalized mso ) constructed in our slim disk model includes the essential physics embodied in the determination of the usual magnetosphere radius . as we argued at the beginning of 2 , for high magnetic systems , a genuine magnetosphere should certainly exist outside the neutron star . for low magnetic systems , however , the accretion flow may well be confined in the disk , and the distinction between the sonic point and the magnetosphere boundary may not exist . we emphasize that our analysis given here is phenomenological . it only takes account of the dynamics of disk under a fixed magnetic field configuration , while a full mhd treatment should include perturbations of both disk fluid and magnetic fields . the usefulness of our analytical result lies in the fact that @xmath117 provides a good approximation to the sonic radius ; and the sonic point is induced by both the gr effect and the magnetic effect . we note that in the presence of disk viscosity and magnetic fields , a fluid element continuously falls inward by the viscous stress and the magnetic stress , and therefore the concept of `` marginally stable orbit '' does not strictly apply . nevertheless , we use the term `` generalized mso '' to refer to the asymptotic sonic radius as determined by our analytical expressions . figure 3 depicts the orbital frequency , rather than the approximate pseudo - newtonian eq . ( [ omegak ] ) . ] at the generalized mso ( approximately the sonic radius ) as a function of the mass accretion rate @xmath3 for several different combinations of model parameters ( @xmath68 and @xmath73 ) . note that once we specify @xmath68 and @xmath73 in units of @xmath122 , the numerical value of @xmath123 depends on the other parameters only through the combination @xmath124 ( see eqs . [ [ kappa2]]-[[smallb ] ] ) . we have therefore used x = m_1.4 ^ 2n-2r_10 ^ -2nm_17b_7 ^ 2 = 0.07317b^2(m_1.4r_10)^(4n-5)/2 [ xlabel]as the @xmath125-variable in fig . 3 ( where @xmath126 is the surface field @xmath2 in units of @xmath127 g ) . for large @xmath128 ( or small @xmath6 ) , @xmath117 approaches @xmath36 and @xmath129 approaches @xmath130 khz . the scaling of @xmath131 with @xmath3 depends mainly on @xmath68 , the index which specifies the shape of the magnetic field lines ( see eq . [ [ bz ] ] ) . note that the sonic point converges to the mso only in the limit of small @xmath90 . thus in reality , the dependence of the sonic - point kepler frequency @xmath132 on @xmath3 may be different if @xmath90 is not small . for example , @xmath133 for radiation - dominated optically - thick disk , and therefore the dependence of @xmath132 on @xmath3 is slightly steeper than what is shown in fig . 3 . we shall discuss some of the applications of fig . 3 to qpos in lmxbs in 6 . as discussed before ( 2.2 ) , for closed field configurations , the quantity @xmath78 measures the total torque on the neutron star due to the accreting matter and the threading magnetic fields . it is of interest to consider how the equilibrium stellar rotation rate @xmath134 , at which @xmath135 , is determined in the slim disk model . we shall restrict to the dipole fields ( @xmath67 ) in this section . first consider the result when the gr effect is neglected . in this case the keplerian disk boundary @xmath37 is determined by the condition @xmath136 , or @xmath137 , where @xmath138 . we find = ^2/7 , [ ghosh](cf . [ [ largeb ] ] ) . the total torque on the star is given by n_tot = m l_0=m l_k(r_m)76 . [ torquen]equilibrium is obtained for @xmath139 . using eq . ( [ ghosh ] ) we then find @xmath140 and @xmath141 , which gives _ s , eq=3.44m_1.4 ^ 1/2r_10 ^ -3/2b^-6/7(khz ) = 1.467m_1.4 ^ 5/7r_10 ^ -18/7m_17 ^ 3/7 ( b_8 ^ 2)^-3/7(khz ) . [ omegeq]this is the standard result that the equilibrium spin frequency is equal to the keplerian frequency at the alfven radius . this result is plotted in fig . 4 . in the slim magnetic disk model , the total torque on the neutron star @xmath78 is obtained from the eigenvalue @xmath53 . in the asymptotic regime discussed in 4 , @xmath53 can be determined from the analytical expressions : @xmath99 is obtained from the condition @xmath142 in eq . ( [ kappa2 ] ) , and then @xmath143 with @xmath110 given by eq . ( [ nbb ] ) ( specialized to @xmath67 ) . it is straightforward to show that in the limit of @xmath144 , we recover the results given in eqs . ( [ ghosh]-[omegeq ] ) . the equilibrium @xmath134 is obtained by requiring @xmath135 . in fig . 4 we plot the equilibrium spin frequency @xmath145 as a function of @xmath6 . clearly , for large @xmath6 , our calculation agrees with the usual nonrelativistic result . relativistic corrections are significant only when @xmath6 is small , for which the sonic point lies close to the neutron star . we note that the discussion in this section is valid only if the magnetic fields are closed throughout the disk . only in these cases does @xmath53 measure the net torque on the neutron star . thus our results in this section are much more restrictive compared to the location of the sonic point ( which depends only on the local field structure ) discussed in 3 - 4 . we have presented in 2 - 4 an unified treatment of the inner region of accretion flow under the combined influences of general relativistic gravity and stellar magnetic fields in lmxbs . we have shown that even relatively weak magnetic fields ( @xmath0 g ) can slow down the orbital motion in the inner disk by taking away angular momentum from the disk , thereby changing the position of the sonic point significantly ( cf . figs . 2 - 3 ) . while the mechanisms responsible for the khz qpos in lmxbs are still uncertain , it is tempting to associate them with orbital motions at a certain preferred radius . if this is the case , then the keplerian frequency at inner radius of the disk , or more precisely the sonic radius is certainly a natural choice ( see paczyski 1987 for an earlier suggestion on the importance of the disk sonic point ) . one can envisage a number of different mechanisms that will lead to qpos at @xmath146 and the beat frequency with the stellar spin ( miller et al . 1996 ; van der klis 1997 ) . the following discussion is based on the hypothesis that the khz qpo frequency corresponds to @xmath131 or the sonic - point kepler frequency . we note that although our treatment of the magnetic field effects in 2 - 4 is rather general ( albeit based on a phenomenological prescription ) , other physical effects might be at work ( such as radiation forces for high - luminosity systems ; see 1 ) . as a result , some of our conclusions below should be considered tentative . \(a ) _ range of qpo frequencies and constraint on the magnetic field strength _ : as emphasized by van der klis ( 1997 ) , similar qpo frequencies ( @xmath25 hz ) have been observed in sources with widely different average luminosities @xmath147 ( from a few times @xmath148 to near @xmath28 , corresponding to @xmath149 between a few times @xmath150 g s@xmath20 to @xmath19 g s@xmath20 ) , while for an individual source @xmath151 often correlates strongly with the x - ray flux . this peculiar lack of correlation between the qpo frequency and @xmath147 can be explained in our model , as long as the star s magnetic field strength correlates with its @xmath149 in such a way as to leave @xmath152 in a certain range ( see fig . 3 ) . for example , if we use the model with @xmath67 ( dipole field ) and @xmath153 hz , then to produce @xmath154 in the range of @xmath25 hz would require @xmath155 ; on the other hand , if we use the model with @xmath156 and @xmath157 ( open field configuration ) , we would require @xmath158 ( we have adopted @xmath159 in these examples ) . indeed , such correlation between the magnetic field strength and the mean mass accretion rate among lmxbs has been suggested independently on the basis of z and atoll source phenomenology ( hasinger & van der klis 1989 ) , although the origin for this correlation is still unclear . we note , however , that the correlation needs not be very strong , considering the wide range of other controlling parameters such as @xmath68 ( the shape of the magnetic field ) and @xmath160 ( stellar rotation ) ( see fig . 3 ) . while there is a natural upper limit to @xmath131 ( corresponding to @xmath161 ; but see ( c ) below ) , the existence of a ( source - dependent ) lower limit to the observed qpo frequency needs an explanation . if we rely on inner disk accretion to explain these qpos ( e.g. , van der klis 1997 ) , then one possibility is that for large @xmath117 ( small @xmath131 ) , the accreting gas can be channeled out of the disk plane by the magnetic field this must happen for sufficiently small @xmath3 ( or sufficiently large @xmath162 , as in the case of accreting x - ray pulsars ) . the precise location where the plasma leaves the disk depends on the near - zone field structure , and is clearly source - dependent . the observed qpo frequencies may already be used to probe the magnetic fields in lmxbs and the nature of the magnetic field disk interactions . as an example , consider the atoll source 4u 0614 + 091 which has extremely small luminosity ( see ford et al . 1997 and references therein ) : to obtain @xmath163 hz requires @xmath164 ( this constraint depends somewhat on the field structure and @xmath160 ; see fig . 3 ) . at @xmath165 , this translates to @xmath166 a stronger magnetic field would push the sonic point ( or the generalized mso ) to a larger radius . we are left with two possibilities : ( i ) if @xmath167 , we would require @xmath168 , i.e. , the dissipation of toroidal fields near the disk must be very efficient ; ( ii ) if @xmath76 ( as expected for open field configurations ) , then we would require @xmath169 . indeed , if the khz qpo sources represent a fair sample of lmxbs , then we might conclude that the magnetic fields in lmxbs are systematically weaker than those in millisecond pulsars . this may indicate that the magnetic field of a neutron star is `` buried '' during the lmxb phase ( e.g. , romani 1990 ; urpin & geppert 1995 ; konar & bhattacharya 1997 ; brown & bildsten 1998 ) , and later regenerates or re - emerges as accretion stops . \(b ) _ scaling of @xmath151 with @xmath3 _ : for most sources , it was found that the khz qpo frequency strongly correlates with the xte count rate ( @xmath170 kev ) , with power - law index greater than unity . however , the scaling relation between the count rate and @xmath3 is not well established , and it has been suggested the flux of the black - body component is a better indicator of qpo frequency ( ford et al . as discussed in 3 , @xmath131 depends primarily on the magnetic field structure near the sonic point , particularly on the `` field shape '' index @xmath68 . if the scaling of @xmath151 with @xmath3 can be established observationally , it may be possible to distinguish a closed field configuration from an open one . for example , if we believe the scaling @xmath171 with @xmath172 , then we may conclude that @xmath173 ( see eq . [ [ largeb ] ] and the discussion following eq . [ [ xlabel ] ] ) , which indicates that magnetic fields in lmxbs do not have dipolar shape , but rather have complex topology ( see arons 1993 for discussion on related issues ) . \(c ) _ the maximum value of @xmath151 _ : it has been suggested ( zhang et al . 1997 ) based on the narrow range of the maximal qpo frequencies ( @xmath174 hz ) in at least six sources that these maximum frequencies correspond to the kepler frequency at @xmath36 , which then implies that the neutron star masses are near @xmath175 ( see also kaaret et al . while we agree that this conclusion seems most natural , we nevertheless add the following cautionary notes : ( i ) the inferred large stellar masses may be problematic : all neutron stars with well - determined masses ( including a few that certainly had accreted mass , although not necessarily in the same accretion mode as in lmxbs ) have masses consistent with being @xmath176 ( e.g. , van kerkwijk et al . in particular , the @xmath177 ms recycled pulsar b1855 + 09 , which is thought to have gone through a lmxb phase ( phinney & kulkarni 1994 ) , has a mass @xmath178 ( kaspi et al . . moreover , accretion of @xmath179 might have spun up the neutron stars to near break - up ( see cook et al . 1994 for calculations of spin - up tracks in the nonmagnetic case ) , in contrary to the observed spin rates ( @xmath180 hz ) . ( ii ) if the maximum @xmath151 is indeed @xmath181 , then the correlation between @xmath151 and @xmath3 should weaken as @xmath3 increases , and eventually @xmath151 should approaches a constant independent of @xmath3 ( see fig . 3 ) . this has not been observed . therefore in our opinion it is premature to identify the maximal @xmath151 with the kepler frequency at @xmath38 . an alternative is that as @xmath151 approaches @xmath174 hz , the rms qpo amplitude decreases as the observations have indicated , making it difficult to detect higher qpo frequencies . \(d ) _ horizontal - branch oscillations ( hbos ) in z - sources _ : in several z - sources ( e.g. , sco x-1 , gx 5 - 1 and gx 17 + 2 ) , hbos with frequencies @xmath182 hz have been detected _ simultaneously _ with the khz qpos ( see van der klis 1997 ) . one standard interpretation of hbos is that they are associated with the beat between the kepler frequency at the magnetosphere boundary and the neutron star spin ( alpar & shaham 1985 ) . since the spin frequencies @xmath160 of these sources ( as determined from the difference in the twin khz qpo frequencies ) lie around @xmath33 hz ( see white & zhang 1997 ) , the putative magnetosphere boundary must be located at a large radius where @xmath183 hz . as we have shown in this paper , such a strong magnetic field must necessarily push the ( generalized ) disk sonic point to a large radius where the kepler frequency drops below the kilo - hertz range . ( recall that for low field systems such as lmxbs , the distinction between the sonic point and the magnetosphere boundary probably does not exist , and the two separate radii are replaced by a single generalized sonic point [ see 3 - 4 ] . ) therefore if the khz qpos are associated with the sonic - point kepler frequency , then the magnetospheric beat frequency model for hbos can not work , and the origin of hbos must lie elsewhere . alternative models for hbos have been discussed by biehle & blandford ( 1993 ) and stella & vietri ( 1997 ) . the author thanks zhiyun li , rob nelson and brian vaughan for valuable discussions and lars bildsten for comment . he also thanks the referee for constructive comments which improved the presentation of the paper . this research is supported by a richard chace tolman fellowship at caltech , nasa grant nag 5 - 2756 , and nsf grant ast-9417371 .

the inner regions of accretion disks of weakly magnetized neutron stars are affected by general relativistic gravity and stellar magnetic fields . even for field strengths sufficiently small so that there is no well - defined magnetosphere surrounding the neutron star , there is still a region in the disk where magnetic field stress plays an important dynamical role . we construct magnetic slim disk models appropriate for neutron stars in low - mass x - ray binaries ( lmxbs ) which incorporate the effects of both magnetic fields and general relativity ( gr ) . the magnetic field disk interaction is treated in a phenomenological manner , allowing for both closed and open field configurations . we show that even for surface magnetic fields as weak as @xmath0 g , the sonic point of the accretion flow can be significantly modified from the pure gr value ( near @xmath1 for slowly - rotating neutron stars ) . we derive an analytical expression for the sonic radius in the limit of small disk viscosity and pressure . we show that the sonic radius mainly depends on the stellar surface field strength @xmath2 and mass accretion rate @xmath3 through the ratio @xmath4 , where @xmath5 measures the azimuthal pitch angle of the magnetic field threading the disk . the sonic radius thus obtained approaches the usual alfven radius for high @xmath6 ( for which a genuine magnetosphere is expected to form ) , and asymptotes to @xmath7 as @xmath8 . we therefore suggest that for neutron stars in lmxbs , the distinction between the disk sonic radius and the magnetosphere radius may not exist ; there is only one `` generalized '' sonic radius which is determined by both the gr effect and the magnetic effect . we apply our theoretical results to the khz quasi - periodic oscillations ( qpos ) observed in the x - ray fluxes of lmxbs . if these qpos are associated with the orbital frequency at the inner radius of the disk , then the qpo frequencies and their correlation with mass accretion rate can provide useful diagnostics on the ( highly uncertain ) nature of the magnetic field disk interactions . in particular , a tight upper limit to the surface magnetic field @xmath2 can be obtained , i.e. , @xmath9 g , where @xmath10 , in order to produce khz orbital frequency at the sonic radius . current observational data may suggest that the magnetic fields in lmxbs have complex topology .

You are an expert at summarizing long articles. Proceed to summarize the following text: following the proposal of paczyski ( 1986 ) , experiments to search for lensing - induced light variations of stars ( microlensing events ) located in the galactic bulge and the magellanic clouds have been or are being conducted by several groups ( macho : alcock et al . 1993 ; eros : aubourg et al . 1993 ; ogle : udalski et al . 1993 ; moa : bond et al . 2001 ; duo : alard & guibert 1997 ) . these experiments have successfully detected a large number of events ( @xmath12 ) , most of which are detected towards the galactic bulge . despite a large number of event detections , the nature of the lenses is still poorly known . this is because the einstein ring radius crossing time @xmath13 ( einstein timescale ) , which is the only observable providing information about the physical parameters of the lens ( lens parameters ) , results from a combination of the lens parameters , i.e.@xmath14^{1/2},\ ] ] where @xmath15 is the einstein ring radius , @xmath16 is the lens mass , @xmath17 is the lens - source transverse speed , and @xmath18 and @xmath19 are the distances to the lens and the source from the observer , respectively . under this circumstance , the only approach one could pursue would be identifying the major lens population by statistically determining the lens mass function based on the observed timescale distribution . however , this approach requires _ a prior _ knowledge about the geometrical distribution of the lens , the lens kinematics , and the functional form of the lens mass function , which are all poorly known . in addition , even if all lensing objects were of the same mass , they would give rise to a broad range of timescale . as a result , it is difficult to identify the major lens population from this approach ( mao & paczyski 1996 ; gould 2001 ) . recently , from the _ hubble space telescope _ ( hst ) images of one of the large magellanic cloud ( lmc ) events ( macho lmc-5 ) taken 6.3 years after the original lens measurement , alcock et al . ( 2001 ) were able to resolve the lens from the lensed source star . by directly imaging the lens , they could identify that the event was caused by a nearby low mass star located in the galactic disk . besides the identification of the lens as a normal star , direct lens imaging is of scientific importance due to following reasons . first , by directly and accurately measuring the lens proper motion with respect to the source , @xmath20 , one can better constrain the physical parameters of the individual lenses . the previous method to determine @xmath20 is based on the analysis of the lensing light curves of events affected by the finite source effect , such as source - transit single lens events and caustic - crossing binary lens events ( gould 1994 ; witt & mao 1994 ; nemiroff & wickramasinghe 1994 ) . by analyzing the part of the light curves near the source transit or the caustic crossing of these events , one can measure the source star angular radius normalized by the angular einstein ring radius @xmath21 , i.e. @xmath22 , where @xmath23 is the angular source star radius . then , the lens proper motion is determined by @xmath24 for the proper motion determination by using this method , however , one should know the source star angular radius , which can only be deduced from an uncertain color - surface brightness relation . as a result , the proper motions determined in this way suffer from large uncertainties . by contrast , if the lens is resolved , the proper motion can be directly and thus accurately measured from the observed image . measuring the proper motion is equivalent to measuring the angular einstein ring radius because @xmath25 , where the event timescale is determined from the light curve . while @xmath13 depends on three lens parameters of @xmath16 , @xmath18 , and @xmath17 , @xmath21 does not depend on @xmath17 , and thus the lens mass can be better constrained . second , if the lens is resolved for an event where the lens - source relative parallax , @xmath26 , was previously measured during the lensing magnification , one can completely break the lens parameter degeneracy and the lens mass is uniquely determined by @xmath27 where @xmath28 ( gould 2001 ) . third , if the source of an event was resolved via either a source transit or a caustic crossing and thus @xmath29 was precisely measured , one can determine the angular source star radius by reversing the process of the classical method of the proper motion determination , i.e. @xmath30 . by measuring @xmath23 , one can determine the effective temperature of the source star , which is important for the accurate construction of stellar atmosphere models ( e.g. , alonso et al . 2000 ) . although the first directly imaged lens was identified for an lmc event , much more numerous direct lens identifications are expected if high resolution followup observations are performed for events detected towards the bulge . there are several reasons for this expectation . first , compared to the total number of lmc events , which is @xmath31 , there are an overwhelmingly large number of bulge events . second , while the majority of lmc events are suspected to be caused by dark ( or very faint ) objects , most bulge events are supposed to be caused by normal stars , for which imaging is possible . third , an important fraction of lenses responsible for bulge events are believed to be located in the galactic disk with moderate distances , and thus more likely to be imaged due to their tendency of being bright and having large proper motions . the goal of this work is to estimate the fraction of galactic bulge events whose lenses can be directly imaged . for this estimation , we first compute the expected distribution of the lens - source proper motions of the currently detected galactic bulge events based on standard models of the geometrical and kinematical distributions of lenses and their mass function ( 2 ) . we then apply realistic detection criteria for lens resolution and the result is presented as a function of the time elapsed after the original lensing measurement , @xmath0 ( 3 ) . based on the result in 3 , we discuss some of the observational aspects of the future high resolution followup lensing observations aimed for direct lens imaging ( 4 ) . we summarize the result and conclude in 5 . the first requirement for direct lens imaging is that the lens should have a large relative proper motion with respect to the source so that it can be widely separated from the lensed source star within a reasonable amount of @xmath0 . in this section , we , therefore , estimate the expected distribution of proper motions of galactic bulge events . with the models of the lens mass function , @xmath32 , the matter density distributions of the lens and source stars along the line of sight towards the galactic bulge field , @xmath33 and @xmath34 , and the kinematical distribution of lens - source transverse velocities , @xmath35 , the distribution of lens proper motions of galactic bulge events is computed by @xmath36 where @xmath37 and @xmath38 are the components of the transverse velocity which are respectively parallel with and normal to the galactic disk plane , @xmath39 is the detection efficiency of events as a function of @xmath13 , and the notation @xmath40 represents the dirac delta function . we note that the factors @xmath17 and @xmath15 are included in equation ( 4 ) to weight the event rate by the transverse speed and the lensing cross section . for the galactic bulge and disk matter density distributions , we adopt the models of dwek et al . ( 1995 ) and bahcall ( 1986 ) , respectively . in the bulge model , the bulge has a triaxial shape and the matter density is represented by an analytic form of @xmath41 ^ 2+(z'/z_0)^4,\ ] ] where @xmath42 kpc , and the coordinates @xmath43 represent the axes of the triaxial bar from the longest to the shortest , and the longest axis is misaligned with the line of sight toward the galactic center by an angle @xmath44 . the bahcall disk model is expressed by a double exponential form of @xmath45,\ ] ] where the radial scale length and the vertical scale height are @xmath46 pc and @xmath47 kpc , respectively . for the transverse velocity distribution , we adopt the model of han & gould ( 1995 ) . in this model , the velocity distributions for both disk and bulge components have a gaussian form of @xmath48 , \qquad i \in y,\ z.\ ] ] the means and the standard deviations of the individual velocity components for events with disk lenses and bulge source stars ( disk - bulge events ) are @xmath49 where @xmath50 corresponds to the rotation speed of the galactic disk and @xmath51 are the adopted velocity dispersions of stars in the solar neighborhood . for events with bulge source stars lensed by another foreground bulge stars ( bulge self - lensing events ) , the means and standard deviations of the transverse velocity distributions are @xmath52 for the barred bulge , the velocity dispersions along the axes of the bar are deduced from the tensor virial theorem ( binney & tremaine 1987 ) , resulting in @xmath53 @xmath54 . due to the projection effect caused by the bar misalignment , the projected velocity dispersions are computed by @xmath55^{1/2 } , \sigma_{z'})= ( 82.5 , 66.3)\ { \rm km}\ { \rm s}^{-1}$ ] , which correspond to the standard deviations in ( 9 ) . we note that the adopted velocity model is a rough approximation in the sense that it does not include factors such as the systematic mean motion of bulge stars discussed by evans & belokurov ( 2002 ) , the figure rotation of the bulge discussed by blum ( 1995 ) , and the possibility of non - gaussian nature of disk star velocity distribution discussed by evans & collett ( 1993 ) , which may actually be important in the resulting distribution of proper motions . since the inclusion of the mean motion or the figure rotation of bulge stars result in high proper motions , we note that the fraction of resolvable lenses predicted by the adopted bulge velocity model is the lower limit . the einstein timescale distribution of events observed by the macho group is claimed to be consistent with the distribution from normal stars ( alcock et al . 2000b ) . we , therefore , model the mass function of lenses based on the present day main - sequence stars determined by kroupa , tout & gilmore ( 1993 ) . the adopted mass function has a three power - law functional form of @xmath56 for the detection efficiency , we adopt the latest determination of the macho experiment ( alcock et al . 2000b ) , whose data were analyzed by using the ` difference image analysis ' method . = 12.0 cm -0.5 cm -0.2 cm = 12.0 cm -0.2 cm in figure 1 , we present the obtained distributions of relative lens - source proper motions for disk - bulge ( solid curve ) and bulge self - lensing ( dashed curve ) events , respectively . due to the large uncertainties in the geometrical and kinematical distributions of lenses , the relative contribution of the disk and bulge lenses to the total event rate is very uncertain . we , therefore , leave the proper motion distributions of the disk - bulge and bulge self - lensing events separately , instead of estimating the distribution of total events by arbitrarily normalizing the ratio between the two populations of events . from the figure , one finds that , as expected , the average proper motion of events caused by disk lenses is larger than that of events caused by bulge lenses . the lens detectability is additionally restricted by the lens brightness . this is not only simply because the lens should be brighter than a detection limit , but also because the threshold lens - source separation for lens detection , @xmath57 , varies depending on the apparent lens / source flux ratio . therefore , we compute the angular threshold as a function of the magnitude difference between the lens and the source , @xmath58 . for the computation of @xmath59 , we first model the point - spread - function ( psf ) of a stellar image as a gaussian , where the standard deviation @xmath60 of the psf characterizes the resolving power of the instrument to be used for followup lensing observations . we then generate the combined image of the lens and the source by normalizing such that the volume under each psf is proportional to the flux of each star . once the combined image is constructed , we then judge whether the lens and the source can be resolved each other . for this judgment , we assume that the lens can be resolved if the sign of the derivative of the combined image s one - dimensional psf profile changes more than two times in the overlapping region between the centers of the lens and the source images ( see figure 2 for illustration ) . then , the angular threshold is defined as the lens - source separation at which the lens is just to be resolved from the source star . in figure 3 , we present the computed angular threshold as a function of @xmath61 . the two curves correspond to the distributions expected when the followup observations are carried out by using two different instruments , whose resolving powers are characterized by @xmath62 ( dashed curve ) and @xmath63 ( solid curve ) , respectively . we note that the advanced camera for surveys ( acs ) recently installed on hst can achieve a resolution of @xmath64 . we also note that the near infrared camera ( nircam ) of _ next generation space telescope _ ( ngst ) , which will have an aperture of 6 7 m , will be sensitive in the wavelength range of 0.6 to 5 microns , and thus can achieve @xmath65 . from the figure , one finds that as the lens becomes fainter , it becomes difficult to resolve the lens from the source star . = 12.0 cm -0.5 cm -0.2 cm with the computed distribution of @xmath59 , we recalculate the distribution of proper motions of events by using equation ( 4 ) , but in this time only for events with detectable lenses . to be detected , the lens should meet the condition of @xmath66 . in addition , we also restrict that detectable lenses should be brighter than a threshold magnitude of @xmath67 . for this computation , we assume that the source star has a fixed brightness of @xmath68 , which corresponds to that of a bulge clump giant , while the lens brightness is deduced from its mass by using the mass - luminosity relation provided by kroupa , tout & gilmore ( 1993 ) . once the absolute magnitudes of the source and the lens are set , their apparent magnitudes are computed considering their distances from the observer , i.e. @xmath19 for the source and @xmath18 for the lens . = 12.0 cm -0.5 cm -0.2 cm in figure 4 , we present the finally determined fractions of events with detectable lenses as a function of @xmath0 . in the figure , the thick and thin curves represent the expected fractions when followup observations are conducted by using instruments with @xmath7 and @xmath69 , respectively . if the instrument has a resolution of @xmath1 , we estimate that lenses can be resolved for @xmath2 and @xmath3 of disk - bulge events and for @xmath4 and @xmath5 of bulge self - lensing events after @xmath6 and 20 years , respectively . the fraction increases substantially with the increase of the resolving power . if followup observations are performed by using an instrument with @xmath7 , we estimate that lenses can be resolved for @xmath8 and @xmath9 of disk - bulge events and for @xmath10 and @xmath11 of bulge self - lensing events after @xmath6 and 20 years , respectively . in the previous section , we estimated the fraction of galactic bulge events for which one can directly image lenses from followup observations by using high resolution imaging instruments and presented the result as a function of a time after original lensing measurements . in this section , based on the result in the previous section we discuss some of the observational aspects of the future followup lensing observations aimed for direct lens imaging . first , from the distributions in fig . 4 , we find that the proper choice of the instrument for the future high resolution followup lensing observations will be ngst . bulge events has been reported since 1993 ( udalski et al . 1994 ; alcock et al . 1995a ) . under a rough assumption that disk - bulge and bulge self - lensing events equally contribute to the total galactic bulge event rate and considering the life expectancy of hst , the fraction of events with resolvable lenses from hst observations will be just @xmath70 even if the followup observations are performed at the end stage of hst for the first generation of lensing events . however , by using ngst , which is scheduled to be launched in 2009 , it will be possible to resolve lenses for a significant fraction of events . second , given that all lensing events will be unable to be followed up due to the limited observation time of ngst , priority of targets should be given to events from which one can obtain extra information about the lens or the source star other than the identification of the lens as a normal star and the measurement of the lens - source proper motion . as mentioned in 1 , one can measure the angular radius of the source star of an event , for which the source was previously resolved , and uniquely determine the lens mass of an event , for which the lens parallax was measured . therefore , these events should be at the top of the target list . in the published literature , we find 10 events , for which the source star was well resolved ( alcock et al . 1997 , 2000a ; albrow et al . 1999 , 2000 , 2001a , b ; afonso et al . 2000 ; an et al . 2002 ) , and 11 events , for which parallax effect was measured ( alcock et al . 1995b ; mao 1999 ; soszyski et al . 2001 ; smith , mao , & woniak 2002 ) . another possible high priority targets will be the longest events with @xmath71 days . these events can not be well explained by the standard models of the geometrical and kinematical distributions of lenses and their mass function ( han & gould 1996 ) . although these events comprise a small fraction ( @xmath72 ) of the total number of events , they are important because their contribution to the total optical depth is important . detection of lenses and the measured proper motions ( or non - detection of lenses ) will allow one to better constrain the nature of these mysterious events . we have estimated the fraction of galactic bulge microlensing events for which the lenses can be directly imaged from future high resolution followup observations by computing the distribution of proper motions of the currently detected bulge events and imposing realistic criteria for lens resolution . from this computation , we find that lens identification will be possible for a significant fraction of bulge events from followup observations using ngst under the assumption that most bulge events are caused by normal stars . besides identifying lenses as stars , direct lens imaging will allow one to accurately determine the lens proper motion , from which the physical parameters of the individual lenses can be better constrained . if lenses are imaged for events where lens parallaxes were measured , the lens parameter degeneracy can be completely broken and the lens mass can be uniquely determined . in addition , high resolution followup observations will provide a valuable chance to measure the angular radii of remote bulge source stars involved with events for which the source was previously resolved via either a source transit or a caustic crossing . afonso , c. et al . , 2000 , , 532 , 340 alard c. , guibert j. , 1997 , , 326 , 1 albrow m. d. et al . , 1999 , , 522 , 1011 albrow m. d. et al . , 2000 , , 534 , 894 albrow m. d. et al . , 2001a , , 549 , 759 albrow m. d. et al . , 2001b , , 550 , l173 alcock c. et al . , 1993 , nature , 365 , 621 alcock c. et al . , 1995a , , 445 , 133 alcock c. et al . , 1995b , , 454 , l125 alcock c. et al . , 1997 , , 491 , 436 alcock c. et al . , 2000a , , 541 , 270 alcock c. et al . , 2000b , , 541 , 734 alcock c. et al . , 2001 , nature , 414 , 617 alonso a. , salaris m. , arribas s. , martinez - roger c. , asensio ramos a. , 2000 , , 355 , 1060 an j. a. et al . , 2002 , , 572 , 521 aubourg e. et al . , 1993 , nature , 365 , 623 bahcall j. n. , 1986 , , 24 , 577 binney j. , tremaine s. , 1987 , galactic dynamics ( princeton : princeton univeristy press ) , 67 blum r. d. , 1995 , , 444 , l89 bond i. et al . , 2001 , , 327 , 868 dwek e. et al . , 1995 , , 445 , 716 evans n. w. , belokurov v. , 2002 , , 567 , l119 evans n. w. , collett j. l. , 1993 , , 264 , 353 gould a. , 1994 , , 421 , l71 gould a. , 2001 , , 113 , 903 han c. , gould a. , 1995 , , 447 , 53 han c. , gould a. , 1996 , , 467 , 540 kroupa p. , tout c. a. , gilmore g. , 1993 , , 262 , 545 mao s. , paczyski b. , 1996 , , 473 , 57 mao s. 1999 , , 350 , l19 nemiroff r. j. , wickramasinghe w. a. d. t. , 1994 , , 424 , l21 paczyski b. , 1986 , , 304 , 1 smith m. c. , mao s. , woniak p. , 2002 , , 332 , 962 soszyski i. et al . , 2001 , , 552 , 731 udalski a. , szymanski m. , kaluzny j. , kubiak m. , krzeminski w. , mateo m. , preston g. w. , paczynski , b. 1993 , acta astron . , 43 , 289 udalski a. , szymanski m. , kaluzny j. , kubiak m. , 1994 , acta astron . , 44 , 1 witt h. j. , mao s. , 1994 , , 430 , 505

recently , from the _ hubble space telescope _ ( hst ) images of one of the large magellanic cloud ( lmc ) events taken 6.3 years after the original lensing measurement , alcock et al . were able to directly image the lens . although the first resolved lens was identified for an lmc event , much more numerous lenses are expected to be resolved for galactic bulge events . in this paper , we estimate the fraction of galactic bulge events whose lenses can be directly imaged under the assumption that all bulge events are caused by normal stars . for this determination , we compute the distribution of lens proper motions of the currently detected galactic bulge events based on standard models of the geometrical and kinematical distributions of lenses and their mass function . we then apply realistic criteria for lens resolution , and the result is presented as a function of the time elapsed after an original lensing measurement , @xmath0 . if followup observations are performed by using an instrument with a resolving power of @xmath1 , which corresponds to that of hst equipped with the new advanced camera for surveys , we estimate that lenses can be resolved for @xmath2 and @xmath3 of disk - bulge events and for @xmath4 and @xmath5 of bulge self - lensing events after @xmath6 and 20 years , respectively . the fraction increases substantially with the increase of the resolving power . if the instrument has a resolution of @xmath7 , which can be achieved by the _ next generation space telescope _ , we estimate that lenses can be resolved for @xmath8 and @xmath9 of disk - bulge events and for @xmath10 and @xmath11 of bulge self - lensing events after @xmath6 and 20 years , respectively . epsf # 1 [ firstpage ] gravitational lensing stars : fundamental parameters

You are an expert at summarizing long articles. Proceed to summarize the following text: young stellar associations ( @xmath11 50 myr ) and open clusters have a crucial importance in advancing our understanding of star formation and the first stages of stellar evolution . since the galactic acceleration does not have a chance to affect the kinematical properties of these young stellar groups , the stellar content of an association is preserved . consequently one can obtain kinematical , dynamical and chemical properties of these young stellar groups by studying their secure members . today nearby associations within the solar neighbourhood have been very well identified . their stellar content of each association has been precisely determined up to a magnitude limit of @xmath1210.5 mag using astrometric data of the hipparcos satellite ( i.e. * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition the new reduced hipparcos catalogue @xcite gave an opportunity to investigate the astrometric data of the stellar content of a large number of open clusters , associations and moving groups more accurately . the first application of the new reduced hipparcos astrometric data is applied to stellar groups within the solar neighbourhood by @xcite . this and forthcoming studies of young stellar groups will improve our understanding of the history of star formation , the initial mass function , chemical and dynamical evolution of the milky way . recent statistical studies , such as @xcite , @xcite and @xcite show a high ratio of multiplicity in stellar formation regions ( sfrs ) and claim that it is not a coincidence , but a characteristic of star formation . the detailed study of multiple systems ( especially those with eclipsing components ) in sfrs will reveal the fundamental stellar parameters more directly and with higher precision compared to those obtained from single stars and thus impose more stringent tests on stellar evolution . critical tests of stellar evolution require masses and radii with a precision better than 3 per cent ( i.e. @xcite ) . however , methods developed for single stars are not capable of delivering masses with a precision better than 5 per cent and radii remain uncertain by a factor of 1.5 . consequently studying single stars does not enable us to obtain accurate dimensions and , therefore , to test the most recent evolutionary models . recent studies on @xmath13 mus by @xcite in lower centaurus - crux association , on v578 mon by @xcite in ngc 2244 , and on ab dor by @xcite in the ab dor association , demonstrate the precision with which age , chemical composition and kinematical properties can be determined by studying such high - mass systems . in the present study , we analyzed the high resolution spectra and _ bvh@xmath0 _ photometric data of i m mon , which is located in the region of ori ob1a association . i m mon is a bright ( v@xmath146.5 mag ) , early - type ( ( _ b - v_)=0.14 mag ) and short orbital period ( p@xmath141.2 days ) eclipsing binary system . its spectroscopic and photometric variations were discovered by @xcite and @xcite respectively . the eccentricity of the spectroscopic orbit obtained by @xcite was commented to be spurious by @xcite in their study on the determination of photometric elements of 14 detached systems . @xcite studied early photometric observations of i m mon , which were collected by @xcite in integral light and by @xcite in @xmath15 and @xmath16 filters . however , due to a large scatter in all photometric observations , which is attributed to the intrinsic variability of one of the components by @xcite , none of the authors was able to find a unique and precise solution for the system . a recent spectroscopic study of @xcite revealed the spectroscopic orbital elements of i m mon and showed that its orbit is circular . in order to reveal more precise absolute dimensions of i m mon and to test its membership to ori ob1a , we included it into our list of eclipsing binaries in the region of ob associations . using all literature based data the orbital period of i m mon is revised in 2.3 . high resolution spectral lines of i m mon are modelled and atmosphere parameters are derived in 3 . the close binary stellar parameters of the system are determined by the analysis of light and radial velocity ( rv ) curves in 4 . in 5 the absolute parameters of the components are derived together with the age and distance of the system . this information enabled us to establish the absolute dimensions of the close binary and properties of the ori ob1a association through the kinematical and dynamical properties of i m mon . finally we summarized our study and present our conclusions in 6 . we collected as many original individual measurements of i m mon as possible from the literature . in table [ listsources ] , all available photometric and spectroscopic observations of i m mon are listed . all the data given in table [ listsources ] are used for ephemeris determination for o c analysis , whereas only relatively more precise photometric data are used for light curve ( lc ) modelling . as a starting orbital period , we adopted @xmath17 , which is published by @xcite and later used by @xcite for their radial velocity analysis . [ cols="^,^,^,^,^,^",options="header " , ] + ( a ) @xcite , ( b ) @xcite , ( c)@xcite , ( d)@xcite , ( e)@xcite , ( f ) @xcite , ( g ) @xcite ob associations are young galactic clusters where star formation is ongoing or has just ended . the study of ob associations yields useful information about the characteristic of star formation such as formation history , binary population and initial mass function . however , this information can be obtained only if the properties of the ob association such as distance , age , metallicity and kinematics are very well established . observing single stars in an ob association does not provide information of sufficient precision unless many of them are observed , which requires a lot of observing time . in this case , eclipsing binaries which are the royal road to the stars can yield precise age , metallicity and kinematics of the medium in which they are embedded , and they do not require much observing time provided that stars with relatively short periods are selected . in this work , using sophisticated modelling tools , we studied the close binary system i m mon together with all its available photometric , spectroscopic , kinematical and dynamical data . the membership of i m mon to ori ob1a sub - group has been established securely by means of comparing the dynamical galactic orbits of 29 ori ob1a members with the galactic orbit of i m mon . the absolute dimensions we derived for i m mon in this study lead to a reliable distance determination . the location of both components of i m mon in the plane of @xmath18 - @xmath19 is fully compatible with the formerly derived ages for this association . the metallicity ( @xmath20=$]0.20(0.15 ) dex ) of i m mon obtained in this study has a large uncertainty which may explain the disagreement between the average metallicity ( @xmath20=-$]0.01(0.04 ) dex ) of the onc and i m mon . using the information derived in the present work , we conclude that ori ob1a is located at a distance of 353(59 ) pc , has an age of 11.5(1.5 ) myr and has a metallicity of 0.20(0.15 ) dex . in summary , i m mon is a secure member of the ori ob1a subgroup . * acknowledgements * + this study is fully supported by the scientific & technological research council of turkey ( tubitak ) with the project code 109t449 . zdenk mikulek is supported by the grants gaav iaa 301630901 and gar 205/08/0003 . we thank s. n. de villiers for valuable comments to our text and the anonymous referee who improved the manuscript by his / her very useful comments . andersen , j. 1991 , a&arv , 3 , 91 baki , v. , baki , h. , eker , z. , & demircan , o. 2007 , mnras , 382 , 609 baki , v. , baki , h. , bilir , s. , soydugan , f. , et al . 2010 , newa . , 15 , 1 bevington , p. r. , robinson , d. k. 2003 , in data reduction and error analysis for the physical sciences , third edition , mcgraw - 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all available photometric and spectroscopic observations were collected and used as the basis of a detailed analysis of the close binary i m mon . the orbital period of the binary was refined to 1.19024249(0.00000014 ) . the roche equipotentials , fractional luminosities ( in _ b , v _ and _ h@xmath0 _ - bands ) and fractional radii for the component stars in addition to mass ratio @xmath1 , inclination @xmath2 of the orbit and the effective temperature @xmath3 of the secondary cooler less massive component were obtained by the analysis of light curves . i m mon is classified to be a detached binary system in contrast to the contact configuration estimations in the literature . the absolute parameters of i m mon were derived by the simultaneous solutions of light and radial velocity curves as m@xmath4 m@xmath5 and 3.32(0.16 ) m@xmath5 , r@xmath6 r@xmath5 and 2.36(0.03 ) r@xmath5 , @xmath7 k and 14500(550 ) k implying spectral types of b4 and b6.5 zams stars for the primary and secondary components respectively . the modelling of the high resolution spectrum revealed the rotational velocities of the component stars as v@xmath8 kms@xmath9 and v@xmath10 kms@xmath9 . the photometric distance of 353(59 ) pc was found more precise and reliable than hipparcos distance of 341(85 ) pc . an evolutionary age of 11.5(1.5 ) myr was obtained for i m mon . kinematical and dynamical analysis support the membership of the young thin - disk population system i m mon to the ori ob1a association dynamically . finally , we derived the distance , age and metallicity information of ori ob1a sub - group using the information of i m mon parameters .

You are an expert at summarizing long articles. Proceed to summarize the following text: weak value amplification ( wva ) @xcite is a concept that has been used under a great variety of experimental conditions @xcite to reveal tiny changes of a variable of interest . in all those cases , a priori sensitivity limits were not due to the quantum nature of the light used ( _ photon statistics _ ) , but instead to the insufficient resolution of the detection system , what might be termed generally as _ technical noise_. wva was a feasible choice to go beyond this limitation . in spite of this extensive evidence , its interpretation has historically been a subject of confusion " @xcite . for instance , while some authors @xcite show that weak - value - amplification techniques ( which only use a small fraction of the photons ) compare favorably with standard techniques ( which use all of them ) " , others @xcite claim that wva does not offer any fundamental metrological advantage " , or that wva @xcite `` does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection '' . however , these conclusions are criticized by others based on the idea that `` the assumptions in their statistical analysis are irrelevant for realistic experimental situations '' @xcite . the problem might reside in here we make use of some simple , but fundamental , results from quantum estimation theory @xcite to show that there are two sides to consider when analyzing in which sense wva can be useful . on the one hand , the technique generally makes use of linear - optics unitary operations . therefore , it can not modify the statistics of photons involved . basic quantum estimation theory states that the post - selection of an appropriate output state , the basic element in wva , can not be better than the use of the input state @xcite . moreover , wva uses some selected , appropriate but partial , information about the quantum state that can not be better that considering the full state . indeed , due to the unitarian nature of the operations involved , it should be equally good any transformation of the input state than performing no transformation at all . in other words , when considering only the quantum nature of the light used , wva can not enhance the precision of measurements @xcite . on the other hand , a more general analysis that goes beyond only considering the quantum nature of the light , shows that wva can be useful when certain technical limitations are considered . in this sense , it might increase the ultimate resolution of the detection system by effectively lowering the value of the smallest quantity that can detected . in most scenarios , although not always @xcite , the signal detected is severely depleted , due to the quasi - orthogonality of the input and output states selected . however , in many applications , limitations are not related to the low intensity of the signal @xcite , but to the smallest change that the detector can measure irrespectively of the intensity level of the signal . a potential advantage of our approach is that we make use of the concept of trace distance , a clear and direct measure of the degree of distinguishability of two quantum states . indeed , the trace distance gives us the minimum probability of error of distinguishing two quantum states that can be achieved under the best detection system one can imagine @xcite . measuring tiny quantities is essentially equivalent to distinguishing between nearly parallel quantum states . therefore we offer a very basic and physical understanding of how wva works , based on the idea of how wva transforms very close quantum states , which can be useful to the general physics reader . here were we use an approach slightly different from what other analysis of wva do , where most of the times the tool used to estimate its usefulness is the fisher information . contrary to how we use the trace distance here , to set a sensitivity bound only considering how the quantum state changes for different values of the variable of interest , the fisher information requires to know the probability distribution of possible experimental outcomes for a given value of the variable of interest . therefore , it can look for sensitivity bounds for measurements by including _ technical characteristics _ of specific detection schemes @xcite . a brief comparison between both approaches will be done towards the end of this paper . one word of caution will be useful here . the concept of weak value amplification is presented for the most part in the framework of quantum mechanics theory , where it was born . it can be readily understood in terms of constructive and destructive interference between probability amplitudes @xcite . interference is a fundamental concept in any theory based on waves , such as classical electromagnetism . therefore , the concept of weak value amplification can also be described in many scenarios in terms of interference of classical waves @xcite . indeed , most of the experimental implementations of the concept , since its first demonstration in 1991 @xcite , belong to this type and can be understood without resorting to a quantum theory formalism . for the sake of example , we consider a specific weak amplification scheme @xcite , depicted in fig . 1 , which has been recently demonstrated experimentally @xcite . it aims at measuring very small temporal delays @xmath0 , or correspondingly tiny phase changes @xcite , with the help of optical pulses of much larger duration . we consider this specific case because it contains the main ingredients of a typical wva scheme , explained below , and it allows to derive analytical expressions of all quantities involved , which facilitates the analysis of main results . moreover , the scheme makes use of linear optics elements only and also works with large - bandwidth partially - coherent light @xcite . in general , a wva scheme requires three main ingredients : a ) the consideration of two subsystems ( here two degrees of freedom : the polarisation and the spectrum of an optical pulse ) that are weakly coupled ( here we make use of a polarisation - dependent temporal delay that is introduced with the help of a michelson interferometer ) ; b ) the _ pre - selection _ of the input state of both subsystems ; and c ) the _ post - selection _ of the state in one of the subsystems ( the state of polarisation ) and the measurement of the state of the remaining subsystem ( the spectrum of the pulse ) . with appropriate _ pre- _ and _ post - selection _ of the polarisation of the output light , tiny changes of the temporal delay @xmath0 can cause anomalously large changes of its spectrum , rendering in principle detectable very small temporal delays . ) splits the input into two orthogonal linear polarisations that propagate along different arms of the interferometer . an additional qwp is introduced in each arm to rotate the beam polarisation by @xmath1 to allow the recombination of both beams , delayed by a temporal delay @xmath0 , in a single beam by the same pbs . after pbs@xmath2 , the output polarisation state is selected with a liquid crystal variable retarder ( lcvr ) followed by a second polarising beam splitter ( pbs@xmath3 ) . the variable retarder is used to set the parameter @xmath4 experimentally . finally , the spectrum of each output beam is measured using an optical spectrum analyzer ( osa ) . ( @xmath5,@xmath6 ) and ( @xmath7,@xmath8 ) correspond to two sets of orthogonal polarisations . figure drawn by one of the authors ( luis - jose salazar serrano).,scaledwidth=70.0% ] let us be more specific about how all these ingredients are realized in the scheme depicted in fig . 1 . an input coherent laser beam ( @xmath9 photons ) shows circular polarisation , @xmath10 , and a gaussian shape with temporal width @xmath11 ( full - width - half maximum , @xmath12 ) . the normalized temporal and spectral shapes of the pulse read @xmath13 the input beam is divided into the two arms of a michelson interferometer with the help of a polarising beam splitter ( pbs@xmath2 ) . light beams with orthogonal polarisations traversing each arm of the interferometer are delayed @xmath14 and @xmath15 , respectively , which constitute the weak coupling between the two degrees of freedom . after recombination of the two orthogonal signals in the same pbs@xmath2 , the combination of a liquid - crystal variable retarder ( lcvr ) and a second polarising beam splitter ( pbs@xmath3 ) performs the post - selection of the polarisation of the output state , projecting the incoming signal into the polarisation states @xmath16 $ ] and @xmath17 $ ] . the amplitudes of the signals in the two output ports write ( not normalized ) @xmath18 \left\{1 + \exp \left [ i ( \omega_0+\omega ) \tau - i\gamma \right ] \right\ } \label{projections1 } \\ & & \phi_v(\tau)=\frac{\psi(\omega)}{2 } \exp \left [ i \left ( \omega_0 + \omega \right)\tau_0 \right ] \left\ { 1-\exp \left [ i ( \omega_0+\omega ) \tau -i\gamma \right ] \right\ } , \label{projections2}\end{aligned}\ ] ] where @xmath19 . ( solid blue line ) and @xmath20 as ( dashed green line ) . in ( a ) the post - selection angle @xmath4 is @xmath21 , so as to fulfil the condition @xmath22 . in ( b ) the angle @xmath4 is @xmath23 . ( c ) shift of the centroid of the spectrum of the output pulse after projection into the polarisation state @xmath7 in pbs@xmath3 , as a function of the post - selection angle @xmath4 . green solid line : @xmath24 as ; dotted red line : @xmath25 as , and dashed blue line : @xmath20 as . label * i * corresponds to @xmath26 [ mode for @xmath20 as shown in ( b ) ] . label * ii * corresponds to @xmath27 , where the condition @xmath22 is fulfiled [ mode for @xmath20 shown in ( a ) ] . it yields the minimum mode overlap between states with @xmath28 and @xmath29 . data : @xmath30 m and @xmath31 fs.,scaledwidth=90.0% ] after the signal projection performed after pbs@xmath3 , the wva scheme distinguishes different states , corresponding to different values of the temporal delay @xmath0 , by measuring the spectrum of the outgoing signal in the selected output port . the different spectra obtained for delays @xmath28 and @xmath20 as , for two different polarisation projections , are shown in figures 2 ( a ) and 2 ( b ) . to characterize different modes one can measure , for instance , the centroid of the spectrum . 2 ( c ) shows the centroid shift of the output signal for @xmath32 , which reads @xmath33 the differential power between both signals ( with @xmath28 and @xmath32 ) reads @xmath34\ ] ] when there is no polarisation - dependent time delay ( @xmath28 ) , the centroid of the spectrum of the output signal is the same than the centroid of the input laser beam , i.e. , there is no shift of the centroid ( @xmath35 ) . however , the presence of a small @xmath0 can produce a large and measurable shift of the centroid of the spectrum of the signal . detecting the presence ( @xmath29 ) or absence ( @xmath28 ) of a temporal delay between the two coherent orthogonally - polarised beams after recombination in pbs@xmath2 , but before traversing pbs@xmath3 , is equivalent to detecting which of the two quantum states , @xmath36 or @xmath37 is the output quantum state which describes the coherent pulse leaving pbs@xmath2 . @xmath38 designates the corresponding polarisations . the spectral shape ( mode function ) @xmath39 writes @xmath40 , \label{modes_input}\ ] ] where @xmath41 is the central frequency of the laser pulse , @xmath42 is the angular frequency deviation from the the center frequency and @xmath43 is the spectral shape of the input coherent laser signal . the minimum probability of error that can be made when distinguishing between two quantum states is related to the trace distance between the states @xcite . for two pure state , @xmath44 and @xmath45 , the ( minimum ) probability of error is @xcite @xmath46 for @xmath47 , @xmath48 . on the contrary , to be successful in distinguishing two quantum states with low probability of error ( @xmath49 ) requires @xmath50 , i.e. , the two states should be close to orthogonal . the coherent broadband states considered here can be generally described as single - mode quantum states where the mode is the corresponding spectral shape of the light pulse . let us consider two single - mode coherent beams @xmath51 where @xmath52 and @xmath53 are the two modes @xmath54 and @xmath55 and @xmath56 are the mean number of photons in modes @xmath52 and @xmath53 , respectively . the mode functions @xmath57 and @xmath58 are assumed to be normalized , i.e. , @xmath59 , reads @xmath60 where we introduce the mode overlap @xmath61 that reads @xmath62^{*}.\ ] ] in order to obtain eq . ( [ overlap1 ] ) we have made use of @xmath63^m |0 \rangle = n ! \rho^n \delta_{nm}$ ] . for @xmath64 ( coherent beams in the same mode but with possibly different mean photon numbers ) we recover the well - known formula for single - mode coherent beams @xcite : @xmath65 . making use of eqs . ( [ modes_input ] ) , ( [ overlap1 ] ) and ( [ overlap2 ] ) we obtain @xmath66 , \label{input_result}\end{aligned}\ ] ] where @xmath67 of the mode functions corresponding to the quantum states with @xmath28 and @xmath20 as , as a function of the post - selection angle @xmath4 ( solid blue line ) . the insertion loss , given by @xmath68 is indicated by the dotted green line . the minimum mode overlap , and maximum insertion loss , corresponds to the post - selection angle @xmath4 that fulfils the condition @xmath22 , which corresponds to @xmath27 . data : @xmath69 m , @xmath31 fs.,scaledwidth=50.0% ] in the wva scheme considered here , the signal after pbs@xmath3 is projected into the orthogonal polarisation states @xmath7 and @xmath8 , and as a result the signals in both output ports are given by eqs . ( [ projections1 ] ) and ( [ projections2 ] ) . making use of eqs . ( [ projections1 ] ) , ( [ projections2 ] ) and ( [ overlap2 ] ) one obtains that the mode overlap ( for @xmath70 ) reads @xmath71}{2\left[1+\cos \gamma \right]^{1/2 } \left[1+\gamma \cos ( \omega_0 \tau-\gamma ) \right]^{1/2}}.\ ] ] for @xmath28 , and therefore @xmath72 , we obtain @xmath64 . 3 shows the mode overlap of the signal in the corresponding output port for a delay of @xmath20 as . the mode overlap has a minimum for @xmath22 , where the two mode functions becomes easily distinguishable , as shown in fig . the effect of the polarisation projection , a key ingredient of the wva scheme , can be understood as a change of the mode overlap ( _ mode distinguishability _ ) between states with different delay @xmath0 . however , an enhanced mode distinguishability in this output port is accompanied by a corresponding increase of the insertion loss , as it can be seen in fig . the insertion loss , @xmath73 $ ] , is the largest when the modes are close to orthogonal ( @xmath74 ) . both effects indeed compensate , as it should be , since wva implements unitary transformations , and the trace distance between quantum states is preserved under unitary transformations . the quantum overlap between the states reads @xmath75,\end{aligned}\ ] ] so @xmath76 \label{output_result},\end{aligned}\ ] ] which is the same result [ see eq . ( [ input_result ] ) ] obtained for the signal after pbs@xmath2 , but before pbs@xmath3 . we can also see the previous results from a slightly different perspective making use of the cramr - rao inequality @xcite . the wva scheme considered throughout can be thought as a way of estimating the value of the single parameter @xmath0 with the help of a light pulse in a coherent state @xmath77 . since the quantum state is pure , the minimum variance that can show any unbiased estimation of the parameter @xmath0 , the cramr - rao inequality , reads @xmath78^{-1 } , \label{cramer1}\ ] ] making use of eq . ( [ state1 ] ) , one obtains that here the cramr - rao inequality reads @xcite @xmath79 where @xmath80 is the rms bandwidth in angular frequency of the pulse . in all cases of interest @xmath81 . the cramr - rao inequality is a fundamental limit that set a bound to the minimum variance that any measurement can achieve . it is unchanged by unitary transformations and only depends on the quantum state considered . inspection of eqs . ( [ input_result ] ) and ( [ output_result ] ) seems to indicate that a measurement after projection in any basis , the core element of the weak amplification scheme , provides no fundamental metrological advantage . notice that this result implies that the only relevant factor limiting the sensitivity of detection is the quantum nature of the light used ( a _ coherent state _ in our case ) . to obtain this result , we are implicitly assuming that a ) we have full access to all relevant characteristics of the output signals ; and b ) detectors are ideal , and can detect any change , as small as it might be , if enough signal power is used . if this is the case , weak value amplification provides no enhancement of the sensitivity . however , this can be far from truth in many realistic experimental situations . in the laboratory , the quantum nature of light is an important factor , but not the only one , limiting the capacity to measure tiny changes of variables of interest . on the one hand , most of the times we detect only certain characteristic of the output signals , probably the most relevant , but this is still partial information about the quantum state . on the other hand , detectors are not ideal and noteworthy limitations to its performance can appear . to name a few , they might no longer work properly above a certain photon number input , electronics and signal processing of data can limit the resolution beyond what is allowed by the specific quantum nature of light , conditions in the laboratory can change randomly effectively reducing the sensitivity achievable in the experiment . surely , all of these are _ technical _ rather than _ fundamental _ limitations , but in many situations the ultimate limit might be _ technical _ rather than _ fundamental_. in this scenario , we show below that weak value amplification can be a _ valuable _ and an _ easy _ option to overcome all of these technical limitations , as it has been demonstrated in numerous experiments . that leaves the interferometer . the two points highlighted corresponds to @xmath82 , which yields @xmath83 , and @xmath84 , which yields @xmath85 . ( b ) number of photons ( @xmath86 ) after projection in the polarisation state @xmath87 $ ] , as a function of the angle @xmath4 . the input number of photons is @xmath84 . the dot corresponds to the point @xmath88 and @xmath89 . pulse width : @xmath11=1 ps ; temporal delay : @xmath0= 1 as.,scaledwidth=90.0% ] let us suppose that we have at hand light detectors that can not be used with more than @xmath90 photons . any limitation on the detection time or the signal power would produce such limitation . the technical advantages of using wva in this scenario has been previously pointed out @xcite . here we make this apparent from a quantum estimation point of view , and quantify this advantage . 4(a ) shows the minimum probability of error as a function of the number of photons ( @xmath9 ) entering ( and leaving ) the interferometer . for @xmath91 , inspection of the figure shows that the probability of error is @xmath92 . this is the best we can do with this experimental scheme and these particular detectors without resorting to weak value amplification . however , if we project the output signal from the interferometer into a specific polarisation state , and increase the flux of photons , we can decrease the probability of error , without necessarily going to a regime of high depletion of the signal @xcite . for instance , with @xmath89 , and a flux of photons of @xmath84 , so that after projection @xmath88 photons reach the detector , the probability of error is decreased to @xmath85 , effectively enhancing the sensitivity of the experimental scheme ( see fig . the probability of error can be further decreased , also for other projections , at the expense of further increasing the input signal @xmath9 . in general , the minimum quantum overlap achievable between the states without any projection is @xmath93,\ ] ] while making use of projection in a weak value amplification scheme is @xmath94 . \label{enhancement}\ ] ] eq . ( [ enhancement ] ) shows that when the number of photons that the detection scheme can handle is limited ( @xmath90 ) , projection into a particular polarisation state , at the expense of increasing the signal level , is advantageous . from a quantum estimation point of view , wva increases the minimum probability of error reachable , since the projection makes possible to use the maximum number of photons available ( @xmath90 ) with a corresponding enhanced mode overlap . notice that the effect of using different polarisation projections can be beautifully understood as reshaping of the balance between signal level and mode overlap . as second example , let us consider that specific experimental conditions makes hard , even impossible , to detect very similar modes , i.e. , with mode overlap @xmath95 . we can represent this by assuming that there is an _ effective _ mode overlap ( @xmath96 ) which takes into account all relevant experimental limitations of a specific set - up , given by @xmath97.\ ] ] fig . 5 shows an example where we assume that detected signals corresponding to @xmath98 can not be safely distinguished due to technical restrictions of the detection system . for @xmath98 , @xmath99 , so the detection system can not distinguish the states of interest even by increasing the level of the signal . on the contrary , for smaller values of @xmath61 , accessible making use of a weak amplification scheme , this limitation does not exist since the detection system can resolve this modes when enough signal is present . the detection system can not distinguish the states of interest . data : @xmath100 and @xmath101.,scaledwidth=50.0% ] up to now , we have used the concept of trace distance to look for the minimum probability of error achievable in _ any _ measurement when using a given quantum state . in doing that , we only considered how the quantum state changes for different values of the variable to be measured , without any consideration of how this quantum state is going to be detected . if we would like to include in the analysis additional characteristics of the detection scheme , one can use the concept of fisher information , that requires to consider the probability distribution of possible experimental outcomes for a given value of the variable of interest . in this approach , one chooses different probability distributions to describe formally _ characteristics _ of specific detection scheme @xcite . let us assume that to estimate the value of the delay @xmath0 , we measure the shift of the centroid ( @xmath102 ) of the spectrum @xmath103 , given by eq . ( [ projections2 ] ) . a particular detection scheme will obtain a set of results @xmath104 , @xmath105 for a given delay @xmath0 . @xmath9 is the number of photons detected . the fisher information @xmath106 provides a bound of @xmath107 for any unbiased estimator when the probability distribution @xmath108 of obtaining the set @xmath109 , for a given @xmath0 , is known . if we assume that the probability distribution @xmath110 is gaussian , with mean value @xmath102 given by eq . ( [ centroid_shift ] ) and variance @xmath111 , determined by the errors inherent to the detection process , the fisher information reads @xcite @xmath112 ^ 2\ ] ] where @xmath113}{2\pi\ , \left ( 1+\gamma \cos \phi\right)^2}\ ] ] and @xmath114 . for @xmath115 , i.e. , the angle of post - selection is @xmath116 , the fisher information is @xmath117 notice that @xmath118 corresponds to considering equal input and output polarization state , i.e. , no weak value amplification scheme . for @xmath119 , where the angle of post - selection is @xmath120 , we have @xmath121 @xmath122 corresponds to considering an output polarisation state orthogonal to the input polarisation state i.e. , when the effect of weak value amplification is most dramatic , as it can be easily observed in fig . the fisher bound for @xmath123 is a factor @xmath124 larger than the bound for @xmath125 , so wva achieves enhancement of the fisher information . this fisher information enhancement effect , which does not happen always , it has been observed for certain wva schemes @xcite . there is no contradiction between the facts that the minimum probability of error , obtained by making use of the concept of trace distance , is not changed by wva , while at the same time there can be enhancement of the fisher information . by selecting a particular probability distribution to evaluate the fisher information , we include information about the detection scheme . in our case , we estimate the value of @xmath0 by measuring the @xmath0-dependent shift of the centroid of the spectrum of the signal in one output port after pbs@xmath3 , which is only part of all the information available , given by the full signal in eqs . ( [ projections1 ] ) and ( [ projections2 ] ) . we also assumed a gaussian probability distribution with a constant variance @xmath111 independent of @xmath0 . the cramr - rao bound we have derived here depends on the full information available ( the quantum state ) before any particular detection . an unitary transformation , as wva is , does not modify the bound . on the contrary , the fisher information , by using a particular probability distribution to describe the possible outcomes in an particular experiment , selects certain aspects of the quantum state to be measured ( _ partial information _ ) , and this bound can change in a wva scheme , although the bound should be always above the cramr - rao bound . in this restrictive scenario , the use of certain polarization projections can be preferable . the existence and nature of these different bounds might possibly explain certain confusion about the capabilities of wva , whether wva is considered to provide any metrological advantage or not . on the one hand , if we consider the trace distance , or the quantum cramr - rao inequality , without any consideration about how the quantum states are detected , post - selection inherent in wva does not lower the minimum probability of error achievable , so from this point of view wva offers no metrological advantage . on the other hand , in certain scenarios , the fisher information , when it takes into account _ information about the detection scheme _ , can be enhanced due to post - selection . in this sense , one can think of wva as an advantageous way to optimize a particular detection scheme . wva schemes makes use of linear optics unitary transformations . therefore , if the only limitations in a measurement are due to the quantum nature ( _ intrinsic statistics _ ) of the light , for instance , the presence of shot noise in the case of coherent beams , wva does not offer any advantage regarding any decrease of the minimum probability of error achievable . this is shown by making use of the trace distance between quantum states or the cramr - rao inequality , which set sensitivity bounds that are independent of any particular post - selection . however , notice that this implicitly assume that full information about the quantum states used can be made available , and detectors are ideal , so they can detect any change of the variable of interest , as small as it might be , provided there is enough signal power . nevertheless , these assumptions are in many situations of interest far from true . these limitations , sometimes refereed as _ technical noise _ , even though not fundamental ( one can always imagine using a better detector or a different detection scheme ) are nonetheless important , since they limit the accuracy of specific detection systems at hand . in these scenarios , the importance of weak value amplification is that by decreasing the mode overlap associated with the states to be measured and possibly increasing the intensity of the signal , the weak value amplification scheme allows , in principle , to distinguish them with lower probability of error . we have explored some of these scenarios from an quantum estimation theory point of view . for instance , we have seen that when the number of photons usable in the measurement is limited , the minimum probability of error achievable can be effectively decreased with weak value amplification . we have also analyzed how weak value amplification can differentiate between _ in practice_-indistinguishable states by decreasing the mode overlap between its corresponding mode functions . finally we have discussed how the confusion about the usefulness of weak value amplification can possibly derive from considering different bounds related to how much sensitivity can , in principle , be achieved when estimating a certain variable of interest . one might possibly say that the advantages of wva _ have nothing to do with fundamental limits and should not be viewed as addressing fundamental questions of quantum mechanics _ @xcite . however , _ from a practical rather than fundamental point of view _ , the use of wva can be advantageous in experiments where sensitivity is limited by experimental ( technical ) , rather than fundamental , uncertainties . in any case , if a certain measurement is _ optimum _ depends on its capability to effectively reach any bound that might exist . * references * 19 aharonov , y. , albert , d. z. & vaidman , l. how the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100 . _ * 60 * , 13511354 ( 1988 ) . hosten , o. & kwiat , p. observation of the spin hall effect of light via weak measurements , _ science _ * 319 * , 787790 ( 2008 ) . zhou , x. , zhou , ling , x. , luo h. , & wen , s. identifying graphene layers via spin hall effect of light . * 101 * , 251602 ( 2012 ) . ben dixon p. , starling , d. j. , jordan , a. n. , & howell , j. c. ultrasensitive beam deflection measurement via interferometric weak value amplification . * 102 * , 173601 ( 2009 ) . pfeifer , m. , & fischer , p. weak value amplified optical activity measurements . express _ * 19 * , 1650816517 ( 2011 ) . howell , j. c. , starling , d. j. , dixon , p. b. , vudyasetu , k. p. & jordan , a. n. precision frequency measurements with interferometric weak values . a _ * 82 * , 063822 ( 2010 ) . egan , p. & stone , j. a. weak - 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way information : an inequality , _ phys . lett . _ * 77 * , 21542157 ( 1996 ) . ou , z. y. complementarity and fundamental limit in precision phase measurement . _ phys . rev . _ * 77 * , 23522355 ( 1996 ) . glauber , r. j. coherent and incoherent states of the radiation field . _ * 131 * , 27662788 ( 1966 ) . let us define @xmath126 . for a coherent product state of the form @xmath127 , where the index @xmath128 refers to different frequency modes , one obtains that @xmath129 , where @xmath130 . if @xmath131 , where @xmath132 is the mean number of photons in frequency mode @xmath128 and only @xmath133 depends on the parameter @xmath0 as @xmath134 , one obtains that @xmath135 , where @xmath136 is the creation operator of the corresponding frequency mode . the unknown parameter of interest has value @xmath0 . after repeated measurements to estimate its value , we obtain a distribution of outcomes @xmath137 which can be characterized by a probability distribution @xmath138 that depends on the value of @xmath0 . the fisher information reads @xmath139 . the variance of any unbiased estimator that makes use of the ensemble @xmath137 is bounded from below by @xmath140 . when the fisher function can be written as @xmath141 $ ] , where @xmath142 is the variable that we measure , the fisher information can be written as @xmath143 . viza , g. i. et al . weak - values technique for velocity measurements . _ * 38 * , 2949 - 2952 ( 2013 ) . combes , j. & ferrie , c. & zhang , j. , and carlton m. caves , c. m. quantum limits on postselected , probabilistic quantum metrology , _ phys . a _ * 89 * , 052117 ( 2014 ) . weak value amplification scheme aimed at detecting extremely small temporal delays . the input pulse polarisation state is selected to be left - circular by using a polariser , a quarter - wave plate ( qwp ) and a half - wave plate ( hwp ) . a first polarising beam splitter ( pbs@xmath2 ) splits the input into two orthogonal linear polarisations that propagate along different arms of the interferometer . an additional qwp is introduced in each arm to rotate the beam polarisation by @xmath1 to allow the recombination of both beams , delayed by a temporal delay @xmath0 , in a single beam by the same pbs . after pbs@xmath2 , the output polarisation state is selected with a liquid crystal variable retarder ( lcvr ) followed by a second polarising beam splitter ( pbs@xmath3 ) . the variable retarder is used to set the parameter @xmath4 experimentally . finally , the spectrum of each output beam is measured using an optical spectrum analyzer ( osa ) . ( @xmath5,@xmath6 ) and ( @xmath7,@xmath8 ) correspond to two sets of orthogonal polarisations . spectrum measured at the output . ( a ) and ( b ) : spectral shape of the mode functions for @xmath28 ( solid blue line ) and @xmath20 as ( dashed green line ) . in ( a ) the post - selection angle @xmath4 is @xmath21 , so as to fulfil the condition @xmath22 . in ( b ) the angle @xmath4 is @xmath23 . ( c ) shift of the centroid of the spectrum of the output pulse after projection into the polarisation state @xmath7 in pbs@xmath3 , as a function of the post - selection angle @xmath4 . green solid line : @xmath24 as ; dotted red line : @xmath25 as , and dashed blue line : @xmath20 as . label * i * corresponds to @xmath26 [ mode for @xmath20 as shown in ( b ) ] . label * ii * corresponds to @xmath27 , where the condition @xmath22 is fulfiled [ mode for @xmath20 shown in ( a ) ] . it yields the minimum mode overlap between states with @xmath28 and @xmath29 . data : @xmath30 m and @xmath31 fs . mode overlap and insertion loss as a function of the post - selection angle . mode overlap @xmath61 of the mode functions corresponding to the quantum states with @xmath28 and @xmath20 as , as a function of the post - selection angle @xmath4 ( solid blue line ) . the insertion loss , given by @xmath68 is indicated by the dotted green line . the minimum mode overlap , and maximum insertion loss , corresponds to the post - selection angle @xmath4 that fulfils the condition @xmath22 , which corresponds to @xmath27 . data : @xmath69 m , @xmath31 fs . reduction of the probability of error using a weak value amplification scheme . ( a ) minimum probability of error as a function of the photon number @xmath9 that leaves the interferometer . the two points highlighted corresponds to @xmath82 , which yields @xmath83 , and @xmath84 , which yields @xmath85 . ( b ) number of photons ( @xmath86 ) after projection in the polarisation state @xmath87 $ ] , as a function of the angle @xmath4 . the input number of photons is @xmath84 . the dot corresponds to the point @xmath88 and @xmath89 . pulse width : @xmath11=1 ps ; temporal delay : @xmath0= 1 as .

weak value amplification ( wva ) is a concept that has been extensively used in a myriad of applications with the aim of rendering measurable tiny changes of a variable of interest . in spite of this , there is still an on - going debate about its _ true _ nature and whether is really needed for achieving high sensitivity . here we aim at solving the puzzle , using some basic concepts from quantum estimation theory , highlighting what the use of the wva concept can offer and what it can not . while wva can not be used to go beyond some fundamental sensitivity limits that arise from considering the full nature of the quantum states , wva can notwithstanding enhance the sensitivity of _ real _ detection schemes that are limited by many other things apart from the quantum nature of the states involved , i.e. _ technical noise_. importantly , it can do that in a straightforward and easily accessible manner .

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Proceed to summarize the following text: computer simulations have developed into a powerful tool to predict and optimize material properties . however , even given the ever increasing computational power , relevant time and length scales , in particular for biological and synthetic macromolecules @xcite , will remain prohibitively long for fully atomistic simulations , and multiscale methods are crucial to make progress . while structure - based coarse - graining @xcite has been quite successful @xcite , challenges remain like transferability ( _ e.g. _ , from bulk to surfaces ) , and in particular the correct treatment ( or _ a posteriori _ deduction ) of the materials _ dynamical _ properties through the coarse - grained dynamics . for the latter , markov state models ( msms ) @xcite have been used successfully to tackle the evolution of large proteins towards their native state , bridging the gap from molecular dynamics on nanoseconds to folding on milliseconds @xcite . the dynamics of msms is a discrete - time master equation with transition probabilities obey detailed balance on a network of long - lived ( metastable ) mesostates . here we describe a systematic approach to construct msms with dynamics that break detailed balance . we employ this method to study a model polymer in shear flow . the rheology of dilute flexible polymers has been studied extensively due to their fundamental and practical relevance @xcite . examples include biomolecules such as the von willebrand factor in blood plasma and dna in steady shear flow @xcite . the shear drag can overcome the entropic forces favoring coiled or globular configurations and stretch the polymers , which might be a continuous or even discontinuous transition @xcite . motion of dna tethered to a planar surface has been described as _ cyclic _ in experiments @xcite and computer simulations @xcite . driving a system away from thermal equilibrium implies a non - vanishing entropy production . in a steady state , this entropy is produced exclusively in _ cycles _ since there are no sources and sinks for the probability . this poses novel challenges for constructing coarse - grained models in non - equilibrium , which only very recently have been begun to be addressed @xcite . reducing the complexity by removing states implies a reduced entropy production , which severely influences dynamical properties and fluctuations @xcite . in ref . , we have introduced non - equilibrium markov state modeling ( ne - msm ) performing the coarse - graining in cycle space instead of collecting configurations in metastable basins as for conventional msm . the analog of these basins are now _ communities _ of cycles with similar properties . the coarse - graining procedure preserves the entropy production of these communities , which makes our approach consistent with stochastic thermodynamics @xcite . we study a single model polymer with @xmath0 beads in shear flow inspired by ref . . we employ brownian dynamics ( bd ) simulations with @xmath1 for the bead positions @xmath2 , where @xmath3 represents the shear flow . interactions with solvent particles are modeled by a random force with correlations @xmath4 , where upper indices label vector components . the potential energy @xmath5 is split into the non - bonded short - ranged lennard - jones pair potentials @xmath6 $ ] and bonds @xmath7 that connect the nearest neighboring beads . here , @xmath8 is the distance between the @xmath9-th and @xmath10-th bead , @xmath11 determines the strength of the non - bonded potential , and @xmath12 is the effective spring constant . all quantities have been non - dimensionalized by rescaling lengths with the bead diameter @xmath13 and timescales with the characteristic monomer diffusion time @xmath14 . numerical values for the strain rate @xmath15 thus correspond to the weissenberg number . the polymer is driven into a non - equilibrium steady state through simple shear flow . while some scaling relations depend on hydrodynamic interactions @xcite , the qualitative behavior of the cyclic motion does not and in the following we neglect hydrodynamic interactions . as flow profile we choose @xmath16 where @xmath15 is the strain rate and @xmath17 is the @xmath18 component of the center of mass of the polymer . we found that this shift of the flow stabilizes the globular and stretched configurations as it increases the effective barrier for folding / unfolding and thus leads to a better separation between globular and extended states . qualitatively , the same effect would be expected when including hydrodynamic interactions with the wall . our conclusions do not depend on this detail . the polymer is grafted onto a repulsive planar surface ( the @xmath19-@xmath20 plane with @xmath21 ) by fixing the position of the first bead to @xmath22 . although simplified , this model reproduces the cyclic dynamics found in experiments @xcite . a reasonable order parameter describing the folding and unfolding of the polymer is the relative end - to - end distance @xmath23 , where @xmath24 corresponds to a straight line of touching beads . we perform bd simulations for multiple values of @xmath25 , see fig . [ fig : polymer ] . we find different behaviors of @xmath26 that we categorize into three regimes . for @xmath27 the polymer remains collapsed , while for @xmath28 it is dominantly found in elongated conformations . for intermediate strain rates the polymer exhibits transitions between globular and elongated conformations , which was also found in similar simulations for free polymers @xcite and grafted polymers @xcite under shear . the exemplary time series for @xmath29 in the inset of fig . [ fig : polymer ] shows a clear separation of both states with random lifetimes and fast transitions . the average folding time @xmath30 is much larger than the intrinsic rouse time @xmath31 of the polymer . as a function of strain rate @xmath15 . for @xmath29 ( yellow / bright circle ) the inset shows an exemplary time series . also shown are two snapshots for the globule ( bottom ) and extended polymer ( top ) . ] our goal is to construct a dynamically consistent ne - msm with as few discrete states as possible . every discrete state @xmath32 is a set of @xmath33 particle positions , which we collect in the vector @xmath34 with @xmath35 entries . these states , or _ centroids _ , represent discrete volumes of configuration space , _ i.e. _ , many configurations with slightly different positions . following ref . , the outline of the algorithm is as follows : the first step is to generate many ( specifically @xmath36 ) centroids using the bd simulation data as input employing a spatial clustering algorithm ( for technical details , see ref . and appendix [ sec : spatial ] ) . from the counting matrix of transitions in the bd data we construct an approximate markov process and identify cycles . the crucial step is to group cycles into _ communities _ ( _ i.e. _ , clusters of cycles with similar properties defined through suitable order parameters ) and determine one cycle representative for each community . in the actual coarse - graining , only representatives and the states that they visit are retained . the final step is then to rescale the markov dynamics so that it preserves the entropy production of every community as well as the total entropy production . entropy production arises from non - vanishing probability currents @xmath37 with probability fluxes @xmath38 along transitions @xmath39 . in thermal equilibrium the condition of detailed balance holds , @xmath40 , which implies zero currents . the mean entropy production rate @xcite is @xmath41 with _ affinities _ @xmath42 . in a steady state , kirchhoff s law implies that probability flows in cycles . a cycle @xmath43 is defined as an ordered set of states , at the end of which the starting state is reached again and all other states are visited exactly once . cycles that differ only in the cyclic permutation of their states are considered identical . we extend the concept of affinities to cycles , yielding the cycle affinities @xmath44 , where the sum is over all edges along cycle @xmath45 . traversing a full cycle , the entropy produced equals the cycle affinity . graph theoretical results allow us to decompose the probability fluxes @xcite @xmath46 into cycle fluxes with non - negative weights @xmath47 . the summation is over all cycles @xmath45 that include the edge @xmath48 . the number of cycles quickly becomes very large . here we employ the algorithm introduced in ref . and summarized in appendix [ sec : decomp ] to efficiently determine a subset of cycles with non - vanishing weights @xmath49 . these cycles have non - negative cycle affinities ( and thus entropy production ) , which makes our approach conceptually different from the well - known schnakenberg theory @xcite . inserting eq . into eq . , we obtain @xmath50 , which clearly shows that all entropy production is encoded in cycles . at this point we have determined all entropy - producing cycles . to make progress , we now need to find similarities between cycles . one approach is to consider the connectivity between cycles @xcite . for the globule - stretch transition , we found this approach to be less suitable since it groups cycles that are located in different parts of configuration space . instead , here we describe an alternative , general method to reduce the complexity of the high - dimensional configuration space to a few variables based on the topology of cycles . as our main tool , we perform a principal component analysis ( pca ) @xcite , which is an orthogonal linear transformation returning the eigenvectors ( the principal components ) and eigenvalues of the covariance matrix of the centroid positions . the principal component @xmath51 corresponding to the largest eigenvalue coincides with the direction in configuration space exhibiting the largest variance ; in the present case it already accounts for @xmath52% of the observed variance . in fig . [ fig : cnf ] the normalized projections @xmath53 of centroids onto first and second principal component are plotted , where the normalization @xmath54 ensures that @xmath55 $ ] . centroids with small ( negative ) values for @xmath56 correspond to globular configurations ( they contribute little to the observed variance of positions ) , large ( positive ) values correspond to stretched configurations . the second component indicates the variance within these states . both globular and stretched configurations show larger fluctuations while the intermediate states with @xmath57 exhibit less fluctuations . hence , the pca reproduces the expected , typical picture of two basins with intermediate transition states . ) . scatter plot of the normalized projections of all centroids onto the two largest principal components . colors indicate the end - to - end distance @xmath58 . ] going back to cycles , we now define two variables : ( i ) the cycle centers @xmath59 with @xmath60 the number of centroids in cycle @xmath45 and ( ii ) the cycle diameters @xmath61 the set of points @xmath62 is plotted in fig . [ fig : cyc]a , where every point now represents a single cycle . these points are clearly not random . many cycles have a small diameter @xmath63 but different cycle centers . we identify these cycles as _ local _ because the centroids @xmath64 in these cycles have similar @xmath56 and thus belong to a compact region in configuration space . there is a second group of cycles with large diameter , which , consequently , we identify as _ global _ cycles . basically the same structure is recovered when plotting @xmath65 shown in fig . [ fig : cyc]b , indicating that local cycles have low affinities and global cycles have large affinities . ) . ( a ) scatter plot of cycle center vs. the cycle diameter , and ( b ) cycle affinity vs. cycle center . the cycle diameter and centers are computed using a pca ( see text for details ) . colors represent cycle communities while gray points indicate no community . the filled black circles indicate the cycle representatives . ( c , d ) the same plots as for ( a , b ) but for three cycle communities . ( e ) the fuzzy partition coefficients computed for multiple communities . the best result is obtained for 3 communities followed by a slightly lower value for 5 communities . ] we can turn these insights into a more quantitative statement by partitioning the cycle space into @xmath9 communities . to this end we employ an implementation of the fuzzy @xmath66-means algorithm ( an implementation is available in ref . ) , which assigns to each cycle a probability for belonging to a specific community . first , we normalize all three features ( cycle diameters , cycle centers , and affinities ) by their variance to make them comparable . these features are then used as an input for the fuzzy c - means clustering algorithm returning membership degrees @xmath67 which express the probability that observation @xmath32 belongs to community @xmath68 . to obtain an indicator of how good the clustering results are we compute the fuzzy partition coefficient ( fpc ) that is defined as the frobenius norm of the membership matrix @xmath69 here @xmath9 is the number of chosen communities and @xmath70 the number of observations ( cycles in our case ) . the closer the fpc gets to one the better the cycle space can be partitioned into the chosen number of clusters . the advantage of fuzzy partitioning is that some cycles might not belong to any cycle community while others match well in multiple ones . the minimal number of cycle communities to account for the collective folding and unfolding dynamics is @xmath71 ( see fig . [ fig : cyc]e ) , but to capture the full dynamics we found @xmath72 cycle communities to be more appropriate . we identify three local communities colored in blue , green , and red , as well as two global communities colored in magenta and cyan , see fig . [ fig : cyc]a , b . the blue community represents cycles that correspond to globular configurations while the green and red communities represent similar dynamics for intermediate and fully stretched configurations , respectively . the global cycle communities connect two ( cyan ) or all three ( magenta ) local communities . next , we replace each cycle community by one cycle that we will refer to as _ representative _ @xcite . we find appropriate representatives by mapping mean first - passage times between states that belong to different local cycle communities . after selecting representatives , we delete all states not belonging to any of the representatives and rescale the remaining transition rates with restrictions that : the total entropy production rate @xmath73 , all remaining cycle affinities @xmath74 and all remaining edge affinities @xmath75 are preserved . for details see ref . and appendix [ sec : repr ] . at this stage the coarse - grained msm still contains many states since a single cycle can traverse hundreds of states . the important point is , however , that the coarse - grained model lost much of its original complexity as it now contains only a few cycles . we can thus further reduce the number of states . to this end we identify two dominant motifs , which we refer to as bridge and triangle states . both motifs build on states that have exactly two neighbors . for bridge states the neighbors are not connected to each other @xcite . triangle states complete cycles that do not exhibit positive entropy productions and thus are not part of the decomposition eq . . we iteratively search and remove these states ( for details see appendix [ sec : bridge ] ) until no more are found , which completes our coarse - graining scheme . ( a ) transition network of the polymer dynamics with 15 centroids ( filled circles ) and 5 cycles . the colored states green , blue , and red correspond to the colors of the local cycle communities shown in fig . [ fig : cyc ] . states with a black border are structurally very similar and constitute the transition ensemble . the arrows point in the direction of probability currents ( net flux ) , while the arrow widths represent the magnitude of currents . on average , the polymer dynamics follows the direction of the arrows . arrow colors cyan and magenta correspond to the global cycles . ( b ) transition network in configuration space using the normalized projections onto the two largest principal components . symbols are the configurations while lines indicate the cycles with the arrows pointing in the direction of the net flux . ] the final ne - msm for strain rate @xmath29 is shown in fig . [ fig : tnetwork]a . after removing the bridge and triangle states the transition network contains 15 states ( centroids ) . the collective rates for folding and unfolding of this coarse - grained model agree with those obtained from the bd simulations by construction . moreover , the remaining five cycles now allow for detailed insights into the relevant pathways in non - equilibrium . the three local cycles are composed of three states , the minimal number for a non - trivial , entropy - producing cycle . the red and blue cycle connect stable globular and stretched configurations , respectively . the green cycle represents a metastable intermediate of half - stretched configurations that do not unfold correctly but quickly fold back to the intermediate . the global cyan cycle also contains half - stretched configurations ( structurally similar to the green cycle ) but here the unfolding reaches the final stretched states before returning to their half - stretched origin . finally , the magenta cycle represents the full transition from globule to stretched configurations and back . the five states @xmath76 describing intermediate , half - stretched configurations are very close in configuration space ( see fig . [ fig : tnetwork]b ) . they constitute the analog of the transition ensemble through which the folding / unfolding has to proceed . in non - equilibrium , however , the folding and unfolding processes follow different paths through this narrow region of configuration space . the globule to half - stretched transition proceeds along @xmath77 ( with state 4 belonging to the green cycle ) whereas the reverse half - stretched to globule transition proceeds along @xmath78 , with states 13 and 14 belonging to the cyan cycle . the cycle topology thus reveals the dynamical trapping of the polymer in an intermediate , which can not be captured by structural information alone . employing order parameters like distances in configuration space or lifetimes to identify mesostates will clearly miss this important feature of cyclic non - equilibrium dynamics . another question that we can address is dissipation , the role of which for bio - processes has been investigated recently , _ e.g. _ , for self - replication @xcite and in the activation of signaling proteins @xcite . the rate of dissipated heat @xmath79 created in each cycle community is proportional to their respective entropy production rate @xmath80 . our analysis reveals that both the blue and red cycles are equally responsible for about 30% of the total dissipated heat , while the green , cyan , and magenta communities produce 5% , 15% , and 50% , respectively . the latter is caused by the large conformational changes ( folding and unfolding process ) of the polymer . the blue and red cycles , on the other hand , do not exhibit large conformational changes , therefore , the conformational changes must be on very short timescales , which is confirmed by the large probability currents shown in fig . [ fig : tnetwork]a . we finally comment on using a pca , which is not necessarily the best method to reduce the dimensionality of configuration space , and more advanced methods such as the time - lagged independent correlation analysis exist @xcite . moreover , it is common practice to first apply a dimensionality reduction to the continuous molecular dynamics ( or bd ) data and subsequently discretize the reduced configuration space into finite sets . we did not follow this approach because ne - msms built from reduced configuration space exhibit a significantly lower entropy production rate than ne - msms built from full configuration space ( assuming the same number of centroids is used ) . recent results @xcite suggest that the fluctuations of general currents are bounded by the entropy production , which directly links dynamic with thermodynamic consistency . to conclude , we have presented a general method to systematically construct coarse - grained models composed of a few discrete states from molecular data of steadily driven systems . specifically , we have studied the globule - stretch transition of a simple model polymer , but the method can in principle be applied to more realistic and complex molecules ( such as f@xmath81-atpase @xcite ) , delivering minimal and thermodynamically consistent markov state models . while these models can be employed to bridge time scales , they also allow insights into the relevant transition pathways . here we have shown that different directions can pass through the same region of configuration space , which we believe might be a general property of transitions in driven systems . we thank f. schmid for helpful comments . we acknowledge financial support by the dfg through the collaborative research center trr 146 ( project a7 ) . the configuration space is discretized by employing the popular @xmath9-means clustering algorithm which divides the continuous configuration space into @xmath9 volumes and returns their centroids ( center of each volume ) . next the bd trajectories are projected onto the centroids , storing its dynamical information as simple sequences of centroid indices . we extract the dynamical information by counting the number of transitions @xmath82 for a given lag time @xmath83 ( time in between transitions ) . the count matrix @xmath84 is used to approximate the transition matrix @xmath85 by @xmath86 which is also the maximum probability estimator for the true transition operator @xcite . we additionally require @xmath87 to have two important properties : first , @xmath87 must be ergodic , i.e. each state can be reached from every other state in finite time or the transition network spanned by @xmath87 is connected . second , all occurring transitions are reversible , i.e. if @xmath88 then @xmath89 . . ] if the process described by @xmath87 is time - homogeneous its entries @xmath90 are ought to be time - independent . to test this we perform a lag time analysis @xcite shown in fig . [ lagtime_analysis ] . for equilibrium systems the eigenvalues @xmath91 of the transition matrix @xmath87 can be converted into relaxation timescales of the system @xmath92 with slowest modes corresponding to the largest @xmath91 . if the largest eigenvalues are time - independent , also the slowest modes are time - independent yielding a good estimation of the minimal lag time that can be used . in ne - msm the eigenvalues can become complex , which leaves their physical interpretation still open for discussion . however , for the polymer dynamics the largest eigenvalues sorted by either their real part or absolute value exhibit no imaginary part . for a continuous - time markov process with rate matrix @xmath93 , the transition probability can be expressed as @xmath94 . while conventional markov state models have discrete - time dynamics , for our purposes we require a rate matrix , which we approximate from the computed transition matrix through @xmath95 all off - diagonal elements of @xmath93 are non - negative @xmath96 while the diagonal elements are @xmath97 . following the perron - frobenius theorem , @xmath87 and thus @xmath93 have an unique largest real eigenvalue @xmath98 with a corresponding eigenvector that has strictly positive entries . this eigenvector represents the steady - state probability distribution with elements @xmath99 . finally , the probability fluxes are obtained by @xmath100 . here we describe an algorithm that decomposes a given probability flux matrix @xmath101 into cycle fluxes . each element of @xmath101 transforms through @xmath102 with @xmath103 . the sum adds the cycle weights of all cycles containing the edge @xmath39 . for the cycle decomposition we start by dividing @xmath101 into a symmetric detailed - balance part @xmath104 and a non - negative current part @xmath105 so that @xmath106 here @xmath105 is obtained by @xmath107 with all negative elements set to zero . the symmetric part follows as @xmath108 . to make progress , we identify all non - zero elements of @xmath104 yielding a set of trivial cycles , i.e. cycles with only two different states ( @xmath109 ) . their cycle weights are identical to their corresponding non - zero entry @xmath110 . the algorithm to decompose the current part @xmath105 is split into two parts . first , searching for a non - trivial cycles ( cycle with more than 2 different states ) and second determining its cycle weights . theoretically , the number of possible non - trivial cycles grows exponentially with the number of non - zero entries of @xmath105 . however , if both steps run alternately the decomposition becomes computationally affordable even for a large number of states . to detect a non - trivial cycle , we propose the following steps : 1 . find the position of the largest element of @xmath105 , @xmath111 . search for the shortest path ( smallest number of transitions ) from state @xmath68 leading back to state @xmath32 ( only following the non - zero transitions ) . this step can be efficiently achieved by applying a breadth - fist / depth - first search @xcite . 3 . return the non - trivial cycle , i.e. ( @xmath112found path ) . to determine the corresponding cycle weight @xmath113 , we take all flux values along cycle @xmath114 and determine their smallest value becoming the cycle weight @xmath115 summing up both steps , the final algorithm reads 1 . find a non - trivial cycle 2 . compute its cycle weight @xmath113 3 . update the current matrix by subtracting @xmath113 along @xmath114 , @xmath116 and repeat with the first step 4 . the algorithm stops when the residuum @xmath117 has become smaller than a threshold . general considerations @xcite show that the maximum number of needed non - trivial cycles is bounded by @xmath118 with @xmath119 being the number of non - zero elements of @xmath105 and @xmath120 its rank . once the cycle communities are found , the idea for coarse - graining the msm is to pick one cycle for each community we refer to it as representative , delete all states not belonging to any representative and , finally , rescale the remaining transition rates @xmath121 . to be thermodynamically consistent the newly computed transition rates have to preserve four quantities : the total entropy production rate @xmath80 , the cycle affinities of the representatives @xmath74 , all remaining edge affinities @xmath75 and the dissipated heat along the remaining edges @xmath122 . for a detailed description of the coarse - graining algorithm we refer the reader to our previous publication @xcite . since we know now how to coarse - grain a given set of cycle representatives , we address the question of how to select `` appropriate '' representatives . any set of cycle representatives is thought to be appropriate if the graph spanned by their coarse - grained transition matrix is ergodic and the mean first passage times ( mfpt ) between local communities are preserved . for example , the polymer dynamics for @xmath71 communities as illustrated in fig . [ fig : cyc]c , d exhibits 2 local and 1 global community . to compute the mfpts we identify all states of the red community , say as set @xmath123 , and all states belonging to the blue one as set @xmath124 . any appropriate set of representatives needs to preserve mfpt@xmath125 and mfpt@xmath126 . especially , the conservation of mfpts is of particular importance as it ensures the coarse - grained msm to express the correct timescales . so far no determinstic algorithm exists that returns a reliable set of representatives . for this reason we formulate a stochastic algorithm that picks candidates for cycle representatives randomly and checks for ergodicity and mfpts . the algorithm is outlined as follows : 1 . choose one representative per cycle community by drawing a random number . 2 . check if set of representatives span ergodic transition network . if yes , compute coarse - grained msm , else go back to step ( 1 ) . 3 . compute mfpts of the coarse - grained msm and compare to mfpts of full msm . if mfpts match , return coarse - grained msm , else go back to step ( 1 ) . are modified . the arrows point in the direction of probability currents . ] to coarse - grain bridge states ( state 1 in fig . [ bridgestate]a ) we form a new connection between the two neighboring states ( state 0 and 2 in fig . [ bridgestate]a ) . the new transition rates @xmath127 and @xmath128 have to preserve three characteristics : ( i ) edge affinities @xmath129 , ( ii ) probability currents @xmath130 and ( iii ) dissipated heat @xmath131 . using conditions ( i ) and ( ii ) , the new probability fluxes follow as @xmath132 to fulfill condition ( iii ) we demand that the ratio between any two probabilities of the full network are preserved @xmath133 , which includes @xmath134 , and thus @xmath135 condition ( iii ) is also the main difference between our approach and the one discussed in ref . . in our adaptation the probability distribution of the complete network is changed , while altaner _ _ change it only locally ( @xmath136 and @xmath137 ) and hence absorb @xmath138 into @xmath136 and @xmath137 . the disadvantage of the latter is that condition ( iii ) is not preserved for the full network and , when using the coarse - graining approach iteratively , accumulation of probability in single states might occur , which leads to unphysical results . for the coarse - graining of triangle states ( state 1 in fig . [ bridgestate]b ) we consider all cycles that contain the edges @xmath139 . we modify these cycles by replacing the edges @xmath139 with a new edge @xmath140 . to be thermodynamically consistent the modified cycles have to ( i ) preserve the cycle entropy production rate @xmath141 , ( ii ) preserve the edge affinities @xmath142 and ( iii ) dissipated heat @xmath143 . to rescale the transition rates we use the same rescaling algorithm as for the rescaling of the cycle representatives . note that through restriction ( i ) it is not necessarily possible to coarse - grain all found triangle structures . assume , for instance , that the modified cycle coincides with an already existing cycle , then , the rescaling is not unique anymore and entropy production is destroyed as only one out of two cycles survives . 42ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1038/nsb0902 - 646 [ * * , ( ) ] link:\doibase 10.1146/annurev - physchem-032210 - 103335 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1021/ma8018624 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.116.058302 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1016/j.sbi.2010.10.006 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1038/nature02261 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1122/1.1835336 [ * * , ( ) ] link:\doibase 10.1126/science.283.5408.1724 [ * * , ( ) ] link:\doibase 10.1063/1.1681018 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.84.4769 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1063/1.3149860 [ * * , ( ) ] @noop * * , ( ) @noop _ _ , ( , , ) @noop ( ) link:\doibase 10.1088/1742 - 5468/2010/05/p05015 [ ( ) ] @noop * * , ( ) \doibase doi:10.1088/0034 - 4885/75/12/126001 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.97.128301 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreve.85.041133 [ * * , ( ) ] @noop _ _ , vol . ( , ) @noop * * , ( ) link:\doibase 10.3934/jcd.2015.2.1 [ * * , ( ) ] @noop _ _ ( , ) @noop @noop * * , ( ) link:\doibase 10.1063/1.4818538 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.114.158101 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.116.120601 [ * * , ( ) ] link:\doibase 10.1073/pnas.1524720113 [ * * , ( ) ] @noop * * , ( ) @noop _ _ ( , )

we describe a systematic approach to construct coarse - grained markov state models from molecular dynamics data of systems driven into a non - equilibrium steady state . we apply this method to study the globule - stretch transition of a single tethered model polymer in shear flow . the folding / unfolding rates of the coarse - grained model agree with the original detailed model . we demonstrate that the folding / unfolding proceeds through the same narrow region of configuration space but along different cycles .

You are an expert at summarizing long articles. Proceed to summarize the following text: cosmology deals with the large - scale structure of the universe . a global description of this structure employs an averaging and smoothing of all the objects of astrophysics from solar systems to galaxies and clusters of galaxies . the result is a fluid model of the cosmos goverened by gravitation , as it was proposed by einstein @xcite , see @xcite pp . 160 - 164 and @xcite pp . 704 - 710 . this fluid model is governed by einstein s differential equations which relate the space - time metric to the energy - momentum tensor . to device such a model one has to make assumptions about the global structure and topology of the underlying space - time manifold . einstein in his initial analysis of 1917 assumed cosmic 3-space in form of a 3-sphere . his assumption implies an average curvature @xmath0 . the topology of cosmic 3-space has found new attention in relation to the cosmic microwave background ( cmb ) radiation , which is supposed to originate from an early stage of the universe . the fluctuations of the incoming cmb radiation are well described by a standard model , except for rather low amplitudes at the lowest multipole orders . this raised the question if multipole selection rules could be due to a particular topology of 3-space . * genesis of the mbius strip . * left : as rectangular cell of the mbius crystal * cm * with broken glide reflection line , see fig . [ fig : cm+8cellb ] , middle : twisted by @xmath1 along the glide line , right : bended into a circle and glued by homotopy.,scaledwidth=100.0% ] * mbius crystal and 8-cell . * left : the red mbius strip tiles its cover , the plane , into the cells of a mbius crystal with crystallographic symbol * cm * , with a broken glide line and two pointed mirror reflection lines . the vertical edges are glued by homotopy , fig.[fig : moebi ] , right : a spherical red cube tiles its cover , the 3-sphere , into the 8 spherical cubes of the 8-cell , shown in a projection.,scaledwidth=90.0% ] a familiar paradigm for topology is the mbius strip . in @xcite we explain its topological properties and relate it to the crystallographic space group * cm*. a cell of this crystal is shown on the left - hand side of fig . [ fig : cm+8cellb ] . any spherical topological 3-manifold @xmath2 can be viewed as a prototile on einstein s 3-sphere with pairs of faces glued according to the prescriptions of a group of homotopies . among the spherical 3-manifolds are the platonic ones . their homotopies were derived by everitt @xcite . in @xcite and papers @xcite , @xcite quoted therein we derived from the homotopy or fundamental group of a platonic manifold @xmath2 the corresponding groups deck(@xmath2 ) of deck operations acting on the 3-sphere . these groups generate by fixpoint - free action the tiling from the prototile . each group of deck operations for a platonic 3-manifold @xmath2 is constructed in @xcite as a subgroup of a coxeter group @xmath3 . finally from the unimodular subgroups of the coxeter groups we construct three new spherical 3-manifolds . to study actions on the 3-sphere we pass from the set of four cartesian coordinates @xmath4 in euclidean 4-space to a matrix description , @xmath5,\ ; z_1= x_0-ix_3,\ : z_2=-x_2-ix_1,\ : z_1\overline{z}_1+z_2\overline{z}_2=1.\ ] ] with the help of the pauli matrices @xmath6 and @xmath7 and the trace @xmath8 we recover the cartesian coordinates in the form @xmath9 the wigner polynomials @xcite , @xcite are homogeneous polynomials @xmath10 in the four complex variables @xmath11 of total degree @xmath12 , see @xcite appendix a. in the familiar euler half - angles @xmath13 , the wigner polynomials take the form @xmath14 we look at the set of harmonic wigner polynomials by starting from the integer or half - integer pairs of numbers @xmath15 . these pairs of numbers can be viewed as points from two nested lattices on a plane , see fig.[fig : gridn2n3 ] . for given integer or half - integer values of @xmath16 , one finds @xmath17 we say that any lattice point @xmath15 in this plane carries a _ tower _ , labelled by @xmath18 , of wigner polynomials @xmath19 according to eq . [ eq5 ] . the wigner polynomials form a complete orthogonal system of polynomial functions on the 3-sphere . moreover they are harmonic , that is , vanish under the application of the laplacian acting on functions on euclidean 4-space , see @xcite . therefore they are a basis for harmonic analysis on the 3-sphere . the isometric rotations of the group @xmath20 can be written in the form @xmath21 and act on the coordinates @xmath22 as @xmath23 the elements of the form @xmath24 generate a subgroup @xmath25 acting on @xmath22 by conjugation . the 3-sphere can be written as the homogeneous space @xmath26 . we write the action eq . [ eq6 ] of @xmath27 on the wigner polynomials as @xmath28 here we used the representation property of the wigner polynomials . a general pair @xmath29 can always be brought to diagonal form @xmath30 we interprete the transformation @xmath31 as a transformation to new coordinates @xmath32 . in these new coordinates , @xmath33 are diagonal with diagonal entries @xmath34 , and the action eq . [ eq6a ] with eq . [ eq7 ] takes the form @xmath35 now we can go to a lattice description of the harmonic analysis for the two spherical cubic 3-manifolds : the basis for the harmonic analysis consists of the towers of wigner @xmath19 polynomials on top of a sublattice in the @xmath15-plane as given in fig . [ fig : gridn2n3 ] . as examples we shall choose the two spherical cubic manifolds @xmath36 . the first homotopy groups of the platonic spherical polyhedra were given in everitt @xcite . in @xcite and work cited therein we construct from everitt s results the groups of deck transformations . these act on the 3-sphere and tile it into platonic polyhedra . in @xcite one finds by an enumeration the homotopic face and edge gluings for this manifold . the tiling of the 3-sphere is the 8-cell shown on the right in fig [ fig : cm+8cellb ] . the cubic 3-manifold we take as the central spherical cube in the 8-cell . the two different spherical cubic 3-manifolds differ in their face and edge gluing . in fig . [ fig : twocubetwist ] we have marked the faces with numbers @xmath37 by triangles with the colors yellow , blue , and red . the homotopic self - gluing of the initial 3-manifold is converted in the 8-cell tiling into a gluing of neighbour cubes sharing a face . this is illustrated in colors in fig . [ fig : cubeglueb ] . in fig . [ fig : twocubetwist ] we use the same color coding to illustrate the two different next neighbour cubes for the two 3-manifolds @xmath36 . conversion of homotopic face gluing @xmath38 into a deck operation of two cubic tiles for the manifold n2 . the half - faces of @xmath39 are marked by the colors yellow , blue , red respectively . for gluings of other faces see the left of fig . [ fig : twocubetwist],scaledwidth=50.0% ] * the cubic manifolds @xmath40 and @xmath41*. the cubic prototile and three neighbour tiles sharing its faces * f1 , f2 , f3*. the four cubes are replaced by their euclidean counterparts and separated from one another . visible half - faces * f1 , f2 , f3 * are marked by the colors yellow , blue and red . the actions transforming the prototile into its three neighbours generate the deck transformations and the 8-cell tiling of @xmath42.,scaledwidth=100.0% ] algebraically , the deck operations , being rotations , contain an even number of weyl reflections and can be written in terms of elements @xmath43 . we now construct by projection the linear combinations of wigner polynomials that span the harmonic analysis on the two cubic 3-manifolds . for @xmath40 , the group @xmath44 of deck transformations of the 8-cell tiling from @xcite is a cyclic group @xmath45 . its generator @xmath46 is given in eq . we start from the set of wigner polynomials and use their representation under @xmath27 , in terms of irreducible representations @xmath19 of @xmath47 . this allows to apply to them the projection operator to the identity representation of the group @xmath48 , @xmath49.\ ] ] in general , eq . [ eq10 ] gives a linear combination of wigner polynomials . in case of the platonic manifold @xmath40 , @xmath48 is a cyclic group . by a transformation @xmath50 of coordinates we can reduce the action of this cyclic group to diagonal form . we start from the generator @xmath51 of the group @xmath52 , @xmath53,\ : g_r=\left [ \begin{array}{ll } 0&\overline{a}\\ -a&0\\ \end{array } \right],\ : a=\exp(i\frac{\pi}{4}).\ ] ] we diagonalize @xmath54 as @xmath55 c^{-1},\ : c= \frac{1}{\sqrt{2 } } \left [ \begin{array}{ll } -1&-a\\ \overline{a}&-1\\ \end{array } \right],\ : c^{-1}=c^{\dagger},\ : { \rm det}(c)=1.\ ] ] next we define new coordinates @xmath32 by @xmath56 so that @xmath51 acts on @xmath32 by diagonal matrices from left and right as @xmath57 \ : u'\ : \left [ \begin{array}{ll } a^2&0\\ 0&-a^2\\ \end{array } \right ] = \delta_l^{-1}u'\delta_r.\ ] ] these relations allow to apply eq . it follows that we can reduce the representation of @xmath48 to diagonal form . invariance under @xmath48 now gives a certain linear relation between @xmath58 and @xmath59 . this relation singles out on the full lattice of points @xmath15 all the points of the sublattice with points @xmath60 a basis of this sublattice is @xmath61 , compare fig . [ fig : gridn2n3 ] . the harmonic analysis on @xmath40 is now given by the towers of wigner polynomials on top of the black sublattice points . for the cubic 3-manifold @xmath41 we construct three glue generators @xmath62 in table [ tablen3a ] from the homotopy group @xcite . the result for three faces is shown on the right of fig . [ fig : twocubetwist ] . the deck operations corresponding to the gluings generate the quaternionic group @xmath63 , @xcite p. 134 . it acts exclusively by left action on the 3-sphere . @xmath64=-\mathbf{k}&e\\ \hline 2 & ( x_2,-x_3,-x_0,x_1 ) & \left [ \begin{array}{ll } 0&-1\\ 1&0\\ \end{array } \right]=-\mathbf{j}&e\\ \hline 3 & ( x_3,x_2,-x_1,-x_0 ) & \left [ \begin{array}{ll } -i&0\\ 0&i\\ \end{array } \right]=-\mathbf{i}&e\\ \hline \end{array}$ ] for the harmonic analysis on @xmath41 we employ the projection eq . [ eq10 ] for the quaternion group @xmath65 , @xmath66 \left[1+(-1)^{m_1}\right ] \left[d^j_{m_1,m_2}(u)+(-1)^jd^j_{-m1,m_2}(u)\right].\ ] ] * lattice representation of @xmath40 and @xmath41 basis : * any lattice point @xmath15 carries the countable tower @xmath10 , @xmath67 of wigner polynomials . harmonic analysis on cubic manifolds , left @xmath40 , right @xmath41 ( with vertical mirror line ) , selects the towers @xmath68 on sublattices marked by black points with sublattice bases @xmath69 . only these obey the homotopic boundary conditions.,scaledwidth=80.0% ] in fig . [ fig : gridn2n3 ] , we display the sublattices for the two cubic spherical platonic 3-manifolds . for the manifold @xmath41 we put the symmetry eq . [ eq16 ] , @xmath70 as a vertical mirror line . for the harmonic analysis it follows that an orthogonal basis is given by the collection of all towers of wigner polynomials on top of the sublattice points . any basis function can be characterized by a sublattice point @xmath15 and by a number @xmath71 . in this section we discuss the algebraic tools for analysing incoming cmb radiation in terms of the harmonic bases for a chosen topology . for the harmonic analysis on spherical 3-manifolds we use the spherical harmonics in the form of wigner polynomials . these polynomials in the coordinates @xmath72 are often expressed in terms of euler angle coordinates eq . [ eq4 ] . an alternative system of polar coordinates is used by aurich et al . @xcite . here @xmath73\\ \\ \hline \end{array}\ ] ] we shall see in eq . [ eq21 ] that these polar coordinates are adapted to the analysis of incoming radiation in terms of its direction . the cmb radiation as observed is given as a function of polar angles @xmath74 for its direction . to compare with an expansion of the harmonic basis of a given 3-manifold , we must rewrite the wigner polynomials in terms of polar angles . in terms of representation theory , this can be achieved by reducing the representations of @xmath27 into irreducible representations of its subgroup @xmath25 , see eq.[eq19 ] . we relate our analysis algebraically to this description . to adapt the wigner polynomials to a multipole expansion , we transform them for fixed degree @xmath75 by use of wigner coefficients of @xmath47 , @xcite pp . 31 - 45 , into the new harmonic polynomials @xmath76 whereas the index @xmath77 of the wigner polynomials can be integer or half - integer , the multipole index @xmath78 takes only integer values . for fixed @xmath78 we have @xmath79 , and for fixed @xmath75 : @xmath80 . using representation theory of @xmath47 it can be shown from eq . [ eq6a ] that the conjugation action @xmath81 of the group @xmath25 acts by a rotation @xmath82 only on the coordinate triple @xmath83 , and the new polynomials eq . [ eq18 ] transform as @xmath84 like the spherical harmonics @xmath85 . we therefore adopt eq . [ eq19 ] as the action of the usual rotation group for cosmological models covered by the 3-sphere , and eq . [ eq19 ] qualifies @xmath78 as the multipole index of incoming radiation . the basis transformation eq . [ eq18 ] is inverted with the help of the orthogonality of the wigner coefficients @xcite to yield @xmath86 the result eq . [ eq20 ] can be further elaborated by use of the alternative coordinates @xmath87 eq . [ eq17 ] . from @xcite , . 9 - 17 we find @xmath88 where @xmath89 is a gegenbauer polynomial . the alternative coordinates admit the separation of the new basis into a part depending on @xmath90 and a standard spherical harmonic as a function of polar coordinates @xmath74 . @xmath91 @xmath92 any platonic spherical 3-manifold is distinguished by a specific point symmetry group m which stabilizes its center point . there arises the following enigma : the point group stabilizes the center point , but the deck group acts fixpoint - free . so the two groups can never mix . can they nevertheless be brought together , and what happens to the harmonic analysis ? to examine this question we turn to the cubic spherical manifolds @xmath36 . their coxeter group from @xcite has the coxeter diagram @xmath93 the point group is the full cubic rotation group @xmath94 of order @xmath95 . the group of deck transformations for @xmath41 is @xmath96 , the quaternion group with eight elements . the cubic tiling of the 3-sphere is the 8-cell tiling of einstein s 3-sphere , see @xcite p. 178 . the relation between the deck and point groups is addressed in appendix c of @xcite . in @xcite one finds selection rules from point symmetry for the multipole orders @xmath97 . for the cubic spherical manifold @xmath41 we found there : * prop 1 * : the cubic point group @xmath98 of @xmath41 under conjugation leaves invariant the group h = deck(n3)=q , the quaternion group , and with q forms a semidirect group @xmath99 , which turns out to be @xmath100 , the rotational unimodular subgroup of the coxeter group , generated by an even number of weyl reflections . the relation between point and deck group resembles the case of symmorphic space groups in euclidean crystallography . there the commutative infinite translation group acts fixpoint - free , and a cubic cell has again the cubic group as its point group . the difference is that , when going from euclidean 3-space to the 3-sphere , the deck group is finite and no longer commutative . if we consider first of all only the cubic point group @xmath94 as a subgroup of the rotation group @xmath101 in euclidean 3-space , there are well - known results from molecular physics for the multiplicity of its representation in a given representation of @xmath101 with angular momentum @xmath78 , see @xcite p. 438 . for the identity representation denoted by @xmath102 of @xmath94 , the lowest non - zero angular momentum is @xmath103 . the cubic invariant linear combination of standard spherical harmonics for lowest values of @xmath78 are given in table [ table : cubicl ] . now we wish to include the quaternion group @xmath63 of deck transformations . from the semidirect product property it follows that the projectors on the identity representation for @xmath94 and @xmath65 commute with one another . this allows for the following procedure : we take a cubic @xmath94-invariant linear combination of spherical harmonics and combine it according to eq . [ eq21 ] with the lowest possible function of the angle @xmath90 . then we transform this linear combination back by eq . [ eq20 ] into wigner polynomials and apply the projector eq . [ eq16 ] to @xmath65-invariant form . next we transform back with eq . [ eq18 ] to the basis adapted to the multipole analysis . the resulting linear combination must still be @xmath94-invariant but may contain new @xmath94-invariant linear combinations of spherical harmonics . by use of the cubic invariants from table [ table : cubicl ] we obtain the fully @xmath104-invariant polynomials of table [ table : cubicinv ] . the construction requires only the wigner coefficients of @xmath47 and can easily be continued to higher polynomial degree . for the physics on the cubic spherical manifold with point symmetry , there follows from table [ table : cubicinv ] a special and observable property : different multipole orders of spherical harmonics must be linearly combined to assure the overall invariance . what happens with the first cubic spherical manifold @xmath40 under cubic point symmetry ? here we have from @xcite the following universality : * prop 2 * : any particular homotopy of a regular polyhedron with fixed geometric shape implies a pairwise homotopic boundary condition on its faces . if full rotational symmetry is applied , all the faces and also all their edges are on the same footing . this implies that any particular homotopic boundary condition is automatically fulfilled . for the two cubic spherical manifolds @xmath36 it follows that the same rules apply to their @xmath100-invariant basis whose lowest part we give in table [ table : cubicinv ] . the order of the semidirect product group @xmath99 is @xmath105 . this is half the order @xmath106 of the coxeter group . it means that we are projecting to the identity representations of @xmath100 . the new 3-manifold @xmath107 as fundamental domain of the unimodular coxeter group s@xmath108 , glued from two coxeter simplices.,scaledwidth=50.0% ] the new 3-manifold @xmath109 as fundamental domain of the unimodular coxeter group s@xmath110 , glued from two coxeter simplices.,scaledwidth=60.0% ] the new 3-manifold @xmath111 as fundamental domain of the unimodular coxeter group s@xmath112 , glued from two coxeter simplices.,scaledwidth=60.0% ] we give a re - interpretation of the results of the previous section , we recall that in @xcite we constructed the spherical 3-manifolds from four coxeter groups generated by weyl reflections . table [ table : table2 ] gives the data . in the last section we found that under inclusion of the cubic point group , the group @xmath113 extends into the unimodular subgroup s@xmath3 of the cubic coxeter group @xmath3 , generated by an even number of weyl reflections . when we introduce in addition to topology the point symmetry of the spherical cube , we can define a fundamental subdomain on the cube under the cubic point group . this subdomain may be taken as the cone , shown in euclidean form in yellow on fig . [ fig : cubus1b ] . the cone is formed as a double simplex from two coxeter simplices of the cubic coxeter group @xmath3 , with the second simplex the image of the first one under reflection in the weyl plane perpendicular to weyl vector @xmath114 . in the 8-cell tiling that covers the 3-sphere , the double simplex is a fundamental domain with respect to the unimodular coxeter group s@xmath3 , of volume fraction @xmath115 . this double simplex on the 3-sphere forms a new topological 3-manifold @xmath109 with the group @xmath116 . with its small volume fraction it is an attractive candidate for cosmic topology . the first polynomials invariant under @xmath117 are the entries of table [ table : cubicl ] . turning to the tretrahedral and octahedral 3-manifolds discussed in @xcite , their unimodular coxeter groups admit the analogeous construction . the unimodular subgroup for the tetrahedron is the even subgroup @xmath118 . its analysis in @xcite takes up work with marcos moshinsky @xcite on permutational symmetry . we name the new 3-manifolds @xmath119 , show their double simplices in figs . [ fig : tetra1b ] , [ fig : cubus1b ] , and [ fig : octa1b ] , and give their main data in table [ table : table1 ] , extended from table 1 in @xcite . the double simplices are spherical counterparts to the notion of asymmetric units as used in classical euclidean crystallography . since we know the geometry and the deck groups for these new manifolds , it should not be hard to determine the corresponding homotopies . the harmonic analysis for @xmath120 still has to be done . the harmonic bases will be invariant in particular under the point group of the tetrahedron and octahedron respectively . the lowest non - zero multipole index is @xmath121 for the tetrahedron and @xmath103 for the cubes . if we wish to accomodate lower multipole order we must reduce the point symmetry of the manifold . we conclude : * prop 3 * : the harmonic analysis on the three new 3-manifolds @xmath122 strictly obeys the multipole selection rules given in @xcite table 3 for the tetrahedral and cubic point groups respectively . @xmath123@xmath124||@xmath125m@xmath124h = deck(m)@xmath124|h|@xmath126 - -- @xmath127120@xmath128n1@xmath124c_5@xmath1295@xmath130 --@xmath124384@xmath131n2@xmath132c_8@xmath1298@xmath133n3@xmath132q@xmath1348@xmath135 - - @xmath1241152@xmath136n4@xmath137c_3q@xmath12424@xmath138n5@xmath137b@xmath13424@xmath138n6@xmath137t^*@xmath13724@xmath139 - - @xmath124120120@xmath140n1@xmath124j^*@xmath137120@xmath141 - -- @xmath142120@xmath143n8@xmath144@xmath14260@xmath145 --@xmath142384@xmath146n9@xmath144@xmath142192@xmath145 -- @xmath1421152@xmath146n10@xmath144@xmath142576@xmath147 on the example of the cubic spherical 3-manifolds , we have explained the construction of the harmonic analysis from topology and its transformation into an expansion for the cmb radiation , ordered by the multipole index @xmath78 . we implemented the additional assumption of point symmetry for spherical manifolds . this assumption yields strong selection rules , including a lowest non - trivial multipole order @xmath78 . similar rules apply to the other platonic spherical manifolds analyzed in @xcite . these strong selection rules are easier to test from the fluctuation spectrum of the cmb radiation . moreover we have shown that the inclusion of the unimodular coxeter groups s@xmath3 yields @xmath149 new topological 3-manifolds which cover rather small fractions of the 3-sphere . this meeting is devoted to the memory of our great teacher and good friend , professor marcos moshinsky . for over five decades i had the chance to share with him his insight into groups , their representations and applications in physics . the initial steps of the present analysis were discussed with him in 2008 in mexico . we all miss marcos , and we shall never forget him .

low multipole amplitudes in the cosmic microwave background cmb radiation can be explained by selection rules from the underlying multiply - connected homotopy . we apply a multipole analysis to the harmonic bases and introduce point symmetry . we give explicit results for two cubic 3-spherical manifolds and lowest polynomial degrees , and derive three new spherical 3-manifolds . address = institut fr theoretische physik der universitt tbingen , + germany

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Proceed to summarize the following text: it has long been hoped that x - ray binaries and active galactic nuclei would prove to be good laboratories for testing general relativity . the two most promising lines of attack for disentangling relativistic effects from the physics of accretion disks and of radiative transfer are high resolution spectroscopy of emission lines ( see e.g. reynolds & nowak 2003 for a review ) and studies of high frequency quasi - periodic oscillations ( see e.g. van der klis 2004 for a review ) . recently , high frequency quasi - periodic oscillations ( hfqpos ) have been found in low mass x - ray binaries for which the black hole masses are reasonably well measured . in one case in particular , a frequency has been identified which is too large to be a keplerian orbit around a schwarzschild black hole ( strohmayer 2001 ) . furthermore , a pattern has begun emerging where these high frequency qpos are found in pairs with 2:3 ratios of frequencies . this was first pointed out by abramowicz & kluzniak ( 2001 ) , based on combining observational results from strohmayer ( 2001 ) and remillard et al . more recent work from miller et al . ( 2001 ) , remillard et al . ( 2003 ) , remillard et al . ( 2004 ) and homan et al . ( 2004 ) has lent more weight to the idea that 2:3 frequency ratios are quite common in these systems . a variety of theoretical models have been developed to explain these qpos , most requiring a spinning black hole , but often requiring rather different values of the spin ( compare , e.g. abramowicz & kluzniak 2001 ; rezzolla et al . 2003 ; li & narayan 2004 ) . therefore , there is still much `` astrophysics '' ( i.e. physics of disk structure and stability and physics of radiative transfer ) that must be understood before the fundamental physics can be probed in these systems , but there is strong cause for optimism that these systems really will ultimately tell us something profound about spinning black holes . a key first step to disentangling the `` astrophysics '' is , of course , to develop models which not only match the important frequencies , but also include radiation mechanisms such that the observed x - rays would actually be modulated at that frequency . such has been done recently for a particular realization of the case of the parametric resonance model of abramowicz & kluzniak ( 2001 ) , by considering the possibility of hot spots that form at the resonant radii in the accretion disk ( schnittman & bertschinger 2004 ; schnittman 2004 ) . these authors have found that different sets of model parameters can produce the same fourier power density spectrum with dramatically different qualitative appearances to the light curves . in this letter , we will show that higher order variability statistics , particularly the bispectrum , can break this degeneracy . one key way to distinguish between different mechanisms which produce the same power spectra from qualitatively different light curves is to study the non - linearity of the variability . linear variability is that in which the phases at the different frequencies in the fourier spectrum of a time series are uncorrelated with one another , while time series with non - linear variability show fourier spectra with correlations between the phases at different frequencies . some particularly useful tools for studying non - linearity are the bispectrum and the closely related bicoherence . the bispectrum computed from a time series broken into @xmath0 segments is defined as : @xmath1 where @xmath2 is the frequency @xmath3 component of the discrete fourier transform of the @xmath4-th time series ( e.g. mendel 1991 ; fackrell 1996 and references within ) . it is a complex quantity that measures the strength of the phase coupling of different fourier frequencies in a light curve and has a phase of its own which is the sum of the phases at the two lower frequencies minus the phase at the highest frequency . its value is unaffected by additive gaussian noise , although its variance will increase for a noisy signal . a related quantity , the bicoherence is the vector magnitude of the bispectrum , normalised to lie between 0 and 1 . defined analogously to the cross - coherence function ( e.g. nowak & vaughan 1996 ) , it is the vector sum of a series of bispectrum measurements divided by the sum of the magnitudes of the individual measurements . if the biphase ( the phase of the bispectrum ) remains constant over time , then the bicoherence will have a value of unity , while if the phase is random , then the bicoherence will approach zero in the limit of an infinite number of measurements . mathematically , the bicoherence @xmath5 is defined as : @xmath6 this quantity s value is affected by gaussian noise , but it can be considerably more useful than the bispectrum itself for determining whether two signals are coupled non - linearly . in an astronomical time series analysis context , it has been previously applied to the broad components in the power spectra of cygnus x-1 and gx 339 - 4 , in both cases finding non - linear variability through the presence of non - zero bicoherences over a wide range of frequencies ( maccarone & coppi 2002 ) . since that work , we have become aware of a correction which is , in principle , important for studying aperiodic variability with the bicoherence , namely that the maximum value of the bicoherence is suppressed by smearing of many frequencies into a single bin in the discrete fourier transform . this suppression can not be calculated in a straightforward way ( see e.g. greb & rusbridge 1988 ) . however , we also note that since comparisons in maccarone & coppi ( 2002 ) were made only with model calculations made with the same time binning as the real data , these effects , whatever they may be , are the same for the real data and the simulated data , and hence the conclusions of that paper are not affected substantially . there exist , at the time of this paper s writing , at least four basic concepts for producing the high frequency quasi - periodic oscillations seen from accreting black holes . in historical order , these are diskoseismology ( e.g. okazaki , kato & fukue 1987 ) , relativistic coordinate frequencies ( e.g. stella & vietri 1999 ) , rayleigh - taylor instabilities ( e.g. titarchuk 2002 , 2003 ; li & narayan 2004 ) and oscillating tori ( rezzolla et al . we will briefly summarize the properties of the models in a different order in an effort to smooth the flow of the paper . the first model proposed was based on diskoseismology - the excitation of various trapped modes in the inner region of a keplerian accretion disk in a relativistic potential ( e.g. okazaki , kato & fukue 1987 ; nowak et al . this model seems not to be directly applicable to the data , at least for the cases where small integer ratios of frequencies exist ; it would require considerable fine tuning in the different mass and spin values for the black holes to produce routinely a 2:3 frequency ratio . chen & taam ( 1995 ) showed that a slim disk ( see e.g. abramowicz et al . 1988 ) can produce oscillations at a frequency very close to the radial epicyclic frequency of the accretion disk , sometimes with power at twice this frequency ; it also appears that there may be some excess power at 3/2 of the epicyclic frequency in one of the simulations presented in chen & taam ( 1995 ) see figure 5 of that paper but it is likely that a longer hydrodynamic simulation would be needed to confirm this . in a similar vein is the oscillating torus model of rezzolla et al . this model also applies calculations of the frequencies of @xmath7-modes ( i.e. sound waves ) of zanotti , rezzolla & font ( 2003 ) , but in a geometrically thick , pressure supported torus ( as expected at high accretion rates like those where the hfqpos are seen - de villiers , krolik & hawley 2003 ; kato 2004 ) , rather than in a geometrically thin , keplerian accretion disk . in this case , the different overtones are found to be approximately in a series of integer ratios , starting from 2 , so the model is compatible with existing data on high frequency qpos in black holes . the model most recently applied to high frequency qpos from black hole candidates is that of rayleigh - taylor instabilities , although the same basic idea had previously been applied to qpos from accreting neutron stars ( titarchuk 2002 , 2003 ) . in this picture , non - axisymmetric structures can grow unstably at the magnetospheric radius ( presumed to exist also for black holes , as their accretion disks can become magnetically dominated ) with frequencies of integer ratios of the angular frequency at that radius , though the lowest mode will be stable for low gas pressures ( li & narayan 2004 ) . after the first indications that small integer ratios between hfqpo frequencies were likely , it was noted by abramowicz & kluzniak ( 2001 ) that if the relativistic coordinate frequencies determined the frequencies of the quasi - periodic oscillations ( see e.g. stella & vietri 1999 ) then resonances between these different frequencies ( e.g. vertical and radial epicyclic frequencies ) might occur at locations in the accretion disk where these frequencies have small integer ratios . more recently , schnittman & bertschinger ( 2004 ) performed ray tracing calculations of the light curve of sheared hot spots produced with a 1:3 radial to azimuthal epicyclic frequency ratio and found good agreement with the observed locations of the power spectrum peaks and their relative amplitudes , while bursa et al . ( 2004 ) have shown that it is also possible to produce these frequencies from radial and vertical oscillations of a torus located such that the radial epicyclic frequency is 2/3 the orbital frequency . schnittman ( 2004 ) extended the work by considering the effects of multiple hot spots under different conditions in order to broaden the _ periodic _ oscillations computed in schnittman & bertschinger ( 2004 ) into the _ quasi_-periodic oscillations that are actually observed . in this paper , we present bicoherence calculations only for this last model for the simple reason that this is the only model for which simulated light curves are currently available . as simulated light curves for other models become available , we will consider their higher order statistical properties as well . we now apply the bicoherence to the simulated data . we consider two different model calculations from schnittman ( 2004 ) which give nearly identical power spectra . in each case , the quasi - periodic oscillations are produced by a 1:3 resonance between the radial epicyclic frequency and the orbital frequency . the parameters have been chosen such that the orbital frequency is 285 hz , and the radial epicyclic frequency is 95 hz . this corresponds to a black hole mass of 10 @xmath8 and a spin @xmath9 , with the resonance occuring at a radius of 4.89@xmath10 in geometrized units where @xmath11 ; the black hole mass and qpo frequencies compare reasonably well to those observed for xte j 1550 - 564 ( see e.g. miller et al . 2001 ; remillard et al . the disk inclination is also fixed to be 70 degrees ( where 90 degrees is an edge - on disk ) ; this parameter does not affect the frequencies observed , but can affect the amplitudes of the qpos in the context of the model we are considering here ( schnittman & bertschinger 2004 ) . in each case we compute 1000 seconds of simulated data with a binning timescale of the lightcurve of 0.1 msec . we then compute fourier transforms by breaking the data into 2440 segments of 4096 data points , making use of 999.424 seconds of the simulated data . in the first case , short lived hot spots exist with their orbits all at a single radius , being continually created and destroyed with a characteristic lifetime of 4 orbits . in the second case , long - lived ( lifetimes of 100 msec , or about 30 orbits ) hot spots are distributed over a range of radii ( @xmath12 ) . in both cases , the hot spots are on orbits with eccentricities of 0.1 . for each model , the variability appears quasi - periodic , rather than truly periodic , but for different reasons . in the first case , the creation and destruction of hot spots on short timescales leads to a phase jitter in the light curves . these random offsets in phase broaden the observed periodicity . in the second case , the power spectrum is truly showing that there are many periodicities in the system , with coherent phases . the bicoherence easily detects this difference , as can be seen from figure [ sim ] . in case 1 , the bicoherence shows `` circular '' peaks at various combinations of frequencies where there is power at @xmath13 , @xmath14 and @xmath15 in the contour plot , essentially delta function peaks convolved with two - dimensional lorenzians due to the random phase broadening . in case 2 , the bicoherence shows thin elongated peaks , oriented in a variety of directions depending on @xmath13 and @xmath14 . the reason for this difference is straightforward . in the first case , all hot spots have the same geodesic frequencies , so during a hot spot s lifetime , it is phase locked to all the other hot spots , giving a collection of delta function peaks at the coordinate frequencies . the random phase jitter will broaden the @xmath16functions into qpos , with a similar lorentzian width as described in schnittman ( 2004 ) . the hot spots being created and destroyed in the middle of a fourier transform window will thus create leakage in the power of the qpo to frequencies near the central frequency , but there will be a phase relation between the power in these frequencies and the phase in the central frequency . the phase jitter should thus provide a broadening in the bicoherence similar to that in the power spectrum . we note that the peaks do appear to be somewhat elongated , with the direction of elongation such that the sums of the two smaller frequencies equal the centroid frequencies of the highest frequency qpo in the triplet . this is likely because the centroid is the only truly physical frequency in this case , but a rigorous proof of this point is beyond the scope of this paper . in the second case , where there are many frequencies in the power spectrum due to hot spots found over a range of radii , there will be phase coherence between the different harmonics of each individual hot spot , but not with the hot spots at slightly different frequencies . there will thus be bicoherence between the various harmonic frequencies found at any individual radius , but not between frequencies found at different radii . this second case could be especially interesting . we have calculated analytically the ratios expected between different harmonics frequencies if the radius at which the hot spot occurs is allowed to vary , and have plotted them in figure [ analytical ] . if in real data , similar tracks are seen , then , in the context of this model , they would give the relationships between the different relativistic frequencies . since these tracks trace the coordinate frequencies as a function of radial distance from the black hole , they may be used to make precise measurements of the black hole s mass and spin , plus the central radius of the perturbations . since the observed qpos are rather narrow , so this method would probably be of use only with high signal - to - noise data . to consider whether this observational test is really feasible , we have performed simulations with the rms amplitude of the oscillations reduced to realistic levels and with poisson noise added . we consider two count rate regimes - one similar to that detected by _ rxte _ for the typical x - ray transients at about 10 kpc , which is about 10,000 counts per second , and another which would be expected from the same source , but with a 30 m@xmath17 detector . in each case , we allow 6% of the counts to come from the variable component and to have , intrinsically , count rates given by the simulated light curves of schnittman ( 2004 ) , and the remaining 94% of the counts to come from a constant component . we then simulate observed numbers of counts in 100 microsecond segments as poisson deviates ( press et al . 1992 ) of the model count rates . we then compute the bicoherence as above , but with 2440 segments of 4096 data points , for a total of 999.424 seconds of simulated data . for the _ rxte _ count rates , we find that the bicoherence plots show only noise and only the strongest peak in the power spectrum is clearly significant in a 1000 second simulated observation , while marginal detections exist for the qpos at two - thirds of and twice this frequency ( @xmath18 and @xmath19 ) . this is as expected based on real data , which generally requires exposure times much longer than 1000 seconds to detect these qpos . longer simulated light curves have not been computed at this time due to computational constraints . however , since the signal - to - noise in the bicoherence should be substantially worse than the signal - to - noise in the power spectrum , bicoherence measurements should be possible only when a peak in the power spectrum is considerably stronger than the poisson level . for the count rates expected from a 30 m@xmath17 detector , we find that even within 1000 seconds , several of the higher ( i.e. @xmath20 ) harmonics are observable in the power spectrum and show the clear elongation in the bicoherence plot for case 2 , indicating that proposed missions should be capable of making use of the bicoherence for studying hfqpos . a few very weak peaks are seen in the bicoherence in case 1 even in 1000 seconds . the simulated bicoherences for a 30 @xmath21 detector are plotted in figure [ xeus ] . we note that these simulations are a bit over - simplified , in that we have not included the lower frequency qpos and low frequency band - limited noise that are typically observed in conjunction with the hfqpos , but that these variability components should not significantly affect the phase coupling of the high frequency qpos . we also note that it might be possible to make use of the bicoherence even with _ rxte _ if a more nearby x - ray transient goes into outburst , but that in such a case , the deadtime effects we have neglected here might become important . we have shown here that the bicoherence can be useful in breaking the degeneracies between different qpo models which produce the same fourier power spectrum . in particular , we have shown that in the context of a resonance model for the high frequency quasi - periodic oscillations seen from accreting black holes , the bispectrum is capable of distinguishing between broadening due to phase jitter caused by the rapid creation and destruction of hot spots and broadening due to an intrinsic distribution of physical frequencies in a broad range around a central value . in future work , we will examine the properties of the bicoherence for other models for these qpos , such as oscillations in a pressure supported torus ( rezzolla et al . 2003 ) . we note though that this diagnostic is likely to be useful only with new instrumentation ( or the fortuitous outburst of a bright x - ray transient within about 3 kpc of the sun ) . at the typical 10 kpc distances of x - ray transients , @xmath22 is capable of detecting these qpos generally only with rather long intergrations and careful selection of the photon energy bands to maximize the signal - to - noise , but proposed timing missions with considerably larger collections areas , like xtra ( e.g. strder et al . 2004 ) and the relativistic astrophysics explorer ( e.g. kaaret 2002 ) should have considerably greater potential for making use of these statistics . tm wishes to thank marek abramowicz , wlodek kluzniak , shin yoshida and olindo zanotti for stimulating discussions re - motivating his interest in x - ray variability ; phil uttley and simon vaughan for discussions of statistical properties of the bispectrum ; mariano mendez and marc klein wolt for discussions of observational properties of hfqpos ; ron elsner for providing some useful background information on the bispectrum ; and luciano rezzolla and michiel van der klis for comments on the manuscript as well as additional useful discussions . jds would like to thank edmund bertschinger and ron remillard for many useful discussions , and would like to acknowledge support from nasa grant nag5 - 13306 . abramowicz , m.a . , czerny , b. , lasota , j.p . & szuszkiewicz , e. , 1988 , apj , 332 , 646 abramowicz , m.a . & kluzniak , w. , 2001 , a&a , 374l , 19 bursa , m. , abramowicz , m.a . , karas , v. & kluzniak , w. , 2004 , astro - ph/0406586 chen , x. & taam , r.e . , 1995 , apj , 441 , 354 de villiers , j.p . , krolik , j.h . & hawley , j.f . , 2003 , apj , 599 , 1238 fackrell , j 1996 , ph.d . thesis , university of edinburgh greb , u. & rusbridge , m.g . , 1988 , plasma physics and controlled fusion , 30 , 537 homan , j. , miller , j. m. , wijnands , r. , van der klis , m. , belloni , t.,steeghs , d.,lewin , w. h. g. , 2004 , apj , submitted ( astro - ph/0406334 ) kaaret , p. , 2002 , cospar , 1240 kato , y. , 2004 , pasj , in press li , l .- x . & narayan , r. , 2004 , apjl , 601 , 414 maccarone , t.j . & coppi , p.s . , 2002 , mnras , 336 , 817 mendel , j. 1991 , proc ieee , 79 , 278 miller , j.m . , wijnands , r. , homan , j. , belloni , t. , pooley , d. , corbel , s. , kouveliotou , c. , van der klis , m. & lewin , w.h.g . , 2001 , apj , 563 , 928 nowak , m.a & vaughan , b.a . , 1996 , mnras , 280 , 227 nowak , m.a . , wagoner , r.v . , begelman , m.c . & lehr , d.e . , 1997 , apjl , 477 , 91 okazaki , a.t . , kato , s. & fukue , j. , 1987 , pasj , 39 , 457 press , w.h . , flannery , b.p . , teukolsky , s.a . & vetterling , w.t . , 1992 , _ numerical recipes in fortran _ , cambridge university press remillard , r.a . , morgan , e.h . , mcclintock , j.e . , bailyn , c.d . & orosz , j.a . , 1999 , apj , 522 , 397 remillard , r.a . , et al . , 2002 , apj , 580 , 1030 remillard , r.a . , muno , m.p . , mcclintock , j.e . & orosz , j.a . , 2003 , head , 35 , 3003 remillard , r.a . , mcclintock , j.e . , orosz , j.a . & levine , a.m. , 2004 , apj , submitted ( astro - ph/0407025 ) reynolds , c.s . & nowak , m.a . , 2003 , phr , 377 , 389 rezzolla , l. , yoshida , si . , maccarone , t.j . , zannotti , o. , 2003 , mnras , 344l , 37 schnittman , j.d . & bertschinger , e. , 2004 , apj , 606 , 1098 schnittman , j.d . , 2004 , apj , submitted ( astro - ph/0407179 ) stella , l. & vietri , m. , 1999 , phrvl , 82 , 17 strohmayer , t.e . , 2001 , apjl , 552 , 49 strder , l. , barret , d. , fiorini , c . , kendziorra , e. & lechner , p. , 2004 , spie , 5165 , 19 titarchuk , l. , 2002 , apjl , 578 , 71 titarchuk , l. , 2003 , apj , 591 , 354 zanotti , o. , rezzolla , l. & font , j.a . , 2003 , mnras , 341 , 832

we discuss the use of the bicoherence - a measure of the phase coupling of oscillations at different frequencies - as a diagnostic between different models for high frequency quasi - periodic oscillations from galactic black hole candidates . we show that this statistic is capable of finding qualitative distinctions between different hot spot models which produce nearly identical fourier power density spectra . finally , we show that proposed new timing missions should detect enough counts to make real use of this statistic . [ firstpage ] epsf methods : data analysis , methods : statistical , binaries : close , black hole physics , x - rays : binaries , stars : oscillations

You are an expert at summarizing long articles. Proceed to summarize the following text: due to a predicted high curie temperature ( above 300 k ) , diluted magnetic semiconductors ( dms ) are regarded as important materials for spintronics . @xcite combining ferromagnetism with semiconductivity in oxides gives them an additional degree of freedom and functionality for fabricating unique devices with applications ranging from nonvolatile memory to quantum computing . @xcite some theoretical @xcite and experimental @xcite reports have shown room temperature ferromagnetism ( rtfm ) in highly doped zno films . the magnetic impurities are either implanted into the host matrix @xcite or introduced during the growth . @xcite the experimental results regarding rtfm and its origin in zn@xmath0co@xmath1o samples in different forms including thin films , bulk , polycrystalline and nanocrystalline samples differ widely , and there is no consensus on either the nature of the magnetism ( whether the samples are para- , dia- or ferro - magnetic ) or on the origin of the ferromagnetism ( intrinsic , extrinsic , defect induced etc . ) . in the case of cobalt doped zno films , the observed rtfm has been attributed either to co clustering , @xcite some external pollution @xcite or is thought to be intrinsic in nature . @xcite cobalt and lithium co - doped zno films have revealed only co paramagnetism , and oxygen vacancies were considered as the origin for the rtfm . @xcite in bulk samples the fm is attributed to co clustering , @xcite and in hydrogenated samples it is attributed either to the appearance of co clusters @xcite or oxygen vacancies . @xcite in the case of polycrystalline and nanocrystalline samples , some groups have reported paramagnetism . @xcite recently , nanocrystals of zn@xmath0co@xmath1o have been reported to be ferromagnetic @xcite or superparamagnetic , @xcite and nanowires showed a paramagnetic behavior with co concentrations up to @xmath5 . @xcite very recently , thin films and nanoparticles of pure zno have been reported to be ferromagnetic , and their fm is attributed to defects on zn sites and surface point defects . @xcite from a practical point of view , the ultimate goal of investigating materials such as cobalt doped zno is to fabricate semiconductor devices that have a spin - polarized nature to their electronic transport properties . to this end , explorations of the magnetotransport properties are an essential part of studies into cobalt doped zno . the low temperature magnetoresistance ( mr ) of cobalt doped zno is typically tens of percent in magnetic fields of up to several tesla , @xcite and may show a crossover between positive and negative mr @xcite at different temperature and magnetic field values . zno films containing magnetic nanoclusters within a non - magnetic insulating matrix have positive mr values as large as 811% at 5 k in fields of a few tesla , @xcite revealing the rich potential for magnetic - field influenced charge transport in dms . the exploration of single crystals of dms is of great significance as this form is ideally suited for understanding the intrinsic properties of any material . in the present paper we present the structural , magnetotransport and magnetic properties of both pristine zno and cobalt doped zno single crystals , which were grown by the molten salt solvent technique @xcite . in the present case , the molten salt solvent technique has been employed to grow single crystals of both pristine zno and cobalt doped zno . analytical reagent ( ar ) grade zno , koh@xmath66h@xmath7o and cocl@xmath82h@xmath7o were used as precursors . calculated amounts @xcite of zno and cocl@xmath82h@xmath7o ( an appropriate amount was chosen corresponding to the desired concentration of co ) were slowly added to the molten koh . the crucible was then covered with a specially designed silver lid which was fitted with a silver tube at its centre . the other end of the silver tube was tapered into a cone . the crucible with the charge ( solvent + solute ) was then placed in the furnace such that the charge remained in the uniform temperature region of the furnace at a temperature of 480@xmath9c for @xmath1050 hours . when immersed in the molten charge , the conical end of the tube provides suitable sites for the growth of the crystals , allows the heat of solidification to escape and also serves as a sheath for the thermocouple ( one end of which is placed in the tube ) . these conditions result in the nucleation and isothermal growth of transparent , green - colored , needle - shaped cobalt doped single crystals of zno due to dissociation of zincate species unstable at 480@xmath9c , at the cylindrical walls of the silver crucible and the tapered end of the silver tube . @xcite the crystals were extracted by dissolving the solvent in methyl alcohol . dimensions of the needle - like crystals are typically 2.5 to 5 mm in length , 0.5 - 1.0 mm in width and 0.1 to 0.2 mm in thickness . the single crystals of pristine and cobalt doped zno were powdered and their structural properties were studied using a philips ( xpert pro , model pw 3040 ) x - ray diffraction ( xrd ) system with cu k@xmath11 radiation at wavelength 1.54060 . the semcf used for sem micrographs has rontecs edax ( energy dispersive analysis of x - rays ) system model quantax 200 , which is based on silicon drift detector ( sdd ) technology and provides an energy resolution of 127 ev at mn k@xmath11 . the magnetic moment of a small amount of each kind of crystal was measured as a function of both applied magnetic field and temperature , using a quantum design mpms-7 squid magnetometer . the temperature and magnetic field dependent electrical resistance of individual crystals was also measured . the maximum applied magnetic field for these measurements was 4.5 t , and the range of temperatures employed was 2.5 - 350 k. the x - ray diffractograms of both the pristine zno and zn@xmath0co@xmath1o crystals are shown in fig . [ fig1](a ) . all the diffraction peaks correspond to zno in the wurtzite structure . no secondary phases are detected in the xrd spectrum of zn@xmath0co@xmath1o crystals , which implies that the co@xmath4 ions are incorporated into the lattice substitutionally replacing zn@xmath4 . it can also be seen that the xrd peaks of the zn@xmath0co@xmath1o are slightly shifted to lower angles as compared to the pure zno peaks , which implies that the lattice is slightly expanded relative to pure zno . [ fig1](b ) shows the morphology of a zn@xmath0co@xmath1o single crystal as studied using scanning electron microscopy . this depicts the characteristic hexagonal shape of a zno single crystal . the exposed faces are hexagonal * m * 10@xmath120 and hexagonal * p * 10@xmath121 ( cone ) faces . the precise elemental composition was calculated from the edax spectrum ; the concentration of co in the two samples has been estimated to be @xmath13 and @xmath2 . [ cols="<,^,^,^,^,^,^,^,^,^,^,^ " , ] to study the electronic transport properties of the crystals , the temperature dependent resistance behavior was observed for zn@xmath0co@xmath1o single crystals . electrical contacts were made to individual crystals using silver paste . the 300 k resistivity of the pristine zno crystals is 3.6 @xmath14 cm , and the zn@xmath0co@xmath15o crystals 5.4 @xmath14 cm and 6.7 @xmath14 cm with @xmath16=0.10 and @xmath16=0.02 respectively . the exponential decrease of resistance with increase in temperature as shown in fig . [ fig2](a ) is a clear signature that the cobalt doped zno maintains the semiconducting behavior of undoped zno . the two different slopes of the ln @xmath17 vs 1/t ( inset of fig . [ fig2](a ) ) curve in the low ( 2.5 - 100 k ) and high temperature ( 100 - 300 k ) regions are signatures of two different conduction mechanisms being operative in these two temperature regions . at high temperatures , the conduction mechanism is thermally activated band conduction from donor levels near the conduction band , with an activation energy of 10 mev . @xcite at low temperatures there is insufficient thermal energy to promote conduction through delocalized carriers . instead , below we show that the variable range hopping ( vrh ) conduction mechanism is active at low temperatures , where the electrons hop between levels that are close to the fermi level irrespective of their spatial separation . the relation between electrical conductivity and temperature for vrh is given by mott s equation @xcite @xmath18 the parameters @xmath19 and @xmath20 are given by the following expressions @xcite @xmath21 where @xmath22 is the phonon frequency at the debye temperature , @xmath23 is boltzmann s constant , @xmath24 is the density of localized electronic states at the fermi level and @xmath25 is the inverse localization length . the exponent @xmath26 in eq . [ mottvrh ] varies from @xmath27 when the density of states across the fermi level is constant to @xmath28 in the case of significant electron - electron interactions that lead to the formation of a coulomb gap . from eqs . [ mottvrh ] and [ mottvrhcoeff1 ] we expect @xmath29@xmath30 . the nearly linear plot of @xmath29@xmath31 vs @xmath32 in fig . [ fig2](b ) indicates that vrh is the dominant mechanism of conduction in zn@xmath33co@xmath34o below 100 k. we find similar behavior for both undoped zno and zn@xmath35co@xmath36o crystals . the results of fitting the data for all three samples with the vrh model ( eqs . [ mottvrh ] and [ mottvrhcoeff1 ] ) are reported in table [ table1 ] . the mott parameters , i.e. average hopping distance ( @xmath37 ) and average hopping energy ( @xmath38 ) , were calculated from @xmath39 other conditions for vrh conduction are that the value of @xmath25@xmath37 must be greater than 1 , and that @xmath40 . note that both of these conditions are satisfied in the zn@xmath0co@xmath15o crystals . for the zno crystals the conditions are only satisifed above approximately 25 k , and at lower temperatures other conduction mechanisms may compete with vrh . the values of @xmath41 , @xmath42 and @xmath37 determined by fitting to the vrh model are reasonable compared with results which exist for both doped and undoped zno . @xcite thus , the analysis of the temperature dependent electronic transport data for both the zno and zn@xmath0co@xmath15o crystals provides convincing evidence for the vrh conduction process up to approximately 100 k , above which a thermally activated conduction mechanism becomes more appropriate . measurements were also carried out to study the magnetoresistive behavior of the single crystals up to a magnetic field @xmath43 of 4.5 t and at temperatures down to 2.5 k. figs . [ fig3](a ) and [ fig3](d ) show that for the zn@xmath0co@xmath15o crystals the magnetoresistance ( mr ) is weak and negative ( see inset fig . [ fig3](a ) ) down to a certain temperature . below that temperature the mr becomes positive ( i.e. changes its sign ) and becomes larger in magnitude . the crossover in mr from negative to positive occurs at @xmath1020 k in the doped crystals . it may be noted that the mr increases with the decrease in temperature in all cases , and also with increase in the concentration of cobalt . the largest value attained , near saturation of the positive mr , is 20 % ( fig . [ fig3](d ) ) at 2.5 k for zn@xmath33co@xmath34o single crystals , which is at the upper end of mr values reported by other groups for zno - based dms @xcite . a positive mr also arises in pristine zno crystals below 5 k ( not shown ) , but it is weaker than in the cobalt doped crystals , no more than 1% at 2.5 k , which is likely due to residual impurities . there are several mechanisms which are typically cited to account for the mr in dms at low temperatures : * a negative mr at low fields owing to the magnetic field breaking the time - reversal symmetry of scattering paths involved in weak localization ; * a negative mr at large fields from a reduced scattering of the carriers as the magnetic field aligns the moments of the magnetic dopants ; * a positive mr at intermediate fields from a field - induced spin - splitting of the conduction band due to the s - d exchange interaction . the absence of evidence for the first type , i.e. a sharp negative mr at @xmath440.1 t , as seen in a number of works @xcite , is consistent with the temperature dependent resistance measurements of fig . [ fig2 ] , which indicate a hopping type conduction at low temperatures rather than weak localization of the carriers . the second type of mr is seen in thin films of zno - based dms that exhibit ferromagnetism in magnetic measurements , and the absence of this component at the magnetic fields applied is consistent with the paramagnetic character of the single crystals as measured by squid magnetometry ( discussed below ) . a phenomenological model for the third mechanism of mr has been developed by khosla and fischer @xcite , where the positive mr is proportional to @xmath45 . this model has been used elsewhere to explain the mr behavior of doped zno thin films @xcite . the predicted @xmath45 dependence appears to account for the data of figs . [ fig3](b ) and [ fig3](e ) . however , the khosla - fischer model considers the mr to arise from a field - induced splitting of bands of extended states . in the zn@xmath0co@xmath15o crystals of the present report , conduction is via localized states rather than through bands of extended states . hence , none of the mechanisms of mr commonly applied to dms are applicable in our case , and we must consider an alternate mechanism to explain the low temperature positive mr , based on the vrh conduction process . at low temperatures , where hopping distances are large enough , a positive mr arises due to the shrinking of the wavefunctions of the states available for an electron hop . @xcite in low magnetic fields , the positive mr can then be described with @xmath46 where @xmath47 is the carrier localization length and @xmath26 is the vrh exponent of eq . [ mottvrh ] . this may account for the @xmath45 dependence at low fields as shown in figs . [ fig3](b ) and [ fig3](e ) . at higher fields , there is a change to an approximate @xmath48 behavior , as seen before in thin films of cobalt doped zno in the low temperature hard gap regime of electronic transport @xcite . note that at the lowest temperatures and highest fields , especially for the higher co concentration crystals , the mr is more weakly field dependent than in the low field region . the data of fig . [ fig4 ] show that , at 2.5 k in fields of @xmath491.5 t , the mr of the zn@xmath50co@xmath51o crystals can be approximately described by @xmath52@xmath53 , which is very close to the expected high - field mr in the vrh model of @xmath52@xmath54 @xcite . we thus find that the mr properties of the zn@xmath0co@xmath15o crystals are completely explained as the effect of the magnetic field on the vrh conduction process . the results of magnetic measurements on each crystal type are shown in fig . the magnetization vs field ( @xmath55-@xmath56 ) measurements were carried out at room temperature as well as at 2.5 k. the @xmath55-@xmath56 curves of fig . [ fig5](a ) for cobalt doped samples at 2.5 k , normalized to their respective values at 4.5 t for each sample , exhibit neither remanence nor coercivity . the maximum values of magnetization of the zn@xmath35co@xmath36o and zn@xmath33co@xmath34o crystals in 4.5 t applied field are 2.01 and 2.34 emu / g respectively , corresponding to absolute magnetic moment values of 5@xmath57 - 1@xmath57 emu . in addition , the curves do not saturate , even at the highest applied field . a representative sample of the temperature dependent magnetization ( @xmath55-@xmath58 ) measurements for the crystals , measured under both zero - field cooled ( @xmath59 ) and field - cooled ( @xmath60 ) conditions , is plotted in fig . [ fig5](b ) . a ferromagnetic material , besides exhibiting distinct hysteresis in an @xmath55-@xmath56 loop , also demonstrates thermomagnetic irreversibility ( tmi ) , i.e. , @xmath61@xmath59 in @xmath55-@xmath58 curves . the @xmath62 and @xmath63 curves of fig . [ fig5](b ) are identical , showing no sign of tmi , further confirming the lack of ferromagnetic ordering in the sample ; instead the observed behavior is consistent with paramagnetism . it is thus evident that the zn@xmath0co@xmath15o single crystals are paramagnetic , and doping of co does not result in any ferromagnetism . to confirm that the magnetic behavior observed is due to the cobalt doping , the @xmath55-@xmath56 curves of the pristine zno sample recorded at both 2.5 k and room temperature ( 300 k ) are drawn in the inset of fig . [ fig5](a ) . it may be noted that the data recorded at room temperature exhibit a linear variation of magnetization @xmath55 with applied field , with a negative slope . this is a signature of a diamagnetic material . however , at 2.5 k pristine zno crystals reveal a feeble paramagnetism with a magnetization 100@xmath64 weaker than that of the zn@xmath0co@xmath15o . this weak low - temperature paramagnetism is possibly due to the presence of residual impurities as detected in the mr measurements . in paramagnets , a value of the effective magnetic moment per ion , @xmath65 , can be estimated from the slope ( @xmath66 ) of a @xmath67 vs @xmath58 plot ( curie law fit ) , where @xmath68 is the number of moles of co. the inset of fig . [ fig5](b ) shows that the curie law applies over the full range of temperatures measured . the value of @xmath65 was found to be 3.29 @xmath69 for zn@xmath35co@xmath36o , which compares favorably to values determined for other paramagnetic zno : co samples @xcite and is close to the theoretical free co@xmath4 ion spin - only value of 3.87 @xmath69 . however , at the higher co concentration of @xmath2 , @xmath65 has a lower value of 2.16 @xmath69 . as the co concentration increases , the average co - co distance is reduced , which strengthens the direct antiferromagnetic exchange between co neighbors or superexchange via co - o - co bonds , and consequently the average moment per co is reduced from the case of isolated non - interacting co@xmath4 ions . theoretically , the value of the total moment of paramagnetic co@xmath4 ions is @xmath70(=@xmath71+@xmath72)=3 + 3/2=9/2 . however , for transition metals in semiconductors such as zno it is often found that the orbital angular momentum of the unpaired 3@xmath73 electrons is quenched , i.e. @xmath71=0 , and hence the total moment comes only from the spin momentum @xmath72=3/2 . a brillouin function for the field dependence of the moment of isolated , non - interacting paramagnetic co@xmath4 ions with a spin - only moment of 3/2 is plotted in fig . [ fig5](a ) alongside the magnetic moment data measured at 2.5 k. the calculated field dependence for @xmath72=3/2 paramagnetic ions is extremely close to the experimental data from the cobalt doped samples , confirming that the field dependent magnetization is from a majority of co ions in a divalent , paramagnetic state . the antiferromagnetically correlated fraction of co ions in the crystals is insensitive to the magnetic field . finally , we summarize the state of the incorporated co ions in the single crystals of zno , and their effect on the magnetic and magnetotransport properties . the resistivity and mr measurements point to the presence of localized electronic states , whereas the magnetic properties result from isolated co@xmath4 ions in the crystals . at higher co concentrations , a proportion of the co ions experience some antiferromagnetic exchange , likely with near neighboring co ions , which reduces the measured effective moment per incorporated ion . at low temperatures , charge carriers hop between the localized electronic states , and in the presence of a magnetic field the probability of hopping is reduced as the wavefunctions of the localized states shrink . however , at higher magnetic fields , the paramagnetic co ions progressively align with the field , and a spin dependent vrh conduction process may be expected to lower the resistivity at sufficiently high fields @xcite . no such negative mr is seen in fig . [ fig3 ] , even at the lowest temperature and maximum field , where the @xmath55-@xmath56 data of fig . [ fig5](a ) show that magnetic alignment of the co ions is significant . furthermore , the mr is evidently decoupled from the magnetization of the crystals , as demonstrated by the different shapes of the mr in the two crystals as compared to the identical shape of their @xmath55-@xmath56 curves . this independence of the magnetotransport from the magnetization in the zn@xmath0co@xmath15o crystals implies that the localized states involved in the conduction process may not be spin polarized through alignment of the co ion magnetic moments . although the formation of localized electronic states owing to the incorporation of co in zno is not controversial , our results support the idea that the electronic transport properties in this dms may be only weakly related to the magnetic properties . some results on magnetic tunnel junctions fabricated from ferromagnetic cobalt doped zno have non - saturating positive magnetoresistance at low temperatures @xcite , rather than negative magnetoresistance as would be expected from tunneling of spin - polarized charge carriers between layers with their magnetic moments aligned in high fields . additionally , those authors acknowledged that the behavior of the tunneling magnetoresistance does not correlate with the magnetization of these devices . these results point to the possibility that the existence of ferromagnetism in cobalt doped zno is not a sufficient condition for generation of spin - polarized charge carriers . pristine and cobalt doped zno single crystals were grown by a molten salt solvent technique at low temperatures ( 480 @xmath74c ) . the zn@xmath0co@xmath15o crystals are paramagnetic ; there is no sign of ferromagnetism at up to @xmath2 . the paramagnetic properties of the crystals arise from isolated co@xmath4 ions . at temperatures above 100 k , conduction in the crystals is by thermal activation , with a change to mott variable range hopping at lower temperatures . doped and undoped zno crystals show almost negligible negative mr at room temperature , with a crossover to positive mr below 20 k. the positive mr reaches values as large as 20% in the zn@xmath0co@xmath15o crystals at 2.5 k , and is consistent with a field - induced shrinking of the localized state wavefunctions , which reduces the probability of a hop . the magnetotransport is decoupled from the magnetization , and no spin dependent transport is observed at the highest level of magnetic alignment of the co@xmath4 ions . our results indicate that the co incorporated in low temperature grown single crystals simultaneously introduces paramagnetic moments and localized electronic states into the zno electronic structure , however the bulk magnetic and magnetotransport properties are independent of each other . one of the authors , n.s . , gratefully acknowledges support provided by the exchange agreement between the indian institute of technology delhi , india and the ecole polytechnique fdrale de lausanne , switzerland .

long needle - shaped single crystals of zn@xmath0co@xmath1o were grown at low temperatures using a molten salt solvent technique , up to @xmath2 . the conduction process at low temperatures is determined to be by mott variable range hopping . both pristine and cobalt doped crystals clearly exhibit a crossover from negative to positive magnetoresistance as the temperature is decreased . the positive magnetoresistance of the zn@xmath0co@xmath1o single crystals increases with increased co concentration and reaches up to 20 % at low temperatures ( 2.5 k ) and high fields ( @xmath31 t ) . squid magnetometry confirms that the zn@xmath0co@xmath1o crystals are predominantly paramagnetic in nature and the magnetic response is independent of co concentration . the results indicate that cobalt doping of single crystalline zno introduces localized electronic states and isolated co@xmath4 ions into the host matrix , but that the magnetotransport and magnetic properties are decoupled .

You are an expert at summarizing long articles. Proceed to summarize the following text: today it is widely accepted that the formation of cosmic relaxed objects proceeds via gravitational instability from small to large scales . this clustering process can be studied in detail , following accurately the dynamics of collapse by means of n - body simulations , and statistically , by approximate semi - analytical models which enable a swifter and more efficient exploration of the parameter space . the most complete semi - analytical models developed to date are those by lacey & cole ( 1993 , 1994 , hereafter lc93 and lc94 ) , and by manrique & salvador - sol ( 1995 , 1996 , papers i and ii ) . all the quantities in both models are derived from the statistics of the filtered random gaussian density field , @xmath0/\langle\rho\rangle$ ] , of cold matter encountered at an arbitrary initial epoch , @xmath1 , after recombination when fluctuations are still linear . similarly , in each model the dynamics of dissipationless collapse is approximated by the simple spherical model . these two models differ however in the following aspects . lacey & cole s ( lc ) model is based on the press & schechter ( 1974 , ps ) heuristic prescription for the mass functions of halos , duly extended to obtain the conditional mass function of halos at a given epoch subject to having a larger mass at a later time ( bower 1991 ; bond et al . @xmath2-body simulations show that this model gives highly satisfactory predictions ( lc94 ) . in spite of this remarkable success , there remains the caveat that the seeds of halos are arbitrary points and their associated collapsing clouds undetermined ( possibly disconnected ) regions around them ( cf . lc93 ) . besides , the formation and destruction of halos are arbitrarily and ambiguously defined , leading to poorly motivated formation and destruction times ( see paper ii ) . the model developed by manrique & salvador - sol uses the new cusp ( confluent system of peak trajectories ) formalism , which enables one to follow the _ filtering _ evolution of peaks in the density contrast vs. filtering scale ( @xmath3 vs. @xmath4 ) diagram at @xmath1 , assumed to trace the mass evolution of halos . this model is better founded than the lc model insofar as high peaks are good seeds of massive halos ( bond & myers 1996 ) and do follow spherical collapse ( bernardeau 1994 ) . on the other hand , it provides a natural distinction between accretion and merger which makes the formation and destruction of dark halos well - defined and uniquely determined . as shown in papers i and ii both models provide very similar results for the mass function , the mass increase rate , and the merger rate at large captured masses . however , notable departures were observed in this latter rate at intermediate captured masses . this discrepancy was suspected as being caused by the rather crude approximation used in the cusp model for the density of nested peaks , in connection with the cloud - in - cloud correction . in the present paper we seek to verify this point . we improve the nesting correction by taking into account the correlation among peaks and investigate the effects this has on the cusp model . in 2 we review the foundations of the cusp model developed in papers i and ii . in 3 we derive a more accurate approximation for the density of nested peaks and implement it in 4 . our results are summarized in 5 . comoving lengths are assumed throughout the paper . based on the spherical collapse model , the peak ansatz states that there is a correspondence between non - nested peaks of fixed linear overdensity @xmath5 in the density field smoothed on scale @xmath4 at any arbitrary initial epoch @xmath1 and quasi - steady halos of mass @xmath6 at a time @xmath7 . the mass @xmath6 of the spherical collapsing cloud , or equivalently , of the final halo associated with a peak is an increasing function of the filtering scale @xmath4 , whereas @xmath5 is a decreasing function of @xmath8 . the mass function of halos at @xmath8 , @xmath9 , is therefore simply given by the density of peaks with @xmath5 on scales @xmath4 to @xmath10 , @xmath11 , properly corrected for nesting and transformed to the variables @xmath6 and @xmath8 through the relations @xmath12 and @xmath13 . the approximate expression for the nesting correction used in paper i was ( see next section for justification ) @xmath14 in equation ( [ 1 ] ) , @xmath15 is the mean density of the universe , @xmath16 is the volume , to 0th order in @xmath3 , of the collapsing cloud associated with a peak with @xmath3 on scale @xmath17 , and @xmath18 is the conditional density of peaks with @xmath3 on scales @xmath4 to @xmath10 subject to being located at a point with @xmath3 on scale @xmath19 . this equation is a volterra - type integral equation of the second kind for the scale function @xmath20 of non - nested peaks , which can be readily solved in the standard way from the known functions @xmath21 and @xmath22 derived in paper i ( eqs . [ 11 ] and [ 17 ] ) . the functions @xmath12 and @xmath13 required to obtain @xmath23 from @xmath20 are fixed by simple consistency arguments ( see paper i ) @xmath24 in the above equation @xmath25 is the linear growth factor at the considered collapse time @xmath8 and @xmath26 is the density contrast for collapse at @xmath8 linearly extrapolated to @xmath8 , both functions being dependent on the particular cosmogony assumed . constants @xmath27 and @xmath28 are therefore the only free parameters of the model . for each pair of values of these parameters , the filtering process translates into a different mass evolution of halos , with only one couple of values ( approximately ) recovering the evolution encountered in real gravitational clustering . the correct values of these two parameters can be obtained by adjusting the mass function predicted by the model to the mass function resulting from @xmath2-body simulations at some given time , for example at @xmath29 ( this automatically leads to a similarly good fit at any other time ; see 4 ) . the remaining statistical quantities predicted by the model rely on the definitions of accretion and merger , and halo formation and destruction . since in hierarchical clustering halos grow constantly by capturing other halos , the fact that they are quasi - steady systems implicitly presumes that some of the captures are tiny enough for the steady state of the capturing systems not to be essentially altered . notable captures yielding a transient departure from steadiness establish then the frontier between the initial and final quasi - steady systems . accretion is therefore defined as any apparently continuous and derivable mass increase along the temporal series of quasi - steady systems subtending at each step the mass of those which precede them . in contrast , a merger is any appreciable discontinuity in the mass increase along such a temporal series causing a significant departure from the steadiness of the growing system at that step . a halo is said to survive as long as it evolves by accretion , whereas when it merges it is said to be destroyed . note that when a halo is captured by one that is more massive it merges . however , the capturing halo may just accrete it if the relative captured mass is small enough . only those events in which _ all _ the initial halos merge and are destroyed give rise to the formation of new halos . as shown in paper ii , the assumed correspondence between halos and non - nested peaks allows one to naturally identify the filtering processes at @xmath1 tracing all the preceding processes , as well as to derive their respective rates and characteristic times ( see 4 ) . in the original version of the cusp model the density distribution around peaks was approximated by top - hat spheres . in addition , the conditional density of peaks with @xmath3 on scales @xmath4 to @xmath10 , having @xmath30 on @xmath19 and being located within a peak on that larger scale , was approximated by the conditional density of peaks with @xmath3 on scales @xmath4 to @xmath10 simply having @xmath30 on scale @xmath17 . under these circumstances , the density of peaks with @xmath3 on scales @xmath4 to @xmath10 located within non - nested peaks with @xmath30 on scales @xmath17 to @xmath31 ( @xmath19 ) takes the simple form @xmath32 equation ( [ nnest ] ) leads , for @xmath33 , to the scale function of non - nested peaks ( eq . [ 1 ] ) and , for @xmath34 , to the density of peaks which become nested in an infinitesimal decrement of @xmath3 ( eq . [ nm ] below ) , respectively tracing the mass function of halos and the density of halos merging into more massive ones . any inaccuracy in this expression , therefore , translates into these two quantities and any other that is related . to improve equation ( [ nnest ] ) , we would need to drop the top - hat approximation and take into account the peak - peak correlation . approximate expressions for this latter quantity have been obtained by bardeen et al . ( 1986 ; hereafter bbks ) and regs & szalay ( 1995 ) . unfortunately , these are not useful for our purposes ; being only valid for large separations compared to the filtering scale , while what is needed here is the peak - peak correlation for separations up to @xmath17 . an alternative , more accurate , expression for the density of nested peaks can be obtained , however , from the conditional probability function @xmath35 of finding the density contrast @xmath36 on scale @xmath4 at a separation @xmath37 from a peak with @xmath3 and curvature @xmath38 on that scale ( averaged over the orientations of the second order derivative tensor ) provided by bbks . using this conditional probability function we can write @xmath39 with @xmath37 in units of @xmath40 , and @xmath41 the curvature probability function for non - nested peaks with @xmath30 on scales @xmath17 to @xmath31 . expression ( [ nnestini ] ) takes into account the real density distribution around peaks and , since this includes the distribution of neighboring peaks ( on every scale ) , it also accounts for the peak - peak correlation as desired . it is not exact however , because the conditional mean density of peaks with @xmath3 on scales @xmath4 to @xmath10 subject to having @xmath36 on scale @xmath17 _ at a distance @xmath37 from a peak with @xmath30 and @xmath42 on @xmath17 _ is approximated by @xmath43 , the conditional mean density of peaks with @xmath3 on scales @xmath4 to @xmath10 subject to just having @xmath36 on scale @xmath17 . in the appendix we detail the calculations that allow us to write equation ( [ nnestini ] ) in a compact form identical to equation ( [ nnest ] ) , but for the new function @xmath44 ( eq . [ [ npknest ] ] ) instead of the conditional density function @xmath45 . in figure 1 we plot these two estimates of the density of nested peaks . as can be seen , they are very different for @xmath17 close to @xmath4 while they asymptotically coincide for @xmath46 . this is well understood . there is no correlation among peaks on very different scales . thus the conditional mean density of peaks with @xmath3 on scales @xmath4 to @xmath10 subject to being located at a point with @xmath36 on scale @xmath17 at a distance @xmath37 from a peak with @xmath30 and @xmath42 on the same scale is very well approximated , in this case , by the simple conditional mean density @xmath43 used in equation ( [ nnestini ] ) leading to @xmath44 . on the other hand , the spherical top - hat approximation is also very good in the limit for @xmath46 . hence , not only is the new function @xmath44 entirely accurate at this limit , but it also coincides with the former function @xmath45 . all the quantities predicted by the new version of the cusp model are readily obtained , following the same derivations as in papers i and ii , by simply replacing the function @xmath45 by @xmath47 . this leads to the following expressions . the mass function of halos is @xmath48 with @xmath20 the scale function of non - nested peaks at @xmath3 , solution of the volterra equation @xmath49 in figure 2 we compare this mass function with that obtained in the lc model . the parameters @xmath50 and @xmath27 entering in the cusp model were fixed by adjusting it to the ps mass function in the range @xmath51 m@xmath52 m@xmath53 at @xmath54 , where the latter gives a very good fit to the results of @xmath2-body simulations . both functions are very similar for massive halos at any redshift . only for small masses does the cusp mass function predict slightly fewer halos than that of the ps . note also that the large differences between the new and old approximations for nesting at @xmath55 ( fig . 1 ) have no appreciable effect on the mass function : the original and new solutions being almost exactly superimposed . this indicates that the mass function is particularly insensitive , in the mass range of figure 2 , to the peak - peak correlation . the instantaneous mass accretion rate for halos of mass @xmath6 at @xmath8 is @xmath56 with @xmath57 the second order spectral moment ( eq . [ [ sigmai ] ] ) , and @xmath58 the average inverse curvature for the distribution function @xmath59 . there is no similar prediction in the lc model which does not differentiate accretion from merger . only the total mass increase rates arising from the two models can be compared ( see below ) . in figure 3 we plot the accretion rates obtained from the original and new versions of the cusp model . once again , both solutions are almost fully superimposed showing that this rate is also insensitive to the peak - peak correlation . the instantaneous specific merger rate for halos of mass @xmath6 at @xmath8 per infinitesimal range of the resulting masses @xmath60 is @xmath61 where @xmath62 is the density of peaks with @xmath5 on scales @xmath4 to @xmath10 becoming nested into non - nested peaks with @xmath63 on scales @xmath17 to @xmath31 @xmath64\big|_{\delta_f=\delta}.\label{nm}\ ] ] to compare this specific merger rate with that yielded by the lc model we must take into account the different merger definitions adopted in the two models . in the lc model , merger is any mass capture experienced by halos . in contrast , in the cusp model only notable captures are considered true mergers , while small captures contribute to accretion . as shown in figure 4 , the cusp specific merger rate ( using both the original and new approximations for the density of nested peaks ) fully recovers that of the lc in the asymptotic regime at large @xmath65 , but notably deviates for small @xmath66 . this departure is in the expected direction : for small mass captures the specific merger rate increases monotonically in the lc model , while it shows an abrupt cutoff in the cusp model . from figure 4 we also see that the original and new versions of the cusp model now give markedly different solutions . they coincide at very large @xmath67 , owing to the similarity of the density of nested peaks at @xmath46 , but the old solution begins to decline , deviating from the lc solution , at @xmath67 much greater than unity . this implies that some smooth distinction between tiny and notable captures is already operating at @xmath68 , which is meaningless . this is not the case for the new solution which keeps closer to that of the lc model until @xmath69 , then rapidly falls off implying the existence of a sharp effective frontier between tiny and notable captures at @xmath70 . nonetheless , the new solution is not fully satisfactory : it becomes slightly negative for @xmath67 smaller than the effective threshold for merger , and the latter tends to be unreasonably small or large for extreme values of @xmath6 . this shows the shortcomings of the new approximation used for the density of nested peaks . all the remaining quantities predicted by the cusp model are determined by the three preceding ones which consequently fix the behavior of the whole model . the instantaneous total mass increase rate for halos of mass @xmath6 at @xmath8 , @xmath71 , is equal to the mass accretion rate ( eq . [ mar ] ) plus the mass merger rate , defined as @xmath72 the instantaneous destruction rate ( or global merger rate ) of halos of mass @xmath6 at @xmath8 is @xmath73 while their instantaneous formation rate take the form @xmath74 where @xmath75 is the density of non - nested peaks appearing from @xmath3 to @xmath76 , @xmath77 ( note the error in the expression quoted in paper ii ) , with @xmath78 and @xmath79 defined for a fixed @xmath4 instead of a fixed @xmath3 ( see paper i ) . to see that expressions ( [ fr ] ) and ( [ nfr ] ) can be written in terms of the first three quantities above , one must simply substitute @xmath23 , @xmath80 , and @xmath81 ( eqs . [ mf ] , [ mar ] , and [ dr ] ) into the conservation equation @xmath82-r^{d}[m(t),t]-\partial_m r^a_{mass}(m , t)\bigl|_{m = m(t)},\label{conserv}\ ] ] for the number density of halos per unit mass along mean accretion tracks , @xmath83 , solutions of the differential equation @xmath84 . \label{massr}\ ] ] finally , the distributions ( and typical values ) of formation and destruction times can be readily inferred from the expression giving the cumulative spatial number density of halos at @xmath1 with masses from @xmath85 to @xmath86 surviving until @xmath87 @xmath88\,\delta m(t_i)\,\exp \biggl\{-\int_{t_i}^{t_f } r^{d}[m(t'),t']\,dt'\biggr\ } , \label{nsur}\ ] ] with @xmath83 the mass at @xmath8 of such halos calculated along their mean accretion tracks , and @xmath89 \label{deltam}\ ] ] the element of mass at @xmath1 evolving , through mean accretion tracks , into @xmath90 at @xmath87 . indeed , by taking @xmath91 and @xmath92 in equation ( [ nsur ] ) we obtain the distribution of destruction times of halos at @xmath29 with masses between @xmath93 and @xmath94 , with @xmath95 arbitrarily small , @xmath96\ , \exp\biggl\{-\int_{t_0}^t r^{d}[m(t'),t']\ , dt'\biggr\},\label{phid}\ ] ] while , by taking into account that the cumulative spatial number density of those halos at @xmath29 which preexist at @xmath97 , @xmath98 , coincides with the cumulative spatial number density of halos at @xmath99 surviving until @xmath100 , equations ( [ conserv ] ) and ( [ deltam ] ) lead to the distribution of formation times @xmath101\ , \exp\biggl\{-\int_t^{t_0 } r^{f}[m(t'),t']\,dt'\biggr\}.\label{phif}\ ] ] we have investigated the effects that the inclusion of the spatial correlation among peaks has on the cusp model . the predictions for the mass function of halos , their mass accretion rate , and their specific merger rate for large captured masses , are very similar to those arising from the original version of this model , which were already satisfactory . on the contrary , the specific merger rate at intermediate masses changes appreciably , now becoming more consistent with the corresponding prediction by the lc model . we , therefore , confirm that the poor behavior shown by this latter quantity in the original version of the cusp model was not intrinsic to its foundations , but due to the rather crude approximation originally used for the density of nested peaks . a more accurate expression for the peak - peak correlation than that used here is , however , required for the predicted specific merger rate to be fully satisfactory . an interesting characteristic of the cusp model is that it distinguishes between accretion and merger , which allows one to unambiguously define the formation and destruction of halos respectively as the last and next merger they experience . mergers are those mass captures restructuring the systems , and hence , establishing their morphological properties until a new merger takes place . such a distinction is therefore especially well suited for investigating the origin of the morphology of halos and , by extension , that of galaxies and galaxy clusters . since the ps formalism involves much simpler expressions than those appearing in the peak theory , implementing that distinction in the former would have the added advantage of simplicity . the results of the present work suggest how it should be done . such a modification of the lc model is used , in salvador - sol , solanes , & manrique ( 1997 ) , to study the origin of the correlation between halo concentration and halo mass recently found in cosmological @xmath2-body simulations ( navarro , frenk , & white 1996 ) . the same reasoning leading to equation ( [ nnestini ] ) allows us to write its restriction to peaks with curvature @xmath38 to @xmath102 on scale @xmath4 , @xmath3 on scales @xmath4 to @xmath10 , and @xmath30 on scales @xmath17 to @xmath31 @xmath103 in the preceding expression we have taken the volume of the collapsing clouds associated with peaks delimited by a top - hat window with radius proportional to the filtering scale @xmath4 . likewise , the density function of non - nested peaks with @xmath3 on scales @xmath4 to @xmath10 and curvature between @xmath38 and @xmath102 satisfies the relation @xmath104 similar to equation ( [ corr ] ) . from equations ( [ nnestini ] ) , ( [ b2 ] ) , and ( [ b3 ] ) we have @xmath105 . \label{pxfull } \label{b4}\end{aligned}\ ] ] under the approximation @xmath106 , the curvature moments for this probability function are accurate to within a few percent . since the function @xmath107 is bell - shaped , relatively symmetrical , and close to a gaussian it can be further approximated by a normal distribution with identical mean and variance , @xmath108 and @xmath109 . in taking this approximation we will also extend the domain of integration over @xmath42 in equation ( [ b2 ] ) to the whole range @xmath110 . this does not introduce any appreciable error since @xmath108 is considerably greater than @xmath111 for any reasonable values of @xmath3 and @xmath4 and any realistic spectrum . the distribution function @xmath35 is a normal distribution with mean and variance equal to ( bbks ) @xmath112 and @xmath113 ^ 2= \sigma_0 ^ 2\biggl\{1- { 1\over ( 1-\gamma^2)}\biggl[\psi^2 + \biggl(2\gamma\psi+{\nabla^2\psi\over u^2}\biggr){\nabla^2\psi\over u^2 } \biggr]-5\,\biggl({3\,\psi'\over u^2\,r}-{\nabla^2\psi\over u^2 } \biggr)^2-{3\bigl(\psi'\bigr)^2\over \gamma\,u^2}\biggr\}.\label{var}\ ] ] in the above expressions , @xmath114 , with @xmath37 in units of @xmath40 and @xmath115 the mass correlation function on scale @xmath4 , @xmath116 , @xmath117 , @xmath118 , and @xmath119 with @xmath120 the power spectrum of the density fluctuations . given the normal character of both @xmath121 and @xmath122 , the integration over @xmath42 on the right hand - side of equation ( [ nnestini ] ) can be performed analytically , the result being @xmath123 the new function @xmath124 , equal to the probability function of finding the value @xmath36 at a separation @xmath37 from a peak with @xmath3 in the density field smoothed on scale @xmath4 , is a normal distribution with mean @xmath125 and variance @xmath126 ^ 2= \sigma_0 ^ 2\biggl\{1- { 1\over ( 1-\gamma^2)}\biggl[\psi^2 + \biggl(2\gamma\psi+{\nabla^2\psi\over u^2}\biggr){\nabla^2\psi\over u^2 } \biggr]-5\,\biggl({3\,\psi'\over u^2\,r}-{\nabla^2\psi\over u^2 } \biggr)^2-{3\bigl(\psi'\bigr)^2\over \gamma\,u^2}\nonumber\\ + { ( \delta x)^2\over ( 1-\gamma^2)^2 } \biggl(\gamma\psi+ { \nabla^2\psi\over u^2}\biggr)^2\biggr\}. \label{b7}\end{aligned}\ ] ] this enables us to perform the integration over @xmath36 on the right hand - side of equation ( [ b5 ] ) also analytically , obtaining @xmath127 where @xmath128 \over ( 2\pi)^2r_*^3 } { \displaystyle { \exp\biggl\ { -{[\nu-\epsilon(r)\nu'(r)]^2 \over { 2\,[1-\epsilon^2(r)]}}\biggr\ } } \over { [ 1-\epsilon^2(r)]^{1/2}}}\,{\sigma_2\over \sigma_0}\,r\,dr\label{npkr}\ ] ] is the conditional density of peaks with @xmath3 on scales @xmath4 to @xmath10 subject to being located at a separation @xmath37 from a peak with @xmath30 on scales @xmath17 to @xmath31 . in equation ( [ npkr ] ) @xmath129 , and @xmath130 $ ] is defined just as @xmath131 $ ] in paper i ( eq.[17 ] ) , but for the @xmath37 dependence in @xmath132 and @xmath133 introduced through @xmath134 ^ 2\over\sigma_0 ^ 2(r')}\biggr\}^{1/2 } , \qquad\qquad\nu'(r)={\overline{\delta'(r)}\over \sigma_0(r ' ) } \biggl\{1-{[\delta\delta'(r)]^2\over\sigma_0 ^ 2(r')}\biggr\}^{-1/2}. \label{nur}\ ] ] therefore , by defining @xmath135 equation ( [ b8 ] ) adopts the desired compact form similar to equation ( [ nnest ] ) @xmath136 bardeen , j.m . , bond , j.r . , kaiser , n. , & szalay , a.s . 1986 , , 304 , 15 ( bbks ) bernardeau , f. 1994 , , 427 , 51 bond , j.r . , cole , s. , efstathiou , g. , kaiser , n. 1991 , , 379 , 440 bond , j.r . , & myers , s.t . 1996 , , 103 , 41 bower , r.j . , 1991 , , 248 , 332 lacey , c. , & cole , s. 1993 , , 262 , 627 ( lc93 ) lacey , c. , & cole , s. 1994 , , 271 , 676 ( lc94 ) manrique , a. , & salvador - sol , e. 1995 , , 453 , 6 ( paper i ) manrique , a. , & salvador - sol , e. 1996 , , 467 , 504 ( paper ii ) navarro , j. , frenk , c. , & white , s.d.m . 1996 , , 462 , 563 press , w.h . , & schechter , p. 1974 , , 187 , 425 ( ps ) regs , e. , & szalay , a.s . , 1995 , , 272 , 447 salvador - sol , solanes , j.m . , & manrique , a. 1997 , , submitted

in two previous papers a semi - analytical model was presented for the hierarchical clustering of halos via gravitational instability from peaks in a random gaussian field of density fluctuations . this model is better founded than the extended press - schechter model , which is known to agree with numerical simulations and to make similar predictions . the specific merger rate , however , shows a significant departure at intermediate captured masses . the origin of this was suspected as being the rather crude approximation used for the density of nested peaks . here , we seek to verify this suspicion by implementing a more accurate expression for the latter quantity which accounts for the correlation among peaks . we confirm that the inclusion of the peak - peak correlation improves the specific merger rate , while the good behavior of the remaining quantities is preserved . # 1*#1 *

You are an expert at summarizing long articles. Proceed to summarize the following text: the science of networks is a modern discipline spanning the natural , social and computer sciences , as well as engineering @xcite . networks , or graphs , consist of _ vertices _ and _ edges_. an edge typically connects a pair of vertices . networks occur in an huge variety of contexts . facebook , for instance , is a large social network , where more than one billion people are connected via virtual acquaintanceships . another famous example is the internet , the physical network of computers , routers and modems which are linked via cables or wireless signals ( fig . [ figgennet ] ) . many other examples come from biology , physics , economics , engineering , computer science , ecology , marketing , social and political sciences , etc .. most networks of interest display _ community structure _ , i. e. , their vertices are organised into groups , called _ communities _ , _ clusters _ or _ modules_. in fig . [ figsfi ] we show a collaboration network of scientists working at the santa fe institute ( sfi ) in santa fe , new mexico . vertices are scientists , edges join coauthors . edges are concentrated within groups of vertices representing scientists working on the same research topic , where collaborations are more natural . likewise , communities could represent proteins with similar function in protein - protein interaction networks , groups of friends in social networks , websites on similar topics on the web graph , and so on . identifying communities may offer insight on how the network is organised . it allows us to focus on regions having some degree of autonomy within the graph . it helps to classify the vertices , based on their role with respect to the communities they belong to . for instance we can distinguish vertices totally embedded within their clusters from vertices at the boundary of the clusters , which may act as brokers between the modules and , in that case , could play a major role both in holding the modules together and in the dynamics of spreading processes across the network . community detection in networks , also called _ graph _ or _ network clustering _ , is an ill - defined problem though . there is no universal definition of the objects that one should be looking for . consequently , there are no clear - cut guidelines on how to assess the performance of different algorithms and how to compare them with each other . on the one hand , such ambiguity leaves a lot of freedom to propose diverse approaches to the problem , which often depend on the specific research question and ( or ) the particular system at study . on the other hand , it has introduced a lot of noise into the field , slowing down progress . in particular , it has favoured the diffusion of questionable concepts and convictions , on which a large number of methods are based . this work presents a critical analysis of the problem of community detection , intended to practitioners but accessible to readers with basic notions of network science . it is not meant to be an exhaustive survey . the focus is on the general aspects of the problem , especially in the light of recent findings . also , we discuss some popular classes of algorithms and give advice on their usage . more info on network clustering can be found in several review articles @xcite . the contents are organised in three main sections . section [ sec - comm ] deals with the concept of community , describing its evolution from the classic subgraph - based notions to the modern statistical interpretation . next we discuss the critical issue of validation ( section [ sec - valid ] ) , emphasising the role of artificial benchmarks , the importance of the choice of partition similarity scores , the conditions under which clusters are detectable , the usefulness of metadata and the structural peculiarities of communities in real networks . section [ sec - meth ] hosts a critical discussion of some popular clustering approaches . it also tackles important general methodological aspects , such as the determination of the number of clusters , which is a necessary input for several techniques , the possibility to generate robust solutions by combining multiple partitions , the main approaches to discover dynamic communities , as well as the assessment of the significance of clusterings . in section [ sec - soft ] we indicate where to find useful software . the concluding remarks of section [ sec - ol ] close the work . we start with a subgraph @xmath0 of a graph @xmath1 . the number of vertices and edges are @xmath2 , @xmath3 for @xmath1 and @xmath4 , @xmath5 for @xmath0 , respectively . the adjacency matrix of @xmath6 is @xmath7 , its element @xmath8 equals @xmath9 if vertices @xmath10 and @xmath11 are neighbours , otherwise it equals @xmath12 . we assume that the subgraph is connected because communities usually are . other types of group structures do not require connectedness ( section [ sec - mv ] ) . the subgraph is schematically illustrated in fig . [ figstruct ] . its vertices are enclosed by the dashed contour . the magenta dots are the external vertices connected to the subgraph , while the black ones are the remaining vertices of the network . the blue lines indicate the edges connecting the subgraph to the rest of the network . the _ internal _ and _ external _ degree @xmath13 and @xmath14 of a vertex @xmath10 of the network with respect to subgraph @xmath0 are the number of edges connecting @xmath10 to vertices of @xmath0 and to the rest of the graph , respectively . both definitions can be expressed in compact form via the adjacency matrix @xmath7 : @xmath15 and @xmath16 , where the sums run over all vertices @xmath11 inside and outside @xmath17 , respectively . naturally , the degree @xmath18 of @xmath10 is the sum of @xmath13 and @xmath14 : @xmath19 . if @xmath20 and @xmath21 @xmath10 has neighbours only within @xmath0 and is an _ internal vertex _ of @xmath0 ( dark green dots in the figure ) . if @xmath22 and @xmath21 @xmath10 has neighbours outside @xmath0 and is a _ boundary vertex _ of @xmath0 ( bright green dots in the figure ) . if @xmath23 , instead , the vertex is disjoint from @xmath0 . the _ embeddedness _ @xmath24 is the ratio between the internal degree and the degree of vertex @xmath10 : @xmath25 . the larger @xmath26 , the stronger the relationship between the vertex and its community . the _ mixing parameter _ @xmath27 is the ratio between the external degree and the degree of vertex @xmath10 : @xmath28 . by definition , @xmath29 . now we present a number of variables related to the subgraph as a whole . we distinguish them in three classes . the first class comprises measures based on internal connectedness , i. e. , on how cohesive the subgraph is . the main variables are : * _ internal degree _ @xmath30 . the sum of the internal degrees of the vertices of @xmath0 . it equals twice the number @xmath5 of internal edges , as each edge contributes two units of degree . in matrix form , @xmath31 * _ average internal degree _ @xmath32 . average degree of vertices of @xmath0 , considering only internal edges : @xmath33 . * _ internal edge density _ @xmath34 . the ratio between the number of internal edges of @xmath0 and the number of all possible internal edges : @xmath35 we remark that @xmath36 is the maximum number of internal edges that a simple graph with @xmath4 vertices may have . the second class includes measures based on external connectedness , i. e. , on how embedded the subgraph is in the network or , equivalently , how separated the subgraph is from it . the main variables are : * _ external degree _ , or _ cut _ , @xmath37 . the sum of the external degrees of the vertices of @xmath0 . it gives the number of external edges of the subgraph ( blue lines in fig . [ figstruct ] ) . in matrix form , @xmath38 . * _ average external degree _ , or _ expansion _ , @xmath39 . average degree of vertices of @xmath0 , considering only external edges : @xmath40 . * _ external edge density _ , or _ cut ratio _ , @xmath41 . the ratio between the number of external edges of @xmath0 and the number of all possible external edges : @xmath42 finally , we have hybrid measures , combining internal and external connectedness . notable examples are : * _ total degree _ , or _ volume _ , @xmath43 . the sum of the degrees of the vertices of @xmath0 . naturally , @xmath44 . in matrix form , @xmath45 . * _ average degree _ @xmath46 . average degree of vertices of @xmath0 : @xmath47 . * _ conductance _ @xmath48 . the ratio between the external degree and the total degree of @xmath0 : @xmath49 all definitions we have given hold for the case of undirected and unweighted networks . the extension to weighted graphs is straightforward , as it suffices to replace the number of edges " with the sum of the weights carried by every edge . for instance , the internal degree @xmath50 of a vertex @xmath51 becomes the _ internal strength _ @xmath52 , which is the sum of the weights of the edges joining @xmath51 with the vertices of subgraph @xmath17 . for the internal and external edge densities of eqs . ( [ eq1 ] ) and ( [ eq2 ] ) one would have to replace the numerators with their weighted counterparts and multiply the denominators by the average edge weight @xmath53 , where @xmath54 is the element of the _ weight matrix _ , indicating the weight of the edge joining vertices @xmath10 and @xmath11 ( @xmath55 if @xmath10 and @xmath11 are disconnected ) and @xmath3 the total number of graph edges . in tables [ tab : metricsv ] and [ tab : metrics ] we list all variables we have presented along with their extensions to the case of weighted networks . in directed networks one would have to distinguish between incoming and outgoing edges . extensions of the metrics are fairly simple to implement , though their usefulness is unclear . figure [ figcom ] shows how scholars usually envision community structure . the network has three clusters and in each cluster the density of edges is comparatively higher than the density of edges between the clusters . this can be summarised by saying that communities are dense subgraphs which are well separated from each other . this view has been challenged , recently @xcite , as we shall see in section [ ncp ] . communities may overlap as well , sharing some of the vertices . for instance , in social networks individuals can belong to different circles at the same time , like family , friends , work colleagues . figure [ figoverl ] shows an example of a network with overlapping communities . communities are typically supposed to be overlapping at their boundaries , as in the figure . recent results reveal a different picture , though @xcite ( section [ ncp ] ) . a subdivision of a network into overlapping communities is called _ cover _ and one speaks of _ soft clustering _ , as opposed to _ hard clustering _ , which deals with divisions into non - overlapping groups , called _ partitions_. the generic term _ clustering _ can be used to indicate both types of subdivisions . covers can be _ crisp _ , when shared vertices belong to their communities with equal strength , or _ fuzzy _ , when the strength of their membership can be different in different clusters . the oldest definitions of community - like objects were proposed by social network analysts and focused on the internal cohesion among vertices of a subgraph @xcite . the most popular concept is that of _ clique _ @xcite . a clique is a _ complete graph _ , that is , a subgraph such that each of its vertices is connected to all the others . it is also a maximal subgraph , meaning that it is not included in a larger complete subgraph . in modern network science it is common to call clique any complete graph , not necessarily maximal . triangles are the simplest cliques . finding cliques is an * np*-complete problem @xcite ; a popular technique is the bron kerbosch method @xcite . the notion of cliques , albeit useful , can not be considered a good candidate for a community definition . while a clique has the largest possible internal edge density , as all internal edges are present , communities are not complete graphs , in general . moreover , all vertices have identical role in a clique , while in real network communities some vertices are more important than others , due to their heterogeneous linking patterns . therefore , in social network analysis the notion has been relaxed , generating the related concepts of _ n - cliques _ @xcite , _ n - clans _ and _ n - clubs _ @xcite . other definitions are based on the idea that a vertex must be adjacent to some minimum number of other vertices in the subgraph . @xmath56-plex _ is a maximal subgraph in which each vertex is adjacent to all other vertices of the subgraph except at most @xmath56 of them @xcite . details on the above definitions can be found in specialised books @xcite . for a proper community definition , one should take into account both the internal cohesion of the candidate subgraph and its separation from the rest of the network . a simple idea that has received a great popularity is that a community is a subgraph such that the number of internal edges is larger than the number of external edges " . this idea has inspired the following definitions . an _ @xmath57-set _ @xcite , or _ strong community _ @xcite , is a subgraph such that the internal degree of each vertex is greater than its external degree . a relaxed condition is that the internal degree of the subgraph exceeds its external degree [ _ weak community _ @xcite ] to be a weak community it is not necessary that the number of internal edges @xmath5 exceeds that of external edges @xmath37 . since the internal degree @xmath58 ( section [ sec - var ] ) the actual condition is @xmath59 . ] . a strong community is also a weak community , while the converse is not generally true . a drawback of these definitions is that one separates the subgraph at study from the rest of the network , which is taken as a single object . but the latter can be in turn divided into communities . if a subgraph @xmath0 is a proper community , it makes sense that each of its vertices is more strongly attached to the vertices of @xmath0 than to the vertices of any other subgraph . this concept , proposed by hu et al . @xcite , is more in line ( though not entirely ) with the modern idea of community that we discuss in the following section . it has generated two alternative definitions of strong and weak community . a subgraph @xmath0 is a strong community if the internal degree of any vertex within @xmath0 exceeds the internal degree of the vertex within any other subgraph , i. e. , the number of edges joining the vertex to those of the subgraph ; likewise , a community is weak if its internal degree exceeds the ( total ) internal degree of its vertices within every other community . a strong ( weak ) community la radicchi et al . is a strong ( weak ) community also in the sense of hu et al .. the opposite is not true , in general ( fig . [ figsw ] ) . in particular , a subgraph can be a strong community in the sense of hu et al . even though all of its vertices have internal degree smaller than their respective external degree . the above definitions of communities use _ extensive _ variables : their value tends to be the larger , the bigger the community ( e. g. , the internal and external degrees ) . but there are also variables discounting community size . an example is the internal cluster density @xmath60 of eq . ( [ eq1 ] ) . one could assume that a subgraph @xmath0 with @xmath56 vertices is a cluster if @xmath60 is larger than a threshold @xmath61 . setting the size of the subgraph is necessary because otherwise any clique would be among the best possible communities , including trivial two - cliques ( simple edges ) or triangles . as we have seen in the previous section , traditional definitions of community rely on counting edges ( internal , external ) , in various ways . but what one should be really focusing on is the _ probability _ that vertices share edges with a subgraph . the existence of communities implies that vertices interact more strongly with the other members of their community than they do with vertices of the other communities . consequently , there is a preferential linking pattern between vertices of the same group . this is the reason why edge densities end up being higher within communities than between them . we can formulate that by saying that vertices of the same community have a higher probability to form edges with their partners than with the other vertices . let us suppose that we estimated the edge probabilities between all pairs of vertices , somehow . we can define the groups by means of those probabilities . it is a scenario similar to the classic one we have seen in section [ sec - defs ] , where we add and compare probabilities , instead of edges . natural definitions of strong and weak community are : * a _ strong community _ is a subgraph each of whose vertices has a higher probability to be linked to every vertex of the subgraph than to any other vertex of the graph . * a _ weak community _ is a subgraph such that the average edge probability of each vertex with the other members of the group exceeds the average edge probability of the vertex with the vertices of any other group . ] . the difference between the two definitions is that , in the concept of strong community , the inequality between edge probabilities holds at the level of every pair of vertices , while in the concept of weak community the inequality holds only for averages over groups . therefore , a strong community is also a weak community , but the opposite is not true , in general . now we can see why the former definitions of strong and weak community @xcite are not satisfactory . suppose to have a network with two subgraphs @xmath7 and @xmath62 of very different sizes , say with @xmath63 and @xmath64 vertices ( fig . [ figexsw ] ) . and @xmath62 and edge probabilities @xmath65 between vertices of the subgraphs and @xmath66 between vertices of @xmath7 and @xmath62 . the red circle is a representative vertex of subgraph @xmath7 , the smaller blue circle represents the rest of the vertices of @xmath7 . the subgraphs are both strong and weak communities in the probabilistic sense , but they may be neither strong nor weak communities according to the classic definitions by radicchi et al . @xcite and hu et al . @xcite , if @xmath62 is sufficiently larger than @xmath7 . ] the network is generated by a model where the edge probability is @xmath65 between vertices of the same group and @xmath67 for vertices of different groups . the two subgraphs are communities both in the strong and in the weak sense , according to the probability - based definitions above . the expected internal degree of a vertex of @xmath7 is @xmath68 : since there are @xmath63 possible internal neighbours , but for simplicity one allows for the formation of self - edges , from a vertex to itself . results obtained with and without self - edges are basically undistinguishable , when community sizes are much larger than one . we shall stick to this setup throughout the paper . ] . likewise , the expected external degree of a vertex of @xmath7 is @xmath69 . the expected internal and external degrees of @xmath7 are @xmath70 and @xmath71 . for any two values of @xmath65 and @xmath67 one can always choose @xmath72 sufficiently larger than @xmath63 that @xmath73 , which also implies that @xmath74 . in this setting the subgraphs are neither strong nor weak communities , according to the definitions proposed by radicchi et al . and hu et al .. how can we compute the edge probabilities between vertices ? this is still an ill - defined problem , unless one has a model stating how edges are formed . one can make many hypotheses on the process of edge formation . + for instance , if we take social networks , we can assume that the probability that two individuals know each other is a decreasing function of their geographical distance , on average @xcite . each set of assumptions defines a model . for our purposes , eligible models should take into account the possible presence of groups of vertices , that behave similarly . the most famous model of networks with group structure is the _ stochastic block model _ ( sbm ) suppose we have a network with @xmath2 vertices , divided in @xmath75 groups . the group of vertex @xmath10 is indicated with the integer label @xmath76 . the idea of the model is very simple : the probability @xmath77 that vertices @xmath10 and @xmath11 are connected depends exclusively on their group memberships : @xmath78 . therefore , it is identical for any @xmath10 and @xmath11 in the same groups . the probabilities @xmath79 form a @xmath80 symmetric matrix , called the _ stochastic block matrix_. the diagonal elements @xmath81 ( @xmath82 ) of the stochastic block matrix are the probabilities that vertices of block @xmath56 are neighbours , whereas the off - diagonal elements give the edge probabilities between different blocks between blocks @xmath83 and @xmath84 is fixed ( @xmath85 ) , instead of the edge probabilities . if @xmath86 the two models are fully equivalent if the edge probabilities @xmath87 are chosen such that the expected number of edges running between @xmath83 and @xmath84 coincides with @xmath88 . ] . for @xmath89 , with @xmath90 , we recover community structure , as the probabilities that vertices of the same group are connected exceed the probabilities that vertices of different groups are joined ( fig . [ fig : stylized - hotdog ] ) . it is also called _ assortative structure _ , as it privileges bonds between vertices of the same group . the model is very versatile , though , and can generate various types of group structure . for @xmath91 , with @xmath90 , we have _ disassortative structure _ , as edges are more likely between the blocks than inside them ( fig . [ fig : stylized - bipartite ] ) . in the special case in which @xmath92 we recover _ multipartite structure _ , as there are edges only between the blocks . if @xmath93 , @xmath94 , we have _ core - periphery structure _ : the vertices of the first block ( core ) are relatively well - connected amongst themselves as well as to a peripheral set of vertices that interact very little amongst themselves ( fig . [ fig : stylized - coreper ] ) . if all probabilities are equal , @xmath95 , @xmath96 , we recover the classic random graph la erds and rnyi @xcite ( fig . [ fig : stylized - random ] ) . here any two vertices have identical probability of being connected , hence there is no group structure . this has become a fundamental axiom in community detection , and has inspired some popular techniques like , e. g. , modularity optimisation @xcite ( section [ sec - modopt ] ) . random graphs of this type are also useful in the validation of clustering algorithms ( section [ art - bench ] ) . alternative community definitions are based on the interplay between network topology and dynamics . diffusion is the most used dynamics . random walks are the simplest diffusion processes . a simple random walk is a path such that the vertex reached at step @xmath97 is a random neighbour of the vertex reached at step @xmath98 . a random walker would be spending a long time within communities , due to the supposedly low number of routes taking out of them @xcite . the evolution of random walks does not depend solely on the number or density of edges , in general , but also on the structure and distribution of paths formed by consecutive edges , as paths are the routes that walkers can follow . this means that random walk dynamics relies on _ higher - order structures _ than simple edges , in general . such relationship is even more pronounced when one considers markov dynamics of second order or higher , in which the probability of reaching a vertex at step @xmath99 of the walk does not depend only on where the walker sits at step @xmath97 , but also on where it was at step @xmath98 and possibly earlier @xcite . indeed , one could formulate the network clustering problem by focusing on higher order structures , like _ motifs _ ( e. g. , triangles ) @xcite . the advantage is that one can preserve more complex features of the network and its communities , which typically get lost when one uses network models solely based on edge probabilities , like sbms . the drawback is that calculations become more involved and lengthy . is a definition of community really necessary ? actually not , most techniques to detect communities in networks do not require a precise definition of community . the problem can be attacked from many angles . for instance , one can remove the edges separating the clusters from each other , that can be identified via some particular feature @xcite . but defining clusters beforehand is a useful starting point , that allows one to check the reliability of the final results . in this section we will discuss the crucial issue of validation of clustering algorithms . validation usually means checking how precisely algorithms can recover the communities in benchmark networks , whose community structure is known . benchmarks can be computer - generated , according to some model , or actual networks , whose group structure is supposed to be known via non - topological features ( metadata ) . the lack of a universal definition of communities makes the search for benchmarks rather arbitrary , in principle . nevertheless , the best known artificial benchmarks are based on the modern definition of clusters presented in section [ sec - mv ] . we shall present some popular artificial benchmarks and show that partition similarity measures have to be handled with care . we will see under which conditions communities are detectable by methods , and expose the interplay between topological information and metadata . we will conclude by presenting some recent results on signatures of community structure extracted from real networks . the principle underneath stochastic block models ( section [ sec - mv ] ) has inspired many popular benchmark graphs with group structure . community structure is recovered in the case in which the probability for two vertices to be joined is larger for vertices of the same group than for vertices of different groups ( fig . [ fig : stylized - hotdog ] ) . for simplicity , let us suppose that there are only two values of the edge probability , @xmath65 and @xmath100 , for edges within and between communities , respectively . furthermore , we assume that all communities have identical size @xmath101 , so @xmath102 , where @xmath75 is the number of communities . in this version , the model coincides with the _ planted l - partition model _ , introduced in the context of graph partitioning ) ] . ] the expected internal and external degrees of a vertex are @xmath103 and @xmath104 , respectively , yielding an expected ( total ) vertex degree @xmath105 . girvan and newman @xcite set @xmath106 , @xmath107 ( for a total number of vertices @xmath108 ) and fixed the average total degree @xmath109 to @xmath110 . this implies that @xmath111 and @xmath65 and @xmath112 are not independent parameters . the benchmark by girvan and newman is still the most popular in the literature ( fig . [ figgn ] ) . performance plots of clustering algorithms typically have , on the horizontal axis , the expected external degree @xmath113 . for low values of @xmath113 communities are well separated is higher , to keep the total degree constant . ] and most algorithms do a good job at detecting them . by increasing @xmath113 , the performance declines . still , one expects to do better than by assigning memberships to the vertices at random , as long as @xmath114 , which means for @xmath115 . in section [ detectab ] we will see that the actual threshold is lower , due to random fluctuations . the benchmark by girvan and newman , however , is not a good proxy of real networks with community structure . for one thing , all vertices have equal degree , whereas the degree distribution of real networks is usually highly heterogeneous @xcite . in addition , most clustering techniques find skewed distributions of community sizes @xcite . for this reason , lancichinetti , fortunato and radicchi proposed the _ lfr benchmark _ , having power - law distributions of degree and community size @xcite ( fig . [ figlfr ] ) . the mixing parameters @xmath116 of the vertices ( section [ sec - var ] ) are set equal to a constant @xmath117 , which estimates the quality of the partition is actually only the average of the mixing parameter over all vertices . in fact , since degrees are integer , it is impossible to tune them such to have exactly the same value of @xmath117 for each vertex , and keep the constraint on the degree distribution at the same time . ] . lfr benchmark networks are built by joining stubs at random , once one has established which stubs are internal and which ones are external with respect to the community of the vertex attached to the stubs . in this respect , it is basically a configuration model @xcite with built - in communities . clearly , when @xmath117 is low , clusters are better separated from each other , and easier to detect . when @xmath117 grows , performance starts to decline . but for which range of @xmath117 can we expect a performance better than random guessing ? let us suppose that the group structure is detectable for @xmath118 $ ] . the upper limit @xmath119 should be such that the network is random for @xmath120 . the network is random when stubs are combined at random , without distinguishing between internal and external stubs , which yields the standard configuration model . there the expected number of edges between two vertices with degrees @xmath18 and @xmath121 is @xmath122 , @xmath3 being the total number of network edges . let us focus on a generic vertex @xmath10 , belonging to community @xmath17 . we denote with @xmath123 and @xmath124 the sum of the degrees of the vertices inside and outside @xmath17 , respectively . clearly @xmath125 . in a random graph built with the configuration model , vertex @xmath10 would have an expected internal degree . ] @xmath126 and an expected external degree @xmath127 . since , by construction , @xmath128 and @xmath129 , the community @xmath17 is not real when @xmath130 and @xmath131 , which implies @xmath132 . we see that @xmath123 depends on the community @xmath17 : the larger the community , the lower the threshold is . therefore , not all clusters are detectable at the same time , in general . for this to happen , @xmath117 must be lower than the minimum of @xmath133 over all communities : @xmath134 . if communities are all much smaller than the network as a whole , @xmath135 and @xmath119 could get very close to the upper limit @xmath9 of the variable @xmath117 . however , it is possible that the actual threshold is lower than @xmath119 , due to the perturbations of the group structure induced by random fluctuations ( section [ detectab ] ) . anyway , in most cases the threshold is going to be quite a bit higher than @xmath136 , the value which is mistakenly considered as the threshold by some scholars . the lfr benchmark turns out to be a special version of the recently introduced degree - corrected stochastic block model @xcite , with the degree and the block size distributed according to truncated power laws , which is local , a global parameter @xmath137 is used , estimating how strong the community structure is . ] . the lfr benchmark has been extended to directed and weighted networks with overlapping communities @xcite . the extensions to directed and weighted graphs are rather straightforward . overlaps are obtained by assigning each vertex to a number of clusters and distributing its internal edges equally among them . recently , another benchmark with overlapping communities has been introduced by ball , karrer and newman @xcite . it consists of two clusters @xmath7 and @xmath62 , with overlap @xmath17 . vertices in the non - overlapping subsets @xmath138 and @xmath139 set edges only between each other , while vertices in @xmath17 are connected to vertices of both @xmath7 and @xmath62 . the expected degree of all vertices is set equal to @xmath140 . the authors considered various settings , by tuning @xmath109 , the size of the overlap and the sizes of @xmath7 and @xmath62 , which may be uneven . however , the fact that all vertices have equal degree ( on average ) makes the model less realistic and flexible than the lfr benchmark . following the increasing availability of evolving time - stamped network data sets , the analysis and modelling of temporal networks have received a lot of attention lately @xcite . in particular , scholars have started to deal with the problem of detecting evolving communities ( section [ sec - dynclus ] ) . a benchmark designed to model dynamic communities was proposed by granell et al . it is based on the planted @xmath141-partition model , just like the benchmark of girvan and newman , where @xmath65 and @xmath67 are the edge probabilities within communities and between communities , respectively . communities may grow and shrink ( fig . [ figdb]a ) , they may merge with each other or split into smaller clusters ( fig . [ figdb]b ) , or do all of the above ( fig . [ figdb]c ) . the dynamics unfold such that at each time the subgraphs are proper communities in the probabilistic sense discussed in section [ sec - mv ] . in the merge - split dynamics , clusters actually merge before the inter - community edge probability @xmath112 reaches the value @xmath65 of the intra - community edge probability , due to random fluctuations ( section [ detectab ] ) . . reprinted figure with permission from @xcite . 2015 , by the american physical society . ] in section [ sec - mv ] we have shown why random graphs can not have a meaningful group structure . that means that they can be employed as _ null benchmarks _ , to test whether algorithms are capable to recognise the absence of groups . many methods find non - trivial communities in such random networks , so they fail the test . we strongly encourage doing this type of exam on new algorithms @xcite . the accuracy of clustering techniques depends on their ability to detect the clusters of networks , whose community structure is known . that means that the partition detected by the method(s ) has to match closely the planted partition of the network . how can the similarity of partitions be computed ? this is an important problem , with no unique solution . in this section we discuss some issues about partition similarity measures . more information can be found in @xcite , @xcite and @xcite . let us consider two partitions @xmath142 and @xmath143 of a network @xmath6 , with @xmath144 and @xmath145 clusters , respectively . let @xmath2 be the total number of vertices , @xmath146 and @xmath147 the number of vertices in clusters @xmath148 and @xmath149 and @xmath150 the number of vertices shared by clusters @xmath148 and @xmath149 : @xmath151 . the @xmath152 matrix @xmath153 whose entries are the overlaps @xmath150 is called _ confusion matrix _ , _ association matrix _ or _ contingency table_. most similarity measures can be divided in three categories : measures based on _ pair counting _ , _ cluster matching _ and _ information theory_. pair counting means computing the number of pairs of vertices which are classified in the same ( different ) clusters in the two partitions . let @xmath154 indicate the number of pairs of vertices which are in the same community in both partitions , @xmath155 ( @xmath156 ) the number of pairs of elements which are in the same community in @xmath157 ( @xmath158 ) and in different communities in @xmath158 ( @xmath157 ) and @xmath159 the number of pairs of vertices that are in different communities in both partitions . several measures can be defined by combining the above numbers in various ways . a famous example is the _ rand index _ @xcite @xmath160 which is the ratio of the number of vertex pairs correctly classified in both partitions ( i. e. either in the same or in different clusters ) , by the total number of pairs . another notable option is the _ jaccard index _ @xcite , @xmath161 which is the ratio of the number of vertex pairs classified in the same cluster in both partitions , by the number of vertex pairs classified in the same cluster in at least one partition . the jaccard index varies over a broader range than the rand index , due to the dominance of @xmath159 in @xmath162 , which typically confines the rand index to a small interval slightly below @xmath9 . both measures lie between @xmath12 and @xmath9 . if we denote with @xmath163 and @xmath164 the sets of vertex pairs with are members of the same community in partitions @xmath157 and @xmath158 , respectively , the jaccard index is just the ratio between the intersection and the union of @xmath163 and @xmath164 . such concept can be used as well to determine the similarity between two clusters @xmath7 and @xmath62 @xmath165 the score @xmath166 is also called jaccard index and is the most general definition of the score , for any two sets @xmath7 and @xmath62 @xcite . measuring the similarity between communities is very important to determine , given different partitions , which cluster of a partition corresponds to which cluster(s ) of the other(s ) . for instance , the cluster @xmath149 of @xmath158 corresponding to cluster @xmath148 of @xmath158 is the one maximising the similarity between @xmath148 and @xmath149 , e. g. , @xmath167 . this strategy is also used to track down the evolution of communities in temporal networks @xcite . the rand and the jaccard indices , as defined in eqs . ( [ eqt04 ] ) and ( [ eqt20 ] ) , have the disturbing feature that they do not take values in the entire range @xmath168 $ ] . for this reason , adjusted versions of both indices exist , in that a baseline is introduced , yielding the expected values of the score for all pairs of partitions @xmath169 and @xmath170 obtained by randomly assigning vertices to clusters such that @xmath169 and @xmath170 have the same number of clusters and the same size for all clusters of @xmath157 and @xmath158 , respectively @xcite . the baseline is subtracted from the unadjusted version , and the result is divided by the range of this difference , yielding @xmath9 for identical partitions and @xmath12 as expected value for independent partitions . but there are problems with these definitions as well . the null model used to compute the baseline relies on the assumption that the communities of the independent partitions have the same number of vertices as in the partitions whose similarity is to be compared . but such assumption usually does not hold , in practical instances , as algorithms sometimes need the number of communities as input , but they never impose any constraint on the cluster sizes . adjusted indices have also the disadvantage of nonlocality @xcite : the similarity between partitions differing only in one region of the network depends on how the remainder of the network is subdivided . moreover , the adjusted scores can take negative values , when the unadjusted similarity lies below the baseline . a better option is to use _ standardised indices _ @xcite : for a given score @xmath171 the value of the null model term @xmath116 is computed along with its standard deviation @xmath172 over many different randomisations of the partitions @xmath157 and @xmath158 . by computing the @xmath173-score @xmath174 we can see how non - random the measured similarity score is , and assess its significance . it can be shown that the @xmath173-scores for the jaccard , rand and adjusted rand indices coincide @xcite , so the measures are statistically equivalent . since the actual values @xmath171 of these indices differ for the same pair of partitions , in general , we conclude that the magnitudes of the scores may give a wrong perception about the effective similarity . cluster matching aims at establishing a correspondence between pairs of clusters of different partitions based on the size of their overlap . a popular measure is the _ fraction of correctly detected vertices _ , introduced by girvan and newman @xcite . a vertex is correctly classified if it is in the same cluster as at least half of the other vertices in its cluster in the planted partition . if the detected partition has clusters given by the merger of two or more groups of the planted partition , all vertices of those clusters are considered incorrectly classified . the number of correctly classified vertices is then divided by the number @xmath2 of vertices of the graph , yielding a number between @xmath12 and @xmath9 . the recipe to label vertices as correctly or incorrectly classified is somewhat arbitrary . the fraction of correctly detected vertices is similar to @xmath175 where @xmath176 is the index of the best match @xmath177 of cluster @xmath178 @xcite . a common problem of this type of measures is that partitions whose clusters have the same overlap would have the same similarity , regardless of what happens to the parts of the communities which are unmatched . the situation is illustrated schematically in fig . [ figsimclus ] . partitions @xmath179 and @xmath180 are obtained from @xmath181 by reassigning the same fraction of their elements to the other clusters . their overlaps with @xmath181 are identical and so are the corresponding similarity scores . however , in partition @xmath180 the unmatched parts of the clusters are more scrambled than in @xmath179 , which should be reflected in a lower similarity score . similarity can be also estimated by computing , given a partition , the additional amount of information that one needs to have to infer the other partition . if partitions are similar , little information is needed to go from one to the other . such extra information can be used as a measure of dissimilarity . to evaluate the shannon information content @xcite of a partition , we start from the community assignments @xmath182 and @xmath183 , where @xmath184 and @xmath185 indicate the cluster labels of vertex @xmath10 in partition @xmath157 and @xmath158 , respectively . the labels @xmath186 and @xmath187 are the values of two random variables @xmath188 and @xmath189 , with joint distribution @xmath190 , so that @xmath191 and @xmath192 . the _ mutual information _ @xmath193 of two random variables is @xmath194 , where @xmath195 is the shannon entropy of @xmath188 and @xmath196 is the conditional entropy of @xmath188 given @xmath189 . the mutual information is not ideal as a similarity measure : for a given partition @xmath157 , all partitions derived from @xmath157 by splitting ( some of ) its clusters would all have the same mutual information with @xmath157 , even though they could be very different from each other . in this case the mutual information equals the entropy @xmath197 , because the conditional entropy is zero . it is then necessary to introduce an explicit dependence on the other partition , that persists even in those special cases . this has been achieved by introducing the _ normalized mutual information _ ( nmi ) , obtained by dividing the mutual information by the arithmetic average and @xmath198 @xcite . ] of the entropies of @xmath157 and @xmath158 @xcite @xmath199 the nmi equals @xmath9 if and only if the partitions are identical , whereas it has an expected value of @xmath12 if they are independent . since the first thorough comparative analysis of clustering algorithms @xcite , the nmi has been regularly used to compute the similarity of partitions in the literature . however , the measure is sensitive to the number of clusters @xmath200 of the detected partition , and may attain larger values the larger @xmath200 , even though more refined partitions are not necessarily closer to the planted one . this may give wrong perceptions about the relative performance of algorithms @xcite . a more promising measure , proposed by meil @xcite is the _ variation of information _ ( vi ) @xmath201 the vi defines a metric in the space of partitions as it has the properties of distance ( non - negativity , symmetry and triangle inequality ) . it is a local measure : the vi of partitions differing only in a small portion of a graph depends on the differences of the clusters in that region , and not on how the rest of the graph is subdivided . the maximum value of the vi is @xmath202 , which implies that the scores of an algorithm on graphs of different sizes can not be compared with each other , in principle . one could divide @xmath203 by @xmath202 @xcite , to force the score to be in the range @xmath168 $ ] , but the actual span of values of the measure depends on the number of clusters of the partitions . in fact , if the maximum number of communities is @xmath204 , with @xmath205 , @xmath206 . consequently , in those cases where it is reasonable to set an upper bound on the number of clusters of the partitions , the similarities between planted and detected partitions on different graphs become comparable , and it is possible to assess both the performance of an algorithm and to compare algorithms across different benchmark graphs . we stress , however , that the measure may not be suitable when the partitions to be compared are very dissimilar from each other @xcite and that it shows unintuitive behaviour in particular instances @xcite . so far we discussed of comparing partitions . what about covers ? extensions of the measures we have presented to the case of overlapping communities are usually not straightforward . the _ omega index _ @xcite is an extension of the adjusted rand index @xcite . let @xmath157 and @xmath158 be covers of the same graph to be compared . we denote with @xmath207 the number of pairs of vertices occurring together in exactly @xmath11 communities in both covers . it is a natural generalisation of the variables @xmath159 and @xmath154 we have seen above , where @xmath11 can also be larger than @xmath9 since a pair of vertices can now belong simultaneously to multiple communities . the variable @xmath208 is the fraction of pairs of vertices belonging to the same number of communities in both covers ( including the case @xmath209 , which refers to the pairs not being in any community together ) . the omega index is defined as @xmath210 where @xmath211 is the expected value of @xmath212 according to the null model discussed earlier , in which vertex labels are randomly reshuffled such to generate covers with the same number and size of the communities . the nmi has also been extended to covers by lancichinetti , fortunato and kertsz @xcite . the definition is non - trivial : the community assignments of a cover are expressed by a vectorial random variable , as each vertex may belong to multiple clusters at the same time . the measure overestimates the similarity of two covers , in special situations , where intuition suggests much lower values . the problem can be solved by using an alternative normalisation , as shown in @xcite . unfortunately neither the definition by lancichinetti , fortunato and kertsz nor the one by mcdaid , greene and hurley are proper extensions of the nmi , as they do not coincide with the classic definition of eq . ( [ eqt08 ] ) when partitions in non - overlapping clusters are compared . however , the differences are typically small , and one can rely on them in practice . esquivel and rosvall have proposed an actual extension @xcite . following the comparative analysis performed in @xcite , the nmi by lancichinetti , fortunato and kertsz has been regularly used in the literature , also in the case of regular partitions , without overlapping communities . if covers are fuzzy ( section [ sec - defs ] ) , the similarity measures above can not be used , as they do not take into account the degree of membership of vertices in the communities they belong to . a suitable option is the _ fuzzy rand index _ @xcite , which is an extension of the adjusted rand index . both the fuzzy rand index and the omega index coincide with the adjusted rand index when communities do not overlap . for temporal networks , a nave approach would be comparing partitions ( or covers ) corresponding to configurations of the system in the same time window , and to see how this score varies across different time windows . however , this does not tell if the clusters are evolving in the same way , as there would be no connection between clusterings at different times . a sensible approach is comparing sequences of clusterings , by building a confusion matrix that takes into account multiple snapshots . this strategy allows one to define dynamic versions of various indices , like the nmi and the vi @xcite . in conclusion , while there is no clear - cut criterion to establish which similarity measure is best , we recommend to use measures based on information theory . in particular , the vi seems to have more potential than others , for the reasons we explained , modulo the caveats in refs . there are currently no extensions of the vi to handle the comparison of covers , but it would not be difficult to engineer one , e. g. , by following a similar procedure as in @xcite , though this might cost the sacrifice of some of its nice features . one should keep in mind that the choice of one similarity index or another is a sensitive one , and warped conclusions may be drawn when different measures are adopted . in fig . [ figcompsim ] we show the accuracy of two algorithms on the lfr benchmark ( section [ art - bench ] ) : _ ganxis _ , a method based on label propagation @xcite and _ linkcommunities _ , a method based on grouping edges instead of vertices @xcite ( section [ sec - lc ] ) . the accuracy is estimated with the nmi by lancichinetti , fortunato and kertsz @xcite ( left diagram ) and with the omega index [ eq . ( [ eq_omega1 ] ) ] ( right diagram ) . from the left plot one would think that ganxis clearly outperforms linkcommunities , whereas from the right plot ganxis still prevails for @xmath117 until about @xmath213 ( though the curves are closer to each other than in the nmi plot ) and linkcommunities is better for larger values of @xmath117 . vertices , average degree @xmath214 , maximum degree @xmath215 , exponents @xmath216 and @xmath9 for the degree and community size distributions and range @xmath217 $ ] for the community size.,title="fig : " ] vertices , average degree @xmath214 , maximum degree @xmath215 , exponents @xmath216 and @xmath9 for the degree and community size distributions and range @xmath217 $ ] for the community size.,title="fig : " ] in validation procedures one assumes that , if the network has clusters , there must be a way to identify them . therefore , if we do not manage , we have to blame the specific clustering method(s ) adopted . but are we certain that clusters are always detectable ? most networks of interest are _ sparse _ , i. e. , their average degree is much smaller than the number of vertices . this means that the number of edges of the graph is much smaller than the number of possible edges @xmath218 . a more precise way to formulate this is by saying that a graph is sparse when , in the limit of infinite size , the average degree of the graph remains finite . a number of analytical calculations can be carried out by using network sparsity . many algorithms for community detection only work on sparse graphs . on the other hand , sparsity can also give troubles . due to the very low density of edges , small amounts of noise could perturb considerably the structure of the system . for instance , random fluctuations in sparse graphs could trick algorithms into finding groups that do not really exist ( section [ sec - sign ] ) . likewise , they could make actual groups undetectable . let us consider the simplest version of the assortative stochastic block model , which matches the planted partition model ( section [ art - bench ] ) . there are @xmath75 communities of the same size @xmath219 , and only two values for the edge probability : @xmath65 for pairs of vertices in the same group and @xmath112 for pairs of vertices in different groups . since the graphs are sparse , @xmath65 and @xmath112 vanish in the limit of infinite graph size . so we shall use the expected internal and external degrees @xmath220 and @xmath221 , which stay constant in that limit . by construction , the groups are communities so long as @xmath222 or , equivalently , for @xmath223 . but that does not mean that they are always detectable . in principle , dealing with the issue of detectability involves examining all conceivable clustering techniques , which is clearly impossible . fortunately , it is not necessary , because we know what model has generated the communities of the graphs we are considering . the most effective technique to infer the groups is then fitting the stochastic block model on the data ( _ a posteriori block modelling _ ) . this can be done via the maximum likelihood method @xcite . in recent work @xcite , decelle et al . have shown that , in the limit of infinite graph size , the partition obtained this way is correlated with the planted partition whenever @xmath224 which implies @xmath225 so , given a value of @xmath113 , when @xmath226 is in the range @xmath227 $ ] the probability @xmath228 of classifying a vertex correctly is not larger than the probability @xmath229 of assigning the vertex to a randomly chosen group , although the groups are communities , according to the model . we stress that this result only holds when the graphs are sparse : if @xmath65 and @xmath112 remain non - zero in the large - n limit ( dense graph ) , the classic detectability threshold @xmath114 is correct . a fortiori , no clustering technique can detect the clusters better than random assignment when the inference of the model parameters fails to do so . if communities are searched via the spectral optimisation of newman - girvan s modularity @xcite , one obtains the same threshold of eq . ( [ eqdetect1 ] ) @xcite , provided the network is not too sparse . for the benchmark of girvan and newman ( section [ art - bench ] ) @xcite it has long been unclear where the actual detectability limit sits . girvan - newman benchmark graphs are not infinite , their size being set to @xmath230 , so there is no proper detectability transition , but rather a smooth crossover from a regime in which clusters are frequently detectable to a regime where they are frequently undetectable . for this reason there can not be a sharp threshold separating the two regimes . still it is useful to have an idea of where the pattern changes . in the following we shall still use the term threshold to refer to the crossover point . in the beginning , scholars thought that clusters are detectable as long as they satisfy the definition of strong community by radicchi et al . @xcite ( section [ sec - defs ] ) , i. e. , as long as the expected internal degree exceeds the expected external degree , yielding a threshold @xmath231 @xcite . since the expected total degree of a vertex is set to @xmath110 , communities are detectable as long as @xmath232 . it soon became obvious that the actual threshold should be the one of the modern " definition of community we have presented in section [ sec - mv ] , according to which the condition coincide . ] is @xmath114 , that is @xmath233 . however , numerical calculations reveal that algorithms tend to fail long before that limit . from eq . ( [ eqdetect1 ] ) we see that for the case of four infinite clusters and total expected degree @xmath234 , the theoretical detectability limit is @xmath235 . in fig . [ figdet ] we see the performance on the benchmark of three well - known algorithms : louvain @xcite , a greedy optimisation technique of newman girvan modularity @xcite ( section [ sec - modopt ] ) ; infomap , which is based on random walk dynamics @xcite ( section [ sec - dynmet ] ) ; oslom , that searches for clusters via a local optimisation of a significance score @xcite . the accuracy is estimated via the fraction of correctly detected vertices ( section [ sim - me ] ) . the three thresholds @xmath236 , @xmath237 and @xmath238 are represented by vertical lines . the performance of all methods becomes comparable with random assignment well before @xmath237 . the theoretical limit @xmath238 appears to be compatible with the performance curves . indicates the threshold corresponding to the concept of strong community la radicchi et al . , the dashed line at @xmath239 the threshold according to the probability - based definition of strong community we have given in section [ sec - mv ] . the baseline of random assignment is @xmath240 ( horizontal dashed line ) . all algorithms do not do better than random assignment already before @xmath239 . the theoretical detectability limit is at @xmath241 , in the limit of groups of infinite size . ] graph sparsity is a necessary condition for clusters to become undetectable , but it is not sufficient . the symmetry of the model we have considered plays a major role too . clusters have equal size and vertices have equal degree . this helps to confuse " algorithms . if communities have unequal sizes and the degree of vertices are correlated with the size of their communities , so that vertices have larger degree , the bigger their clusters , community detection becomes easier , as the degrees can be used as proxy for group membership . in this case , the non - trivial detectability limit disappears when there are four clusters or fewer , while it persists up to a given extent of group size inequality when there are more than four clusters @xcite . other types of block structure , like core - periphery , do not suffer from detectability issues @xcite . lfr benchmark graphs are more complex models than the one studied in @xcite and it is not clear whether there is a non - trivial detectability limit , though it is unlikely , due to the big heterogeneity in the distribution of vertex degree and community size . another standard way to test clustering techniques is using real networks with known community structure . knowledge about the memberships of the vertices typically comes from _ metadata _ , i. e. , non - structural information . if vertices are annotated communities are assumed to be groups of vertices with identical tags . examples are user groups in social networks like livejournal and product categories for co - purchasing networks of products of online retailers such as amazon . in fig . [ zach ] we show the most popular of such benchmark graphs , zachary karate club network @xcite . it consists of @xmath242 vertices , the members of a karate club in the united states , who were observed over a period of three years . edges connect individuals interacting outside the activities of the club . eventually a conflict between the club president ( vertex @xmath242 ) and the instructor ( vertex @xmath9 ) led to the fission of the club in two separate groups , whose members supported the instructor and the president , respectively ( indicated by the colours ) . indeed , the groups make sense topologically : vertices @xmath9 and @xmath242 are hubs , and most members are directly connected to either of them . most algorithms of community detection have been tested on this network , as well as others , e. g. , the american college football network @xcite or lusseau s network of bottlenose dolphins @xcite . the idea is that the method doing the best job at recovering groups with identical annotations would also be the most reliable in applications . such idea , however , is based on a questionable principle , i. e. , that the groups corresponding to the metadata are also communities in the topological sense we have discussed in section [ sec - comm ] . communities exist because their vertices are supposed to be similar to each other , in some way . the similarity among the vertices is then revealed topologically through the higher edge probability among pairs of members of the same group than between pairs of members of different groups , whose similarity is lower . hence , when one is provided with annotations or other sources of information that allows to classify vertices based on their similarity , one expects that such similarity - based classes are also the best communities that structure - based algorithms may detect . indeed , for some small networks like zachary s karate club this seems to be the case . but for quite some time scholars could not test this hypothesis , due to the limited number of suitable data sets . over the past few years this has finally become possible , due to the availability of several large network data sets with annotated vertices @xcite . it turns out that the alignment between the communities found by standard clustering algorithms and the annotated groups is not good , in general . in fig . [ tmhric ] we show the similarity between the topological partitions found by different methods and the annotated partitions , for several social , information and technological networks @xcite . the heights of the vertical bars are the values of the normalised mutual information ( nmi ) @xcite . groups of contiguous bars represent the scores for a given data set . to the left of the vertical dashed line we see the results for classic benchmarks , like lfr graphs ( section [ art - bench ] ) @xcite , zachary karate club , etc . , and the scores are generally good . but for the large data sets to the right of the line the scores are rather low , signalling a significant mismatch between topological and annotated communities . for amazon co - purchasing network @xcite , in which vertices are products and edges are set between products often purchased by the same customer(s ) , the similarity is quite a bit higher than for the other networks . this is because the classification of amazon products is hierarchical ( e. g. , ` books / fiction / fantasy ` ) , so there are different levels of annotated communities , and the reported scores refer to the one which is most similar to the structural ones detected by the algorithms , while the other levels would give lower similarity scores . low similarity at the partition level does not rule out that some communities of the structural partition significantly overlap with their annotated counterparts , but precision and recall scores show that this is not the case . results depend more on the network than on the specific method adopted , none of which appears to be particularly good on any ( large ) data set . so the hypothesis that structural and annotated clusters are aligned is not warranted , in general . there can be multiple reasons for that . the attributes could be too general or too specific , yielding communities which are too large or too small to be interesting . moreover , while the best partition of the network delivered by an algorithm can be poorly correlated with the metadata , there may be alternative topological divisions that also belong to a set of valid solutions , according to the algorithm , but happen to be better correlated with the annotations @xcite . the fact that structural and annotated communities may not be strongly correlated has important consequences . scholars have been regularly testing their algorithms on small annotated graphs , like zachary s karate club , by tuning parameters such to obtain the best possible performance on them . this is not justified , in general , as it makes sense only when there is a strong correspondence , which is a priori unknown . also , forcing an alignment with annotations on one data set does not guarantee that there is going to be a good alignment with the annotations of a different network . besides , one of the reasons why people use clustering algorithms is to provide an improved classification of the vertices , by using structure . if one obtained the same thing , why bother ? the right thing to do is using structure _ along with _ the annotations , instead of insisting on matching them . this way the information coming from structure and metadata can be combined and we can obtain more accurate partitions , if there is any correspondence between them . recent approaches explicitly assume that the metadata ( or a portion thereof ) are either exactly or approximately correlated with the best topological partition @xcite . a better approach is not assuming a priori that the metadata correlate with the structural communities . the goal is quantifying the relationship between metadata and community and use it to improve the results . if there is no correlation , the metadata would be ignored , leaving us with the partition derived from structure alone . methods along these lines have been developed , using stochastic block models . newman and clauset @xcite have proposed a model in which vertices are initially assigned to clusters based on metadata , and then edges are placed between vertices according to the degree - corrected stochastic block model @xcite . hric et al . have designed a similar model @xcite , in which the interplay between structure and metadata is represented by a multilayer network ( fig . [ dameta ] ) . the generative model is an extension of the hierarchical stochastic block model ( sbm ) @xcite with degree - correction for the case with edge layers @xcite . at ( 0,0 ) ; at ( 0,0 ) ; here the metadata is not supposed to correspond simply to a partition of the vertices . the majority of data sets contain rich metadata , with vertices being annotated multiple times , and often few vertices possess the exact same annotations and can be thus associated to the same group . in addition , while the number of communities is required as input by the method of newman and clauset , here it is inferred from the data . finally , it is also possible to assess the metadata in its power to predict the network structure , not only their correlation with latent partitions . this way it is possible to predict _ missing vertices _ of the network , i. e. , to infer the connections of a vertex from its annotations only . we stress that neither method requires that all vertices are annotated . applications of the method by hric et al . @xcite reveal that in many data sets there are statistically significant correlations between the annotations and the network structure , while in some cases the metadata seems to be largely uncorrelated with structural clusters . we conclude that network metadata should not be used indiscriminately as ground truth for community detection methods . even when the metadata is strongly predictive of the network structure , the agreement between the annotations and the network division tends to be complex , and very different from the one - to - one mapping that is more commonly assumed . moreover , data sets usually contain considerable noise in their annotations , and some metadata tags are essentially random , with no relationship to structure . artificial benchmark graphs are certainly very useful to assess the performance of clustering algorithms . however , one could always question whether the model of community structure they propose is reliable . how can we assess this ? in order to characterise real " communities we have to find them first . but that can only be done via some algorithm , and different algorithms search for different types of objects , in general . still , one may hope that general properties of communities can be consistently uncovered across different methods and data sets , while other features are more closely tied to the specific method(s ) used to detect the clusters and ( or ) the specific data set at study ( or classes thereof ) . a seemingly robust feature of communities in real networks is the heterogeneity of their size distribution . most clustering techniques find skewed distributions of cluster sizes in many networks . so , there appears to be no characteristic size for a community : small communities usually coexist with large ones . this feature is rather independent of the type of network ( fig . [ figcomsize ] ) . it may signal a hierarchy among communities , with small clusters included in large ones . methods unable to distinguish between hierarchical levels might find blended " partitions , consisting of communities of different levels and hence of very different sizes . the lfr benchmark was the first graph model to take explicitly into account the heterogeneity of community sizes ( section [ art - bench ] ) . another interesting question is how the quality of communities depends on their size . leskovec et al . @xcite carried out a systematic analysis of clusters in many large networks , including traditional and on - line social networks , technological , information networks and web graphs . instead of considering partitions , they focused on individual communities , which are derived by optimising conductance ( section [ sec - var ] ) around seed vertices . we remind that the conductance of a cluster is the ratio between the number of external edges and the total degree of the cluster . minimising conductance effectively combines the two main community demands , i. e. , good separation from the rest of the graph ( low numerator ) and large number of internal edges ( high denominator ) . the measure is also relatively insensitive to the size of the clusters , as both the numerator and the denominator are typically proportional to the number of vertices of the community . therefore one could use it to compare the quality of clusters of different sizes . for any given size @xmath56 leskovec et al . identified the subgraph with @xmath56 vertices with the lowest conductance . this way , for each network one can draw the _ network community profile _ ( ncp ) , showing the minimum conductance as a function of community size . the ncps of all networks studied by leskovec et al . have a characteristic shape : they go downwards till @xmath243 vertices , and then they rise monotonically for larger subgraphs [ fig . [ figlesk ] ( left ) ] . alternative shapes have been recently found for other data sets @xcite . for networks characterised by ncps like the one in fig . [ figlesk ] ( left ) the most pronounced communities are fairly small in size . such small clusters are weakly connected to the rest of the network , often by a single edge ( in this case they are called _ whiskers _ ) , and form the _ periphery _ of the graph . large clusters have comparatively lower quality and melt into a big _ core_. large communities can often be split in parts with lower conductance , so they can be considered conglomerates of smaller communities . a schematic picture of the resulting network structure is shown in fig . [ figlesk ] ( right ) . the shape of the ncp is fairly independent of the specific technique adopted to identify the subgraphs with minimum conductance . the different shapes of the ncps encountered in data suggest that core - periphery is not the only model of group structure of real networks @xcite . the ncp is a signature that can be used to select generative mechanisms of community structure . indeed , many standard models typically yield ncp sloping steadily downwards , at odds with the ones encountered in many social and information networks . stochastic block models ( section [ sec - mv ] ) are sufficiently versatile that they can reproduce the ncp shape of fig . [ figlesk ] ( left ) , by suitably tuning the parameters . in the standard lfr benchmark ( section [ art - bench ] ) the mixing parameters are tightly concentrated about a value @xmath117 by construction , hence all clusters have approximately conductance @xmath117 , yielding a roughly flat ncp , it can be easily shown that the conductance of the subgraph is exactly equal to @xmath117 . ] however , the model can be easily modified by making @xmath117 community - dependent and a large variety of ncps are attainable , including the one of fig . [ figlesk ] ( left ) . the main problem of working with ncps is that they are based on extreme statistics , as one systematically reports the minimum conductance for a given cluster size . how representative is this extremal subgraph of the population of subgraphs with the same size ? there may be just a few clusters of a given size with low conductance . it may happen that many subgraphs have conductance near the minimum corresponding to their size(s ) , which would then be representative . alternatively most subgraphs might have much larger conductance than the minimum but low enough that they can be still considered communities . in this case one should conclude that communities of that size are not of very high quality , on average . the above scenarios might lead to different conclusions about the actual community structure of the system . in general , even if one could produce a version of the ncp where the trend refers to representative samples of communities of equal size ( whatever that means ) , the actual values of the conductance are as important as the shape of the curve . if conductance is sufficiently low for all cluster sizes , it means that there are good communities of any size . the fact that small clusters could be of higher quality does not undermine the role of large clusters . the observation that large clusters consist of smaller clusters of higher quality may just be evidence of hierarchical structure in the network , which is a trademark of many complex systems @xcite . in that case high levels of the hierarchy are not less important than low ones , a priori . in fact , the actual relative importance of communities should not only come from the sheer value of specific metrics , like conductance , but also from their statistical significance ( section [ sec - sign ] ) . that notwithstanding , we strongly encourage analyses like the one by leskovec et al . , as they provide a statistical characterisation of community structure , in a way that is only weakly algorithm - dependent . one has to define operationally what a cluster is , but in a simple intuitive way that allows us to draw conclusions about the structure of the graph . in principle one could do the same by analysing the clusters delivered by any algorithm , but there would be two important drawbacks . first , the clusters may not be easy to interpret , as most clustering algorithms usually do not require a clear - cut definition of community . second , one would have to handle a partition of the network in communities , instead of probing locally the group structure of the network . therefore , for a given vertex one would have only one cluster ( or a handful , if communities overlap ) , while a local exploration allows to analyse a whole population of candidate subgraphs , which gives more information . the local subgraphs recovered this way do not need to be strongly matching the clusters delivered by any algorithm , but they provide useful signatures that allow to restrict the set of possible model explanations for the network s group structure . such investigation can be replicated on any model graph to check whether the results match ( e. g. , whether the ncps coincide ) . [ fig : noover ] + + another approach to infer properties of clusters of real networks is using annotations . while we have shown that annotated clusters do not necessarily coincide with structural ones ( section [ topmet ] ) , general features can be still derived , provided they are consistently found across different data sets and annotations . a recent analysis by yang and leskovec has questioned the common picture of networks with overlapping communities @xcite . scholars usually assume that clusters overlap at their boundaries , hence edge density should be larger in the non - overlapping parts ( fig . [ figoverl ] ) . instead , by analysing the overlaps of annotated clusters in large social and information networks , yang and leskovec found that the probability that two vertices are connected is larger in the overlaps , and grows with the number of communities sharing that pair of vertices . in addition , _ connector vertices _ , i. e. , vertices with the largest number of neighbours within a community , are more likely to be found in the overlaps . these findings suggest that the overlaps may play an important role in the community structure of networks . in fig . [ fig : agm ] we compare the conventional view with the one resulting from the analysis . the _ community - affiliation graph model _ ( agm ) @xcite and the _ cluster affiliation model for big networks _ ( bigclam ) @xcite are clustering techniques based on generative models of networks featuring communities with dense overlaps . the models are based on the principle that vertices are more likely to be neighbours the more the communities sharing them , in line with the empirical finding of @xcite . actual overlapping communities exist in many contexts . however , it is unclear whether soft clustering is statistically founded . a recent analysis aiming at identifying suitable stochastic block models to describe real network data indicate that in many cases hard partitions ought to be preferred , as they give simpler descriptions of the group structure of the data than soft partitions @xcite . this could be due to the fact that the underlying models are based on placing edges independently of each other , neglecting higher order structures between vertices , like motifs . by adopting approaches that take into account higher - order structures things may change and community overlaps might become a statistically robust feature . the pervasive overlaps found by yang and leskovec in annotated data can be found if higher order effects are considered @xcite , without ad hoc hypotheses . there are many algorithms to detect communities in graphs . they can be grouped in categories , based on different criteria , like the actual operational method @xcite , or the underlying concept of community @xcite . in most applications , however , just a few popular algorithms are employed . in this section we present a critical analysis of these methods . we show the advantages of knowing the number of clusters before - hand and how it is possible to derive robust solutions from partitions delivered by stochastic clustering techniques . we discuss approaches to the problem of detecting communities in evolving networks and how to assess the significance of the detected clustering . we conclude by suggesting the methods that currently appear to be most promising . in general , the only preliminary information available to any algorithm is the structure of the network , i. e. , which pairs of vertices are connected to each other and which are not ( possibly including weights ) . any insight about community structure is supposed to be given as output of the procedure . naturally , it would be valuable to have some information on the unknown division of the network beforehand , as one could reduce considerably the huge space of possible solutions , and increase the chance of successfully identifying the communities . among all the possible pre - detection inputs , the number @xmath75 of clusters plays a prominent role . many popular classes of algorithms require the specification of @xmath75 before they run , like methods imported from data clustering or parametric statistical inference approaches ( section [ sec - inference ] ) . other methods are capable to infer @xmath75 as they can choose among partitions into different numbers of communities . but even such methods could benefit from a preliminary knowledge of @xmath75 @xcite . in fig . [ fignumclus ] we report standard accuracy plots of two algorithms on the planted partition model ( section [ art - bench ] ) with two clusters of equal size . the algorithms are modularity optimisation via simulated annealing @xcite and the absolute potts model ( apm ) @xcite ( section [ sec - dynmet ] ) . there are two performance curves for each method : one comes from the standard application of the method , without constraints ; the other is obtained by forcing the method to explore only the subset of partitions with the correct number of clusters @xmath93 . vertices , which are grouped in two equal - sized communities . the accuracy is measured via the fraction of correctly detected vertices ( section [ sim - me ] ) . the horizontal line indicates the accuracy of random guessing , the dashed vertical line the theoretical detectability limit ( section [ detectab ] ) . for each algorithm we show two curves , referring to the results of the method in the absence of any information on the number of clusters , and when such information is fed into the model as initial input . in both cases , knowing the number of clusters beforehand leads to a much better performance . ] we see that the accuracy improves considerably when @xmath75 is known . this is particularly striking in the case of modularity optimisation , which is known to have a limited resolution , preventing the method from identifying the correct scale of the communities , even when the latter are very pronounced @xcite ( section [ sec - modopt ] ) . knowing @xmath75 and constraining the optimisation of the measure to partitions with fixed @xmath75 , the problem can be alleviated @xcite . but how do we know how many clusters there are ? here we briefly discuss some heuristic techniques , for statistically principled methods we defer the reader to section [ sec - inference ] . it has been recently shown that in the planted partition model @xmath75 can be correctly inferred all the way up to the detectability limit from the spectra of two matrices : the _ non - backtracking matrix _ @xmath244 @xcite and the _ flow matrix _ @xmath245 @xcite . they are @xmath246 matrices , where @xmath3 is the number of edges of the graph . each edge is considered in both directions , yielding @xmath247 directed edges and indicated with the notation @xmath248 , meaning that the edge goes from vertex @xmath10 to vertex @xmath11 . their elements read @xmath249 and @xmath250 in eq . ( [ eqfm ] ) @xmath18 is the degree of vertex @xmath10 . so the elements of @xmath245 are basically the elements of @xmath244 , normalised by vertex degrees . this is done to account for the heterogeneous degree distributions observed in most real networks . both matrices have non - zero elements only for each pair of edges forming a directed path from the first vertex of one edge to the second of the other edge . to do that , edges have to be incident at one vertex . as a matter of fact , the non - backtracking matrix @xmath244 is just the adjacency matrix of the ( directed ) edges of the graph . the name of the matrix @xmath251 is due to a connection with the properties of non - backtracking walks . non - backtracking walk _ @xcite is a path across the edges of a graph that is allowed to return to a vertex visited previously only after at least two other vertices have been visited ; immediate returns like @xmath252 are forbidden . the elements of the @xmath56-th power of @xmath251 yield the number of non - backtracking walks of length @xmath56 from a ( directed ) edge of the graph to another and the trace of the power matrix the number of closed non - backtracking walks of length @xmath56 starting from any given ( directed ) edge . a remarkable property of both matrices is that on networks with homogeneous groups ( i. e. , of similar size and internal edge density ) most eigenvalues , which are generally complex , are enclosed by a circle centred at the origin , and that the number of eigenvalues lying outside of the circle is a good proxy of the number of communities of the network @xcite . for @xmath244 the circle s radius is given by the square root @xmath253 of the leading eigenvalue @xmath137 , which may diverge for networks with heterogeneous degree distributions ( e. g. , power laws ) ; for @xmath245 it equals @xmath254 , which is never greater than @xmath9 . unfortunately , computing the eigenvalues of the non - backtracking or the flow matrix is lengthy . both are @xmath246 matrices . the adjacency matrix @xmath255 has @xmath256 elements , so @xmath244 and @xmath245 are larger by a factor of @xmath257 , where @xmath109 is the average degree of the network . an approximate but reliable computation of the spectra requires a time which scales superlinearly ( approximately quadratic ) with the network size @xmath2 . so the problem is intractable for graphs with number of edges of the order of millions or higher . also , if communities have diverse sizes and edge densities , as it happens in most networks encountered in applications , the bulk of eigenvalues may not have a circular shape , and it may become problematic to identify eigenvalues falling outside of the bulk . besides , non - backtracking walks must contain cycles , hence trees dangling off the graph do not affect the spectrum of @xmath251 , which remains unchanged if all dangling trees are removed . this is a disturbing feature , as tree - like regions of the graph may play a role in the network s community structure , and most methods would find different partitions if trees are kept or removed . the spectrum of the flow matrix , instead , changes when dangling trees are kept or removed @xcite . in the limiting case in which the network itself is a tree , all eigenvalues of @xmath251 and @xmath258 are zero and even if there were a community structure one gets no relevant information . the number of clusters can also be deduced by studying how the eigenvectors of graph matrices rotate when the adjacency matrix of the graph is subjected to random perturbations @xcite . on stochastic block models this approach infers the correct value of @xmath75 up to a threshold preceding the detectability limit . the method is also computationally expensive . in general , if one can identify a set ( range ) of promising @xmath75-values , from preliminary information or via calculations like the ones described above or in section [ sec - inference ] , it is better to run constrained versions of clustering methods , searching for solutions only among partitions with those numbers of communities , than letting the methods discover @xmath75 by themselves , which may lead to solutions of lower quality . many clustering techniques are stochastic in character and do not deliver a unique answer . a common scenario is when the desired solution corresponds to extrema of a cost function , that can only be found via approximation techniques , with results depending on random seeds and on the choice of initial conditions . techniques not based on optimisation sometimes have the same feature , when tie - break rules are adopted in order to choose among multiple equivalent options encountered along the calculation . what to do with all these partitions ? sometimes there are objective criteria to sort out a specific partition and discard all others . for instance , in algorithms based on optimisation , one could pick the solution yielding the largest ( smallest ) value of the function to optimise . for other techniques there is no clear - cut criterion . a promising approach is combining the information of the different outputs into a new partition . consensus clustering @xcite is based on this idea . the goal is searching for a _ consensus _ _ partition _ , that is better fitting than the input partitions . consensus clustering is a difficult combinatorial optimisation problem . an alternative greedy strategy @xcite relies on the _ consensus matrix _ , which is a matrix based on the co - occurrence of vertices in communities of the input partitions ( fig . [ figcons ] ) . the consensus matrix is used as an input for the graph clustering technique adopted , leading to a new set of partitions , which produce a new consensus matrix , etc . , until a unique partition is finally reached , which is not changed by further iterations . . the combination of ( i ) , ( ii ) , ( iii ) and ( iv ) yields the weighted consensus matrix illustrated on the right . the thickness of each edge is proportional to its weight . in the consensus matrix the community structure of the original network is more visible : the two communities have become cliques , with `` heavy '' edges , whereas inter - community edges are rather weak . this improvement is obtained despite the presence of two inaccurate partitions in three clusters ( iii and iv ) . reprinted figure with permission from @xcite . 2012 , by the nature publishing group . ] the steps of the procedure are listed below . the starting point is a network @xmath6 with @xmath2 vertices and a clustering algorithm @xmath259 . 1 . apply @xmath259 on @xmath6 @xmath260 times , yielding @xmath260 partitions . compute the consensus matrix @xmath261 : @xmath262 is the number of partitions in which vertices @xmath10 and @xmath11 of @xmath6 are assigned to the same community , divided by @xmath260 . all entries of @xmath261 below a chosen threshold @xmath263 are set to zero and use the full matrix all along @xcite . ] . 4 . apply @xmath259 on @xmath261 @xmath260 times , yielding @xmath260 partitions . if the partitions are all equal , stop . otherwise go back to 2 . since the consensus matrix is in general weighted , the algorithm @xmath259 must be able to handle weighted networks , even if the graph at study is binary . fortunately many popular algorithms have natural extensions to the weighted case . the integration of consensus clustering with popular existing techniques leads to more accurate partitions than those delivered by the methods alone on lfr benchmark graphs @xcite . interestingly , this holds even for methods whose direct application gives poor results on the same graphs , like modularity optimisation ( section [ sec - modopt ] ) . the variability of the partitions , rather than being a problem , becomes a factor of performance enhancement . the outcome of the procedure depends on the choice of the threshold parameter @xmath263 and the number of input partitions @xmath260 , which can be selected by testing the performance on benchmark networks @xcite . consensus clustering is also a promising technique to detect communities in evolving networks ( section [ sec - dynclus ] ) . spectral graph clustering is an approach to detect clusters using spectral properties of the graph @xcite . the eigenvalue spectrum of several graph matrices ( e. g. , the adjacency matrix , the laplacian , etc . ) typically consists of a dense bulk of closely spaced eigenvalues , plus some outlying eigenvalues separated from the bulk by a significant gap . the eigenvectors corresponding to these outliers contain information about the large - scale structure of the network , like community structure . spectral clustering consists in generating a projection of the graph vertices in a metric space , by using the entries of those eigenvectors as coordinates . the @xmath10-th entries of the eigenvectors are the coordinates of vertex @xmath10 in a @xmath56-dimensional euclidean space , where @xmath56 is the number of eigenvectors used . the resulting points can be grouped in clusters by using standard partitional clustering techniques like @xmath56-means clustering @xcite . spectral clustering is not always reliable , however . when the network is very sparse ( section [ sec - mv ] ) the separation between the eigenvalues of the community - related eigenvectors and the bulk is not sharp . eigenvectors corresponding to eigenvalues outside of the bulk may be correlated to high - degree vertices ( hubs ) , instead of group structure . likewise , community - related eigenvectors can be associated to eigenvalues ending up inside the bulk . in these situations , selecting eigenvectors based on whether their associated eigenvalues are inside or outside the bulk yields a heterogeneous set , containing information both on communities and on other features ( e. g. , hubs ) . using those eigenvectors for the spectral clustering procedure renders community detection more difficult , sometimes impossible . unfortunately , many of the networks encountered in practical studies are very sparse and can lead to this type of problems . indeed on sparse networks constructed with the planted partition model spectral methods relying on standard matrices [ adjacency matrix , laplacian , modularity matrix @xcite , etc . ] fail before the theoretical detectability limit ( section [ detectab ] ) @xcite . the non - backtracking matrix @xmath251 of eq . ( [ eqnbtm ] ) was introduced to address this problem @xcite . on the planted partition model the associated eigenvalues of the community - related eigenvectors of @xmath251 are separated from the bulk until the theoretical detectability limit , so spectral methods using the top eigenvectors of @xmath251 are capable to find communities as long as they are detectable , modulo the caveats we expressed in section [ sec - tools ] . soft clustering , where communities may overlap , is an even harder problem than hard clustering , where there is no community overlap . the possibility of having multiple memberships for the vertices introduces additional degrees of freedom in the problem , causing a huge expansion of the space of possible solutions . it has been pointed out that overlapping communities , especially in social networks , reflect different types of associations between people @xcite . two actors could be co - workers , friends , relatives , sport mates , etc .. actor @xmath7 could be a work colleague of @xmath62 and a friend of @xmath17 , so she would sit in the overlap between the community of colleagues of @xmath62 and the community of friends of @xmath17 . for this reason , it has been suggested that an effective way to recover overlapping clusters is to group edges , rather than vertices . in the example above , the edges connecting @xmath7 with @xmath62 and @xmath7 with @xmath17 would be placed in different groups , and since they both have @xmath7 as endpoint , the latter turns out to be an overlapping vertex . moreover , edge clustering is claimed to have the additional advantage of reconciling soft clustering with hierarchical community structure @xcite . if there is hierarchy , communities are nested within each other as many times as there are hierarchical levels . hierarchical structure is often represented via _ dendrograms _ , with the network being divided in clusters , which are in turn divided in clusters , and so on until one ends up with singleton clusters , consisting of one vertex each . but this can be done only if communities do not share vertices . overlapping vertices should be assigned to multiple clusters of lower hierarchical levels , yielding multiple copies of them in the dendrogram . instead , one could build a dendrogram displaying edge communities , where each edge is assigned to a single cluster , but clusters can still overlap because edges in different clusters may share one endpoint @xcite . some remarks are in order . first , there may still be overlapping communities even if there were a single type of association between the vertices . for instance , if we keep only the friendship relationships within a given population of actors , there are many social circles and there could be active actors with multiple ties within two circles , or more . second , in the traditional picture of networks with community structure ( fig . [ figcom ] ) , the edges connecting two different groups may be assigned to one of the communities they join or they could be put together in a separate group . either way , they would signal an overlap between the communities , which is artificial . this happens even in the extreme case of a single edge connecting vertices @xmath7 and @xmath62 of two groups , as that edge will have to be assigned to a group , which inevitably forces @xmath7 and @xmath62 into a common cluster . third , if we rely on the picture emerging from the analysis by yang and leskovec @xcite ( section [ ncp ] ) overlaps between clusters could be much denser than we expect , hence not only vertices but also edges may be shared among different groups , and edge dendrograms would have the same problem as classic vertex dendrograms . fourth , the computational complexity of the calculation can rise substantially , as in networks of interest there are typically many more edges than vertices . finally , there is nothing revealing that there is a conceptual or algorithmic advantage in grouping edges versus vertices , other than works showing that a specific edge clustering technique outperforms some vertex clustering techniques on a specific set of networks . to shed some light on the situation , we performed the following test . we took some network data sets with annotated vertices , giving an indication about what the communities of those networks could be that metadata are not necessarily correlated with topological clusters . we used data sets for which there is some correlation . ] . for each network @xmath6 we derived the corresponding _ line graph _ @xmath264 , which is the graph whose vertices are the edges of @xmath6 , while edges are set between pairs of vertices of @xmath264 whose corresponding edges in @xmath6 are adjacent at one of their endpoints . vertex communities of @xmath264 are then edge communities of the original network @xmath6 . the question is whether by working on @xmath264 the detection improves or not . we searched for overlapping communities with oslom @xcite . we applied oslom on the original graphs and on their line graphs . the covers found on the line graphs were turned into covers of the vertices of @xmath6 , by replacing each vertex of @xmath264 with the pair of vertices of the corresponding edge of @xmath6 . the results can be seen in fig . [ figlg ] , showing how similar the covers found on the original networks and on the line graphs are with respect to the covers of annotated vertices . neither approach is very accurate , as expected ( see section [ topmet ] and fig . [ tmhric ] ) , but vertex communities show a greater association to the annotated clusters than edge communities , except in a few instances where the similarity is very low . analyses carried out on lfr benchmark graphs ( not shown ) lead to the same conclusion . we stress that traditional line graphs have the problem that edges adjacent to a hub vertex in the original graph turn into vertices who are all connected to each other , forming giant cliques , which might dominate the structure of the line graph , misleading clustering techniques . the procedure can be refined by introducing weights for the edges of the line graphs , that can be computed in various ways , e. g. , based on the similarity of the neighbourhoods of adjacent edges in the original network @xcite . still we believe that our tests provide some evidence that edge clustering is no better than vertex clustering , in general . the superiority of algorithms based on either approach should be assessed a posteriori , case by case , and the answer may depend on the specific data sets under investigation . statistical inference provides a powerful set of tools to tackle the problem of community detection . the standard approach is to fit a generative network model on the data @xcite . the stochastic block model ( sbm ) is by far the most used generative model of graphs with communities ( see section [ sec - mv ] and references therein ) . we have seen that it can describe other types of group structure , like disassortative and core - periphery structure ( fig . [ figsbm ] ) . the unnormalised maximum log - likelihood that a given partition @xmath265 in @xmath75 groups of the network @xmath6 is reproduced by the standard sbm reads @xcite @xmath266 where @xmath88 is the number of edges running from group @xmath83 to group @xmath84 , @xmath267 ( @xmath268 ) the number of vertices in @xmath83 ( @xmath84 ) and the sum runs over all pairs of groups ( including when @xmath269 ) . this version of the model , however , does not account for the degree heterogeneity of most real networks , so it does a poor job at describing the group structure of many of them . therefore , karrer and newman proposed the _ degree - corrected stochastic block model _ ( dcsbm ) @xcite , in which the degrees of the vertices are kept constant , on average , via the introduction of additional suitable parameters . the unnormalised maximum log - likelihood for the dcsbm is @xmath270 where @xmath271 ( @xmath272 ) is the sum of the degrees of the vertices in @xmath83 ( @xmath84 ) . the most important drawback of this type of approach is the need to specify the number @xmath75 of groups beforehand , which is usually unknown for real networks . this is because a straight maximisation of the likelihoods of eqs . ( [ eqllsbm ] ) and ( [ eqlldc ] ) over the whole set of possible solutions yields the trivial partition in which each vertex is a cluster ( _ overfitting _ ) . in section [ sec - tools ] we have seen ways to extract @xmath75 from spectral properties of the graph . but it would be better to have statistically principled methods , to be consistent with the approach used to perform the inference . a possibility is _ model selection _ , for instance by choosing the model that best compresses the data @xcite . the extent of the compression can be estimated via the total amount of information necessary to describe the data , which includes not only the fitted model , but also the information necessary to describe the model itself , which is a growing function of the number of blocks @xmath75 @xcite . this quantity , that we indicate with @xmath273 , is called the _ description length_. minimising the description length naturally avoids overfitting . partitions with large @xmath75 are associated to heavy " models in terms of their information content , and do not represent the best compression . on the other hand , partitions with low @xmath75 have high information content , even if the model itself is not loaded with parameters . hence the minimum description length corresponds to a non - trivial number of groups and it makes sense to minimise @xmath273 to infer the block structure of the graph . it turns out that this approach has a limited resolution on the standard sbm : the maximum number of blocks that can be resolved scales as @xmath274 for a fixed average degree @xmath109 , where @xmath2 is the number of vertices of the network . this means that the minimum size of detectable blocks scales as @xmath274 , just as it happens for modularity maximisation ( section [ sec - modopt ] ) . a more refined method of model selection , consisting in a nested hierarchy of stochastic block models , where an upper level of the hierarchy serves as prior information to a lower level , brings the resolution limit down to @xmath202 , enabling the detection of much smaller blocks @xcite . other techniques to extract the number of groups have been proposed @xcite . optimisation techniques have received the greatest attention in the literature . the goal is finding an extremum , usually the maximum , of a function indicating the quality of a clustering , over the space of all possible clusterings . quality functions can express the goodness of a partition or of single clusters . the most popular quality function is the _ modularity _ by newman and girvan @xcite . it estimates the quality of a partition of the network in communities . the general expression of modularity is @xmath275 where @xmath3 is the number of edges of the network , the sum runs over all pairs of vertices @xmath10 and @xmath11 , @xmath8 is the element of the adjacency matrix , @xmath276 is the _ null model term _ and in the kronecker delta at the end @xmath277 and @xmath278 indicate the communities of @xmath10 and @xmath11 . the term @xmath276 indicates the average adjacency matrix of an ensemble of networks , derived by randomising the original graph , such to preserve some of its features . therefore , modularity measures how different the original graph is from such randomisations . the concept was inspired by the idea that by randomising the network structure communities are destroyed , so the comparison between the actual structure and its randomisation reveals how non - random the group structure is . a standard choice is @xmath279 , @xmath18 and @xmath121 being the degrees of @xmath10 and @xmath11 , and corresponds to the expected number of edges joining vertices @xmath10 and @xmath11 if the edges of the network were rewired such to preserve the degree of all vertices , on average . this yields the classic form of modularity @xmath280 other choices of the null model term allow us to incorporate specific features of network structure , like bipartiteness @xcite , correlations @xcite , signed edges @xcite , space embeddedness @xcite , etc .. the extension of eq . ( [ eq : mod ] ) and of its variants to the case of weighted networks is straightforward @xcite . for simplicity we focus on unweighted graphs here , but the issues we discuss are general . because of the delta , the only contributions to the sum come from vertex pairs belonging to the same cluster , so we can group these contributions together and rewrite the sum over the vertex pairs as a sum over the clusters @xmath281 . \label{eq : mod1}\ ] ] here @xmath282 the total number of edges joining vertices of community @xmath17 and @xmath283 the sum of the degrees of the vertices of @xmath17 ( section [ sec - var ] ) . the first term of each summand in eq . ( [ eq : mod1 ] ) is the fraction of edges of the graph falling within community @xmath17 , whereas the second term is the expected fraction of edges that would fall inside @xmath17 if the graph were taken from the ensemble of random graphs preserving the degree of each vertex of the original network , on average . the difference in the summand would then indicate how non - random " subgraph @xmath17 is . the larger the difference the more confident we can be that the placement of edges within @xmath17 is not random ( fig . [ mod - ill ] ) . large values of @xmath284 are then supposed to indicate partitions with high quality . modularity maximisation is np - hard @xcite . therefore one can realistically hope to find only decent approximations of the modularity maximum and a wide variety of approaches has been proposed . due to its simplicity , the prestige of its inventors and early results on the benchmark of girvan and newman ( section [ art - bench ] ) and on small real benchmark networks , like zachary karate club network ( fig . [ zach ] ) , modularity has become the best known and most studied object in network clustering . in fact , soon after its introduction , it seemed to represent the essence of the problem , and the key to its solution . however , it became quickly clear that the measure is not as good as it looks . for one thing , there are high - modularity partitions even in random graphs without groups @xcite . this seems counterintuitive , given that modularity has been designed to capture the difference between random and non - random structure . modularity is a sort of distance between the actual network and _ an average _ over random networks , ignoring altogether the distribution of the relevant community variables , like the fractions of edges within the clusters , over all realisations generated by the configuration model . if the distribution is not strongly peaked , the values of the community variables measured on the original graph may be found in a large number of randomised networks , even though the averages look far away from them . in other words , we should pay more attention to the _ significance _ of the maximum modularity value @xmath285 , than to the value itself . how can we estimate the significance of @xmath285 ? a natural way is maximising @xmath284 over all partitions of every randomised graph . one then computes the average @xmath286 and the standard deviation @xmath287 of the resulting values . the statistical significance of @xmath285 is indicated by the distance of @xmath285 from the null model average @xmath286 in units of the standard deviation @xmath287 , i. e. , by the @xmath173-score @xmath288 if @xmath289 , @xmath285 indicates strong community structure . this approach has problems , though . the main drawback is that the distribution of @xmath290 over the ensemble of null model random graphs , though peaked , is not gaussian . therefore , one can not attribute to the values of the @xmath173-score the significance corresponding to a gaussian distribution , and one ought to compute the statistical significance for the correct distribution . also , the @xmath173-score depends on the network size , so the same values may indicate different levels of significance for networks differing considerably in size . next , it is not true that the modularity maximum always corresponds to the most pronounced community structure of a network . in fig . [ reslim ] we show the well - known example of the ring of cliques @xcite . the network consists of @xmath110 cliques with four vertices each . every clique has two neighbouring cliques , connected to it via a single edge . intuition suggests that the graph has a natural community structure , with @xmath110 communities , each corresponding to one clique . indeed , the @xmath284-value of this partition is @xmath291 , pretty close to @xmath9 , which is the upper bound of modularity . however , there are partitions with larger values , like the partition in @xmath292 clusters indicated by the dashed contours , whose modularity is @xmath293 . this is due to the fact that @xmath284 has a preferential scale for the communities , deriving from the underlying null model and revealed by its explicit dependence on the number of edges @xmath3 of the network [ eq . ( [ eq : mod1 ] ) ] . according to the configuration model , the expected number @xmath294 of edges running between two subgraphs @xmath7 and @xmath62 with total degree @xmath295 and @xmath296 , respectively , is approximately @xmath297 . consequently , if @xmath295 and @xmath296 are of the order of @xmath298 or smaller , @xmath294 could become smaller than @xmath9 . this means that in many randomisations of the original graph @xmath6 , subgraphs @xmath7 and @xmath62 are disconnected and even a single edge joining them in @xmath6 signals a non - random association . in these cases , modularity is larger when @xmath7 and @xmath62 are put together than when they are treated as distinct communities , as in the example of fig . [ reslim ] . the modularity scale depends only on the number of edges @xmath3 , and it may have nothing to do with the size of the actual communities of the network . the resolution limit questions the usefulness of modularity in practical applications @xcite . many attempts have been made to mitigate the consequences of this disturbing feature . one approach consists in introducing a resolution parameter @xmath299 into modularity s formula @xcite . by tuning @xmath299 it is possible to arbitrarily vary the resolution scale of the method , going from very large to very small communities . we shall discuss such multi - resolution approaches in section [ sec - dynmet ] . here we emphasize that multi - resolution versions of modularity do not provide a reliable solution to the problem . this is because modularity maximisation has an additional bias : large subgraphs are usually split in smaller pieces @xcite . this problem has the same source as the resolution limit , namely the choice of the null model . since modularity has a preferential scale for the communities , when a subgraph is too large it is convenient to break it down , to increase the modularity of the partition . so , when there is no characteristic scale for the communities , like when there is a broad cluster size distribution , large communities are likely to be broken , and small communities are likely to be merged . since multi - resolution versions of modularity can only shift the resolution scale of the measure back and forth , they are unable to correct both effects at the same time @xcite . in addition , tuning the resolution parameter in the search for good partitions is usually computationally very demanding , as in many cases the optimisation procedure has to be repeated over and over for all @xmath299-values one desires to investigate . we stress that the resolution limit is a feature of modularity itself , not of the specific way adopted to maximise it . therefore , there is no magic heuristic that can circumvent this issue . the louvain method @xcite has been held as one such magic heuristic . the method performs a greedy optimisation of @xmath284 in a hierarchical manner , by assigning each vertex to the community of their neighbours yielding the largest @xmath284 , and creating a smaller weighted super - network whose vertices are the clusters found previously . partitions found on this super - network hence consist of clusters including the ones found earlier , and represent a higher hierarchical level of clustering . the procedure is repeated until one reaches the level with largest modularity . in the comparative analysis of clustering algorithms performed by lancichinetti and fortunato on the lfr benchmark @xcite , the louvain algorithm was the second best - performing method , after infomap @xcite . this has given the impression that the peculiar strategy of the method solves the resolution problems above , which is not true . the reason why the performance is so good is that lancichinetti and fortunato adopted the lowest partition of the hierarchy , the one with the smallest clusters @xcite . by using the partition with highest modularity performance degrades considerably ( fig . [ louvainl ] ) , as expected . as suggested by the developers of the algorithm themselves , using the lowest level helps avoiding unnatural community mergers ; as an example , they showed that the natural partition of the ring of cliques ( fig . [ reslim ] ) can be recovered this way . and @xmath300 vertices ) and of the range of community sizes ( label @xmath301 indicates that communities have between @xmath302 and @xmath215 vertices , label @xmath62 that they have between @xmath303 and @xmath304 vertices ) . accuracy is calculated via the normalised mutual information ( nmi ) , in the version by lancichinetti , fortunato and kertsz @xcite . results are heavily depending on the hierarchical level one chooses at the end of the procedure . when one picks the top level ( diamonds ) , which is the one with largest modularity , the accuracy is poor , as expected , especially when communities are smaller . when one goes for the bottom level ( squares ) , which has lower modularity and smaller clusters than the top level partition , there is a far better agreement with the planted partition and the performance gets closer to that of infomap ( circles ) . the squares follow the performance curves used in the comparative analysis by lancichinetti and fortunato @xcite . courtesy from andrea lancichinetti . ] however , the bottom level has lower modularity than the top level , so we face a sort of contradiction , in that users are encouraged to use suboptimal partitions , even though one assumes that the best clustering corresponds to the highest value of the quality function , which is what the method is supposed to find . there is no guarantee that the bottom level yields the most meaningful solution . on the other hand , users have the option of choosing among a few partitions and a slightly higher chance to find what they search for . moreover , the modularity landscape is characterised by a larger than exponential of graph vertices . ] number of distinct partitions , whose modularity values are very close to the global maximum @xcite . this explains why many heuristic methods of modularity maximisation are able to come very close to the global maximum of @xmath284 , but it also implies that the global maximum is basically impossible to find . in addition , high-@xmath284 partitions are not necessarily similar to each other , despite the proximity of their modularity scores . the optimal structural partition , which may not correspond to the modularity maximum due to problems exposed above , may however have a large @xmath284-value . therefore the optimal partition is basically indistinguishable from a huge number of high - modularity partitions , which are in general structurally dissimilar from it . the large structural diversity of high - modularity partitions implies that one can not rely on any of them , at least in principle . reliable solutions could be singled out when the domain user imposes some constraints on the clustering of the system , or when she expects it to have specific features . in the absence of additional information or expectations , consensus clustering could be used to derive more robust partitions . indeed , it has been shown that the consensus of many high - modularity partitions , combined with a hierarchical approach , could help to solve resolution problems and to avoid to find communities in random graphs without groups @xcite . as of today , modularity optimisation is still the most used clustering technique in applications . this may appear odd , given the serious issues of the method and the fact that nowadays more powerful techniques are available , like a posteriori stochastic block modelling ( section [ sec - wmt ] ) . indeed newman has proven that optimising modularity is equivalent to maximising the likelihood that the planted partition model reproduces the network @xcite . but the planted partition model is a very specific case of the general stochastic block model , in that the intra - group edge probabilities are all equal to the same value @xmath65 and the inter - group edge probabilities are all equal to the same value @xmath112 . there is no reason to limit the inference to this specific case , when one could use the full model . optimising partition quality functions may lead to resolution problems , just like it happens for modularity , i. e. , the partition found when the detection is performed only on @xmath301 @xcite . this is possible when the coefficients of the summand corresponding to each community does not depend on global properties of the graph . even those functions have their own preferential community scale , though . ] . instead , one could try to optimise _ cluster quality functions_. one starts with some function @xmath305 expressing how community - like " a subgraph is and with a seed vertex @xmath84 . the goal is to build a cluster @xmath306 including @xmath84 such that @xmath307 is maximum . this is usually done by exploring the neighbours of the temporary subgraph @xmath306 , starting from the neighbours of @xmath84 when @xmath306 includes only @xmath84 . the neighbouring vertex whose inclusion yields the largest increase of @xmath75 is added to the subgraph . when a new vertex is included , the structure of the subgraph is altered and the other vertices can be examined again , as it might be advantageous to knock some of them out . the process stops when the quality @xmath307 can not be increased anymore via the inclusion or the exclusion of vertices . the optimisation of cluster quality functions offers a number of advantages over the optimisation of partition quality functions . first , it is consistent with the idea that communities are local structures , which are sensitive to what happens in their neighbourhood , but are fairly unaffected by the rest of the network : the structure of a social circle in europe is hardly influenced by the dynamics of social circles in australia , though they are parts of the same global social network of humans . consequently , if a network undergoes structural changes in one region , community structure is altered and is to be recovered only in that region , while the clustering of the rest of the network remains the same . by optimising partition quality functions , instead , any little change may have an effect on every community of the graph . second , since cluster quality functions do not embody any global scale , severe resolution problems are usually avoided . moreover , one can investigate only parts of the network , which is particularly valuable when the graph is large and a global analysis would be out of reach , computationally . the local exploration of the graph allows to reach vertices already assigned to clusters , so overlaps can be naturally detected . in the last years several algorithms based on the optimisation of cluster quality functions have been designed @xcite . communities can also be identified by running dynamical processes on the network , like diffusion @xcite , spin dynamics @xcite , synchronisation @xcite , etc .. in this section we focus on diffusion and spin dynamics , that inform most approaches . random walk dynamics is by far the most exploited in community detection . if communities have high internal edge density and are well - separated from each other , random walkers would be trapped in each cluster for quite some time , before finding a way out and migrating to another cluster . we briefly discuss two broad classes of algorithms : methods based on vertex similarity and methods based on the map equation . the first class of techniques consists in using random walk dynamics to estimate the similarity between pairs of vertices . for instance , in the popular method _ walktrap _ the similarity between vertices @xmath10 and @xmath11 is given by the probability that a random walker moves from @xmath10 to @xmath11 in a fixed number of steps @xmath97 @xcite . the parameter @xmath97 has to be large enough , to allow for the exploration of a significant portion of the graph , but not too big , as otherwise one would approach the stationary limit in which transition probabilities trivially depend on the degrees of the vertices . if there is a pronounced community structure , pairs of vertices in the same cluster are much more easily reachable by a random walk than pairs of vertices in different clusters , so the vertex similarity is expected to be considerably higher within groups than between groups of the graph , whose element @xmath308 equals the probability that a random walker , sitting at @xmath11 , moves to @xmath10 ; raising the elements of the resulting matrix to a power , such that the larger values are enhanced with respect to the smaller ones , many of which are set to zero to lighten the calculations , while the remaining ones are normalised by dividing them by the sum of elements of their column , yielding a new transfer matrix . the process eventually reaches a stationary state , corresponding to the matrix of a disconnected graph , whose connected components are the sought clusters . ] . in that case , clusters can be readily identified via standard hierarchical or partitional clustering techniques @xcite . this class of methods have a high computational complexity , higher than quadratic in the number @xmath2 of vertices ( on sparse graphs ) , so they can not be used on large networks . besides , they are often parameter - dependent . the map equation stems from a seminal paper by rosvall and bergstrom @xcite , who asked what is the most parsimonious way to describe an infinitely long random walk taking place on the graph . the information content of any description is given by the total number of bits required to indicate the various stages of the process . the simplest description is obtained by listing sequentially all vertices reached by the random walker , each vertex being described by a unique codeword . however , if the network has a community structure , there may be a more compact description , which follows the principle of geographic maps , where there are multiple cities and streets with the same name across regions . vertex codewords could be recycled among different communities , which play the role of regions / states , and vertices with identical name are distinguished by specifying the community they belong to . if clusters are well separated from each other , transitions between clusters are infrequent , so it is advantageous to use the map , with the communities as regions , because in the description of the random walk the codewords of the clusters will not be repeated many times , while there is a considerable saving in the description due to the limited length of the codewords used to denote the vertices ( fig . [ figinfomap ] ) . the map equation yields the description length of an infinite random walk consists of two terms , expressing the shannon entropy of the walk within and between clusters . the best partition is the one yielding the minimum description length . this method , called infomap , can be applied to weighted networks , both undirected and directed . in the latter case , random walk dynamics is modified by introducing a teleportation probability , as in the pagerank process @xcite , to ensure that a non - trivial stationary state is reached . it has been successively extended to the detection of hierarchical community structure @xcite and of overlapping clusters @xcite . in classic random walks the probability of reaching a vertex only depends on where the walker stands , not on where it is coming from . the map equation has also been extended to random walks whose transition probabilities depend on earlier steps too ( higher - order markov dynamics ) @xcite , retaining memory of the ( recent ) past . applications show that in this way one can recover overlapping communities more easily than by using standard first - order random walk dynamics , especially pervasive overlaps , which are usually out of reach for most clustering algorithms ( section [ ncp ] ) . infomap and its variants usually return different partitions than structure - based methods ( e. g. , modularity optimisation ) . this is because they are based on flows running across the system , as opposed to structural variables like number of edges , vertex degrees , etc .. the difference is particularly striking on directed graphs @xcite , where edge directions heavily constrain the possible flows . structural features obviously play a major role on the dynamics of processes running on graphs , but dynamics can not be generally reduced to an interplay of structural elements , at least not simple ones like , e. g. , vertex degrees . sometimes structural and dynamic approaches are equivalent , though . for instance , newman - girvan s modularity is a special case of a general quality function , called _ stability _ , expressing the persistence of a random walk within communities @xcite . the methods we have discussed so far are global , in that they aim at finding the whole community structure of the system . however , random walks along with other dynamical processes can be used as well to explore the network locally , starting from seed vertices @xcite . good communities correspond to bottlenecks of the dynamics and depend on the choice of the seed vertices , the time scale of the dynamics , etc .. such local perspective enables to identify community overlaps in a natural way , due to the possibility of reaching vertices multiple times , even if they are already classified . spin dynamics @xcite are also regularly used in network clustering . the first step is to define a spin model on the network , consisting of a set of spin variables @xmath309 , assigned to the vertices and a hamiltonian @xmath310 , expressing the energy of the spin configuration @xmath311 . for community detection , spins are usually integers : @xmath312 . contributions to the energy are usually given by spin - spin interactions . the coupling of a spin - spin interaction can be _ ferromagnetic _ ( negative ) or _ antiferromagnetic _ ( positive ) , if the energy is lower when the spins are equal or not , respectively . the goal is to find those spin configurations that minimise the hamiltonian @xmath310 . if couplings are all ferromagnetic , the minimum energy would be trivially obtained for the configurations where all vertices have identical spin values . instead , one would like to have identical spins for vertices of the same cluster , and different spins for vertices in different clusters , to identify the community structure . therefore , hamiltonians feature both ferromagnetic and antiferromagnetic interactions [ _ spin glass dynamics _ @xcite ] . a popular model consists in rewarding edges between vertices in the same cluster , as well as non - edges between vertices in different clusters , and penalising edges between vertices of different clusters , along with non - edges between vertices in the same cluster . this way , if the edge density within communities is appreciably larger than the edge density between communities , as it often happens , having equal spin values for vertices in the same cluster would considerably lower the energy of the configuration . on the other hand , to bring the energy further down the spins of vertices in different clusters should be different , as many such vertices would be disjoint from each other , and such non - edges would increase the energy of the system if the corresponding spin variables were equal . a general expression for the hamiltonian along these lines is @xcite @xmath313\delta(s_i , s_j)\ , , \label{eqrbgen}\ ] ] where @xmath8 is the element of the adjacency matrix , @xmath314 are arbitrary coefficients , and the kronecker delta selects only the pairs of vertices with the same spin value . a popular model is obtained by setting @xmath315 and @xmath316 , where @xmath299 is a tunable parameter and @xmath276 a null model term , expressing the expected number of edges running between vertices @xmath10 and @xmath11 under a suitable randomisation of the graph structure . the resulting hamiltonian is @xcite @xmath317 if @xmath318 and @xmath279 , @xmath18 ( @xmath121 ) being the degree of @xmath10 ( @xmath11 ) and @xmath3 the total number of graph edges , the hamiltonian of eq . ( [ eqrb ] ) coincides with the modularity by newman and girvan [ eq . ( [ eq : mod ] ) ] , up to an irrelevant multiplicative constant . consequently , modularity can be interpreted as the hamiltonian of a spin glass as well . by setting @xmath319 and @xmath320 we obtain the _ absolute potts model _ ( apm ) @xcite , whose hamiltonian reads @xmath321\delta(s_i , s_j)\ , . \label{eqapm}\ ] ] here , there is no null model term . the models of eqs . ( [ eqrb ] ) and ( [ eqapm ] ) can be trivially extended to weighted graphs @xcite . they allow to explore the network at different resolutions , by suitably tuning the parameter @xmath299 . however , there usually is no information about the community sizes , so it is not possible to decide _ a priori _ the proper value(s ) of @xmath299 for a specific graph . a common heuristic is to estimate the _ stability _ of partitions as a function of @xmath299 . it is plausible that partitions recovered for a given @xmath299-value will not be disrupted if @xmath299 is varied a little . so , the whole range of @xmath299 can be split into intervals , each interval corresponding to the most frequent partition detected in it . good candidates for the unknown community structure of the system could be the partitions found in the widest intervals of @xmath299 , as they are likely to be more stable ( or robust ) than the other partitions . however , the results of the algorithm do not usually have a linear relationship with @xmath299 , hence the width of the intervals is not necessarily correlated with stability , as intervals of the same width but centred at different values of @xmath299 may have rather different importance . a good operational definition of stability is based on the stochastic character of optimisation methods , which typically deliver different results for the same system and set of parameters , by changing initial conditions and/or random seeds . if a partition is robust in a given range of @xmath299-values , most partitions delivered by the algorithm will be very similar . on the other hand , if one explores a @xmath299-region in between two strong partitions , the algorithm will deliver the one or the other partition and the individual replicas will be , on average , not so similar to each other . so , by calculating the similarity @xmath322 of partitions found by the method at a given resolution parameter @xmath299 ( for different choices of initial conditions and random seeds ) , stable communities are revealed by peaks of @xmath322 @xcite . since clustering in large graphs can be very noisy , peaks may not be well resolved . noise can be reduced by working with consensus partitions of the individual partitions returned by the method for a given @xmath299 ( section [ sec - consensus ] ) . these manipulations are computationally costly , though . besides , multi - resolution techniques may miss relevant cluster sizes , as it happens for multi - resolution modularity @xcite ( section [ sec - modopt ] ) . due to the increasing availability of time - stamped network data , there is currently a lot of activity on the development of methods to analyse temporal networks @xcite . in particular , the problem of detecting dynamic communities has received a lot of attention @xcite . clustering algorithms used for static graphs can be ( and often are ) used for dynamic networks as well . what needs to be established is how to handle the evolution . typically one can describe it as a succession of _ snapshots _ @xmath323 , where each snapshot @xmath324 corresponds to the configuration of the graph in a given time window . there are two possible strategies . the simplest approach is to detect the community structure for each individual snapshot , which is a static graph @xcite . next , pairs of communities of consecutive windows are associated . a standard procedure is finding the cluster @xmath325 in window @xmath97 that is most similar to cluster @xmath326 in window @xmath99 , for instance by using jaccard similarity score [ eq . ( [ eqt20 ] ) ] @xcite . this way every community has an image in each phase of the network evolution and one can track its dynamics . various scenarios are possible . communities may disappear at some point and new communities may appear , following the exclusion or the introduction of vertices and edges , respectively . furthermore , a cluster may fragment into smaller ones or merge with others . however , since snapshots are handled separately , this strategy often produces significant variations between partitions close in time , especially when the data sets are noisy , as it usually happens in applications . it would be preferable to have a unified framework , in which communities are inferred both from the current structure of the graph and from the knowledge of the community structure at previous times . an interesting implementation of this strategy is _ evolutionary clustering _ @xcite . the goal of the approach is finding a partition that is both faithful to the system configuration at snapshot @xmath97 and close to the partition derived for the previous snapshot @xmath98 . a cost function is introduced , whose optimisation yields a tradeoff between such two constraints . there is ample flexibility on how this can be done , in practice . many known clustering techniques normally used for static graphs can be reformulated within this evolutionary framework . some interesting algorithms based on evolutionary clustering have been proposed @xcite . mucha et al . have also presented a method that couples the system s configurations of different snapshots , within a modularity - based framework @xcite . in the resulting quality function ( _ multislice modularity _ ) , all configurations are simultaneously taken into account and the coupling between them is expressed by a tunable parameter . the approach can handle general multilayer networks @xcite , where layers are either networks whose vertices are connected by a specific edge type ( e. g. , friendship , work relationships , etc . , in social networks ) , or networks whose vertices have connections ( interactions , dependencies ) with the vertices of other networks / layers . on the other hand , since the approach is based on modularity optimisation , it has the drawbacks exposed in section [ sec - modopt ] . consensus clustering ( section [ sec - consensus ] ) is a natural approach to find stable dynamic clusterings by combining multiple snapshots . let us suppose we have a time range going from @xmath327 to @xmath328 , that we want to divide into @xmath329 windows of size @xmath330 . for the sake of stability , one should consider sliding windows , i. e. , overlapping time intervals . this way consecutive partitions will be based on system configurations sharing a lot of vertices and edges , and change is ( typically ) smooth . in order to have exactly @xmath329 frames , each of them has to be shifted by an interval @xmath331 with respect to the previous one . so we obtain the windows @xmath332 $ ] , @xmath333 $ ] , @xmath334 $ ] , ... , @xmath335 $ ] . the community structure of each snapshot can be found via any reliable static clustering technique . next , the consensus partition from the clusterings of @xmath83 consecutive snapshots , with @xmath83 suitably chosen , is derived @xcite . again , one could consider sliding windows : for instance , the first window could consist of the first @xmath83 snapshots , the second one by those from @xmath216 to @xmath336 , and so on until the interval spanned by the last @xmath83 snapshots . in fig . [ dyncons ] we show an application of this procedure on the citation network of papers published in journals of the american physical society ( aps ) . years , except at the right end of each diagram : since there is no data after @xmath337 , the last windows must have @xmath337 as upper limit , so their size shrinks ( @xmath338 , @xmath339 , @xmath340 , @xmath341 ) . each vertical bar represents a consensus partition combining pairs of consecutive snapshots . a color uniquely identifies a community , the width of the links between clusters is proportional to the number of papers they have in common . the rapid growth of the field _ complex networks _ is clearly visible , as well as its later split into a number of smaller subtopics , like _ community structure _ , _ epidemic spreading _ , _ robustness _ , etc .. reprinted figure with permission from @xcite . 2012 , by the nature publishing group . ] an alternative way to uncover the evolution of communities by accounting for the correlation between configurations of neighbouring time intervals is to use probabilistic models @xcite . if the system is large and its structure is updated in a stream fashion , instead of working on snapshots one could detect the clustering _ online _ , every time the configuration of the system varies due to new information , like the addition of a new vertex or edge @xcite . an advantage of this approach is that change is due to the effect that the small variation in the network structure has on the system , and it can be tracked by simply adjusting the partition of the previous configuration , which can be usually done rather quickly . let us suppose that we have identified the communities , somehow . are we done ? unfortunately , things are not that simple . in fig . [ figrand0 ] ( top left ) we show the adjacency matrix of the random graph la erds - rnyi illustrated in fig . [ fig : stylized - random ] . the graph has @xmath342 vertices , so the matrix is @xmath343 . black dots indicate the existence of an edge between the corresponding vertices , while missing edges are represented in white . by construction , there is no group structure . however , we can rearrange the elements of the matrix , by reordering the vertex labels . in fig . [ figrand0 ] ( top right ) we see that , by doing that , one can generate a group structure , of the assortative type , with two blocks of equal size . if we increase the number of blocks to three fig . [ figrand0 ] ( bottom left ) and ten fig . [ figrand0 ] ( bottom right ) we can make the matrix look more and more modular . this is why many clustering techniques detect communities in random networks as well , though they should not . where do the groups come from ? since they can not be real by construction , they must be generated by random fluctuations in the network construction process . random fluctuations are particularly relevant on sparse graphs ( section [ detectab ] ) . the lesson we learn from this example is that it is not sufficient to identify groups in the network , but one should also ask _ how significant _ , or non - random , they are . unfortunately , most clustering algorithms are not able to assess the significance of their results . if the groups are compatible with random fluctuations , they are not proper groups and should be disregarded . the lower the chance that they are generated by randomness , the more confident we can be that the blocks reflect some actual group structure . naturally , this can be done only if one has a reliable _ null model _ , describing how the structure of the network at study can be randomised and allowing us to estimate how likely it is that the candidate group structure is generated this way . the configuration model @xcite is a popular null model in the literature . it generates all possible configurations preserving the number of vertices and edges of the network at study , and the degrees of its vertices . one may compute some variables of the original network , and estimate the probability that the model reproduces them , or _ p - value _ , i. e. , the fraction of model configurations yielding values of the variables compatible with those measured on the original graph . if the p - value is sufficiently low ( @xmath344 is a standard threshold ) , one concludes that the property at study can not be generated by randomness only . for community structure , one can compute various properties of the clusters , e. g. , their internal density , and compare them with the model values . some clustering algorithms , like oslom @xcite are based on this principle . along the same lines , @xmath173-scores can be used as well [ see the example of eq . ( [ eq : zscore ] ) ] . degree - corrected stochastic block models @xcite ( section [ sec - inference ] ) also include the configuration model , which corresponds to the case without group structure ) . ] . in this case significance can be estimated by doing model selection between the versions with and without blocks ( section [ sec - inference ] ) . a concept very related to significance is that of _ robustness_. if clusters are significant it means that they are resilient if the network structure is perturbed , to some extent . one way to quantitatively assess this is introducing into the system a controlled amount of noise , and checking how much noise it takes to disrupt the group structure . the greater the required perturbation , the more robust the communities . for instance , a perturbation could be rewiring a fraction of randomly chosen edges @xcite . after the network is perturbed , the community structure is derived and compared to the one of the original network . the trend of the partition similarity shows how the group structure responds to perturbations . a similar approach consists in sampling network configurations from a population which the original network is supposed to belong to ( _ bootstrapping _ ) , and comparing the clusterings found in those configurations , to check how frequently subsets of vertices are clustered together in different samples , which is an index of the robustness ( significance ) of their clusters @xcite . at the end of the day , what most people want to know is : which method shall i use on my data ? since the clustering problem is ill - defined , there is no clear - cut answer to it . popular techniques are based on similar ideas of communities , like the ones we reviewed in sections [ sec - defs ] and [ sec - mv ] . what makes the difference is the way clusters are sought . the specific procedure affects the reliability of the results ( e. g. , because of resolution problems ) and the time complexity of the calculation , determining the scope of the method and constraining its applicability . most methods propose a universal recipe , that is supposed to hold on every data set . in so doing , one neglects the peculiarities of the network at study , which is valuable information that could orient the method towards more reliable solutions . but algorithms are usually not so flexible to account for specific network features . for instance , in some cases , there is no straightforward extension capable to handle high - level features like edge direction or overlapping communities . validation of algorithms , like the comparative analysis of @xcite , have allowed to identify a set of methods that perform well on artificial benchmarks . there are two important issues , though . first , we do not know how well real networks are described by currently used benchmark models . therefore , there is no guarantee that methods performing well on benchmarks also give reliable results on real data sets . structural analyses like the ones discussed in section [ ncp ] might allow to identify more promising benchmark models . second , if we rely so much on current benchmarks , which are versions of the stochastic block model ( sbm ) , we already know what the best method is : a posteriori block modelling , i. e. , fitting a sbm on the data . indeed , there are several advantages to this approach . it is more general , it does not only discover communities but several types of group structures , like disassortative groups ( fig . [ fig : stylized - bipartite ] ) and core - periphery structure ( fig . [ fig : stylized - coreper ] ) . it can also capture the existence of hierarchies among the clusters . moreover , it yields much richer results than standard clustering algorithms , as it delivers the entire set of parameters of the most likely sbm , with which one can construct the whole network , instead of just grouping vertices . sbms are very versatile as well . they can be extended to a variety of contexts , e. g. , directed networks @xcite , networks with weighted edges @xcite , with overlapping communities @xcite , with multiple layers @xcite , with annotations @xcite . besides , the procedure can be applied to any network model with group structure , not necessarily sbms . the choice between alternative models can be done via model selection . a posteriori block modelling is not among the fastest techniques available . networks with millions of vertices and edges could be investigated this way , but very large networks remain out of reach . fortunately , many networks of interest can be attacked . the biggest problem of this class of methods , i. e. , the determination of the number of clusters , seems to be solvable ( section [ sec - inference ] ) . we recommend to exploit the power of this approach in applications . algorithms based on the optimisation of cluster quality functions should be considered as well ( section [ sec - modopt ] ) , because they may avoid resolution problems and explore the network locally , which is often the only option when the system is too large to be studied as a whole . algorithms based on the optimisation of partition quality functions , like modularity maximisation , are plagued by the problems we discussed in section [ sec - modopt ] . nevertheless , if one knows , or discovers , the correct number of clusters @xmath75 , and the optimisation is constrained on the subset of partitions having @xmath75 clusters , such algorithms become competitive @xcite . we also encourage to use approaches based on dynamics ( section [ sec - dynmet ] ) . in principle , the resulting clustering depends on the specific dynamics adopted . in practice , there often is a substantial overlap between the clusters found with different dynamics . an important question is whether dynamics may uncover groups that are not recoverable from network structure alone . differences in the clusterings found via dynamical versus structural approaches could be due to the fact that dynamical processes are sensitive to more complex structural elements than edges ( e. g. , paths , motifs ) @xcite ( section [ sec - mv ] ) . however , even if that were true , dynamical approaches could be more natural ways to handle such higher - order structures , and to make sense of the resulting community structure . in general , however , the final word on the reliability of a clustering algorithm is to be given by the user , and any output is to be taken with care . intuition and domain knowledge are indispensable elements to support or disregard solutions . in this section we provide a number of links where one can find the code of clustering algorithms and related techniques and models . * _ artificial benchmarks_. code to generate lfr benchmark graphs ( section [ art - bench ] ) can be found here . the code for the dynamic benchmark by granell et al . @xcite is available at . * _ partition similarity measures_. many partition similarity measures have their own function in r , python and matlab and are easy to find . the variant of the nmi for covers proposed by lancichinetti et al . @xcite can be found at , the one by esquivel and rosvall @xcite at . * _ consensus clustering_. the technique proposed by lancichinetti and fortunato @xcite to derive consensus partitions from multiple outputs of stochastic clustering algorithms can be downloaded from . * _ spectral methods_. the spectral clustering method by krzakala et al . @xcite , based on the non - backtracking matrix ( sections [ sec - tools ] and [ sec - spectral ] ) , can be downloaded here : . * _ edge clustering_. the code for the edge clustering technique by ahn et al . @xcite can be found here : . the link to the stochastic block model based on edge clustering by ball et al . @xcite is provided below . * _ methods based on statistical inference_. the code to perform the inference of the degree - corrected stochastic block model we have pointed to techniques to infer the number of clusters beforehand . ] by karrer and newman is available at . the weighted stochastic block model by aicher et al . @xcite can be found at . the code for the overlapping stochastic block model based on edge clustering by ball et al . @xcite is at . the model combining structure and metadata by newman and clauset @xcite is coded at . the program to infer the bipartite stochastic block model by larremore et al . @xcite can be found at . + the algorithms for the inference of community structure developed by tiago peixoto are implemented within the python module graph - tool and can be found at . they allow us to perform model selection of various kinds of stochastic block models : degree - corrected @xcite , with overlapping groups @xcite , and for networks with layers , with valued edges and evolving in time @xcite . the hierarchical block model of @xcite , that searches for clusters at high resolution , is also available . all such variants can be combined at ease by selecting a suitable set of options . + the algorithms for the inference of overlapping communities via the community - affiliation graph model ( agm ) @xcite and the cluster affiliation model for big networks ( bigclam ) @xcite ( section [ ncp ] ) can be found in the package . * _ methods based on optimisation_. there is a lot of free software for modularity optimisation . in the igraph library ( ) there are several functions , both in the @xmath345 and in the python package : cluster_fast_greedy ( r ) and community_fastgreedy ( python ) , implementing the fast greedy optimisation by clauset et al . @xcite ; cluster_leading_eigen ( r ) and community_leading_eigenvector ( python ) for the optimisation based on the leading eigenvector of the modularity matrix @xcite ; cluster_louvain ( r ) and community_multilevel ( python ) are the implementations of the louvain method @xcite ; cluster_optimal ( r ) and community_optimal_modularity turn the task into an integer programming problem @xcite ; cluster_spinglass ( r ) and community_spinglass ( python ) optimise the multi - resolution modularity proposed by reichardt and bornholdt @xcite . + some methods based on the optimisation of cluster quality functions are also available . the code for the optimisation of the local modularity by clauset @xcite can be found at . the code for oslom is downloadable from the dedicated website . * _ methods based on dynamics_. infomap @xcite is currently a very popular algorithm and its code can be found in various places . it has a dedicated website , where several extensions can be downloaded , including hierarchical community structure @xcite , overlapping clusters @xcite and memory @xcite . infomap has also its own functions on igraph , both in the r and in the python package ( cluster_infomap and community_infomap , respectively ) . _ walktrap _ @xcite , another popular method based on random walk dynamics , is available on igraph , via the functions cluster_walktrap ( r ) and community_walktrap ( python ) . the local community detection algorithms proposed in @xcite can be downloaded from . * _ dynamic clustering_. the code to optimise the multislice modularity by mucha et al . @xcite is available at . detection of dynamic communities can be performed as well with consensus clustering ( section [ sec - consensus ] ) and via stochastic block models ( section [ sec - inference ] ) . links to the related code have been provided above . as long as there will be networks , there will be people looking for communities in them . so it is of uttermost importance to have a set of reliable concepts and principles guiding scholars towards promising solutions for network clustering . we have presented established views of the main aspects of the problem , and exposed the strengths as well as the limits of popular notions and approaches . what s next ? we believe that there will be a trend towards the development of domain - dependent algorithms , exploiting as much as possible information and peculiarities of network data sets . generalist methods could still be used to get first indications about community structure and orient the investigation in promising directions . some existing approaches are sufficiently flexible to accommodate various features of networks and community structure ( section [ sec - wmt ] ) . at the same time , we believe that it is necessary to find accurate models of networks with community structure , both for the purpose of designing realistic benchmark graphs for validation , and for a more precise inference of the groups and of their features . investigations of real networks at the level of subgraphs , along the lines of those discussed in section [ ncp ] , are instrumental to the definition of such models . while benchmark graphs can be improved , there is one test that one can rely on to assess the performance of clustering algorithms : applying methods on random graphs without group structure . we know that many popular techniques find groups in such graphs as well , failing the test . on a related note , it is critical to determine how non - random the clusters detected on real networks are , i. e. , to estimate their significance ( section [ sec - sign ] ) . we stress that this exposition is by no means complete . the emphasis is on the fundamental aspects of network clustering and on main stream approaches . we discussed works and listed references which are of more immediate relevance to the topics discussed . a number of topics have not been dealt with . still we hope that this work will help practitioners to design more and more reliable methods and domain users to extract useful information from their data . we thank alex arenas , florian kimm , tiago peixoto , mason porter and martin rosvall for a careful reading of the manuscript and many valuable comments . we gratefully acknowledge multiplex , grant no . 317532 of the european commission .

community detection in networks is one of the most popular topics of modern network science . communities , or clusters , are usually groups of vertices having higher probability of being connected to each other than to members of other groups , though other patterns are possible . identifying communities is an ill - defined problem . there are no universal protocols on the fundamental ingredients , like the definition of community itself , nor on other crucial issues , like the validation of algorithms and the comparison of their performances . this has generated a number of confusions and misconceptions , which undermine the progress in the field . we offer a guided tour through the main aspects of the problem . we also point out strengths and weaknesses of popular methods , and give directions to their use .

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Proceed to summarize the following text: ultra - luminous infrared galaxies ( ulirgs ) radiate most of their extremely large , quasar - like luminosities ( @xmath2 ) as infrared dust emission , and dominate the bright end of the galaxy luminosity function in the nearby universe @xcite . recent studies have revealed that the bulk of the cosmic sub - mm background emission has been resolved into discrete sources , similar to nearby ulirgs @xcite , and for this reason data on nearby ulirgs have been extensively used to derive information on star - formation rates , dust content , and metallicity in the early universe @xcite . understanding the nature of nearby ulirgs is therefore of particular importance both locally and cosmologically . in contrast to the majority of less infrared luminous ( @xmath3 10@xmath4 ) galaxies , it has been suggested that energetically important , dust - obscured agns are present in ulirgs ( veilleux , kim , & sanders 1999a ; fischer 2000 ) . if the geometry of the dust distribution is toroidal and/or the amount of dust along our line of sight is not too great , signatures of such dust - obscured agns can be found in the optical and/or near - infrared wavelength range at @xmath5 2 @xmath0 m @xcite . however , detecting agns that are deeply embedded in a spherical dust shell ( hereafter buried agns ) and estimating their energetic importance is very difficult at @xmath5 2 @xmath0 m , even though they are energetically important , since spectral tracers are dominated by less strongly obscured star - formation activity . to detect such buried agns and estimate their energetic importance , we must observe at wavelengths where the effects of dust extinction are smaller . at wavelengths 313 @xmath0 m , dust extinction is lower . furthermore , emission is dominated by radiation from dust , the dominant emission mechanism of ulirgs . thus , discussions of the energy sources of ulirgs are more straightforward than those based on observations at x - ray or radio wavelengths , which are also potentially capable of detecting buried agns , but are dominated by emission mechanisms other than dust emission . finally , by using spectral features at 313 @xmath0 m , we can distinguish the energy sources of individual galaxies : while star - formation - dominated galaxies show strong polycyclic aromatic hydrocarbon ( pah ) emission features , buried agns should show dust absorption features below smooth continuum emission @xcite . using spectral features , particularly the 7.7 @xmath0 m pah emission , in the rest - frame 511 @xmath0 m spectra obtained with the _ infrared space observatory _ ( iso ) , it has been argued that the majority of ulirgs are star - formation powered @xcite . however , the determination of the continuum level with respect to which emission and absorption features should be measured in _ iso _ spectra is highly uncertain , due to insufficient wavelength coverage longward of these features . distinguishing between star - formation and buried agn activity therefore remains controversial @xcite . as discussed in @xcite , observations at 34 @xmath0 m have two important advantages : firstly , dust extinction is as small as that at 78 @xmath0 m @xcite ; and secondly , the 3.3 @xmath0 m pah emission and 3.4 @xmath0 m carbonaceous dust absorption features can be used to distinguish between the different energy sources of ulirgs , without serious uncertainty in the continuum determination . ground - based 34 @xmath0 m spectroscopy is thus potentially a powerful tool to investigate the energetic importance of agns buried in the compact nuclei of ulirgs , and to settle the controversy discussed above . our particular interest is the search for buried agns in ulirgs with cool far - infrared colors and/or non - seyfert optical spectra , which are typically taken to be star - formation - dominated , based on _ iso _ studies ( genzel & cesarsky 2000 ; taniguchi et al . 1999 ; lutz , veilleux , & genzel 1999 ) . ugc 5101 is such a ulirg ( table 1 in this paper ; veilleux et al . 1999a ) , and has actually been diagnosed as being star - formation powered @xcite . in this letter , we report on the results of 34 @xmath0 m spectroscopic observations of ugc 5101 , and the discovery of evidence for an energetically important buried agn in this source . throughout this paper , @xmath6 @xmath7 75 km s@xmath8 mpc@xmath8 , @xmath9 = 0.3 , and @xmath10 = 0.7 are adopted . we used the nsfcam grism mode @xcite to obtain a 34 @xmath0 m spectrum of ugc 5101 at irtf on mauna kea , hawaii on the night of 2001 april 9 ( ut ) . sky conditions were photometric throughout the observations , and the seeing was measured to be 0@xmath1170@xmath118 in full - width at half maximum . the detector was a 256 @xmath12 256 insb array . the hkl grism and l blocker were used with the 4-pixel slit (= 1@xmath13 ) . the resulting spectral resolution was @xmath14150 at 3.5 @xmath0 m . the spectrum of ugc 5101 was obtained toward the flux peak at 34 @xmath0 m . the position angle of the slit was 0 degree east of north . a standard telescope nodding technique with a throw of 12@xmath15 was employed along the slit to subtract background emission . hr 4112 ( f8v , v=4.8 ) was observed with an airmass difference of @xmath3 0.1 to correct for the transmission of the earth s atmosphere , and provide flux calibration . standard data analysis procedures were employed , using iraf . initially , bad pixels were replaced with the interpolated signals of the surrounding pixels . bias was subsequently subtracted from the obtained frames and the frames were divided by a spectroscopic flat frame . finally the spectra of ugc 5101 and hr 4112 were extracted . wavelength calibration was performed taking into account the wavelength - dependent transmission of the earth s atmosphere . since we set each exposure time to 1.2 sec for the ugc 5101 observation , to reduce observing overheads , data at @xmath16 4 @xmath0 m were affected by the non - linear response of the detector . we excluded these data from our analysis . the ugc 5101 spectrum was then divided by the observed spectrum of hr 4112 and multiplied by the spectrum of a blackbody with a temperature appropriate to f8v stars ( 6200k ) . by adopting @xmath17 = 3.4 for hr 4112 based on @xmath18 = 1.4 @xcite , a flux - calibrated spectrum was produced . a flux - calibrated 34 @xmath0 m spectrum of ugc 5101 is shown in figure [ fig1 ] . at 34 @xmath0 m , ugc 5101 was spatially very compact , and its spatial extent along the slit direction was indistinguishable from a point source . the spatially - unresolved red near - infrared nucleus reported by @xcite and interpreted as a possible agn by these authors is the probable origin of the bulk of the detected 34 @xmath0 m continuum in our spectrum . our spectrum gives a value of @xmath17 ( 3.55 @xmath0 m ) of 10.1 mag , which is similar to the value of @xmath19 ( 3.7 @xmath0 m ) of 9.8 mag measured with a 5@xmath15 aperture @xcite . given that ugc 5101 has a red near - infrared color ( @xmath20 = 1.3 ; sanders et al . 1988a ) , so that @xmath21 should be @xmath16 0 mag , our slit loss for the continuum emission is less than 0.3 mag . the dust continuum emission at 1020 @xmath0 m @xcite , pa@xmath22 emission ( genzel et al . 1998 , the top - middle of fig 8) , and 618 cm radio continuum emission @xcite all show a centrally peaked morphology , with spatially - extended emission presumably originating in weakly obscured star - formation activity . our 1@xmath112 slit covers the bulk of this emission , so that most of the pah emission should also be covered with our slit . any missing pah flux is very likely to be much smaller than the detected pah flux , and will not seriously affect our quantitative discussion of the energy source of ugc 5101 ( @xmath23 4.1 ) . the spectrum in fig . [ fig1 ] shows a remarkable deviation at 3.33.7 @xmath0 m from the smooth continuum emission . ( a linear continuum level is shown as the solid line in fig . [ fig1 ] . ) we attribute the emission feature to 3.3 @xmath0 m pah emission , since the flux peak at 3.42 @xmath0 m is consistent with the expected peak wavelength of the redshifted pah emission ( 3.29 @xmath0 m @xmath12 1.040 ) . based on our adopted linear continuum level , the observed flux and rest - frame equivalent width of the 3.3 @xmath0 m pah emission are estimated to be f@xmath24 = 1.3 @xmath12 10@xmath25 ergs s@xmath8 @xmath26 and ew@xmath24 = 0.025 @xmath0 m , respectively . at wavelengths longward of the pah emission , signals at 3.53.7 @xmath0 m are suppressed compared to the continuum level . the flux level suddenly and steeply decreases at @xmath143.7 @xmath0 m , compared to the extrapolation from longer wavelengths , so that a strong absorption feature is undoubtedly present . we attribute this flux suppression to the 3.4 @xmath0 m carbonaceous dust absorption @xcite , because the observed maximum of absorption at @xmath143.53 @xmath0 m is consistent with the redshifted absorption peak wavelength of the carbonaceous dust absorption feature ( 3.4 @xmath0 m @xmath12 1.040 ) . its observed optical depth is @xmath27(observed ) @xmath14 0.7 . the 3.3 @xmath0 m pah emission profile almost fades out at 3.333.34 @xmath0 m @xcite , while the 3.4 @xmath0 m carbonaceous dust absorption profile is just beginning at this wavelength range @xcite . at _ z _ = 0.040 , this wavelength range , where neither emission nor absorption is particularly important , is redshifted to 3.463.47 @xmath0 m . it will be seen that in fig . [ fig1 ] the adopted linear continuum intersects the observed data points at this wavelength , providing further evidence that the continuum determination is reasonable . an alternative hypothesis is represented by the curved continuum shown as the dashed line in fig . if this model were adopted , the observed flux and rest frame equivalent width of the 3.3 @xmath0 m pah emission would increase to f@xmath24 = 2.0 @xmath12 10@xmath25 ergs s@xmath8 @xmath26 and ew@xmath24 = 0.045 @xmath0 m , respectively , and the observed optical depth of the 3.4 @xmath0 m dust absorption would decrease to @xmath27(observed ) @xmath14 0.6 . however , the level of this curved continuum seems to be too low ; it requires that the absorption optical depth be close to zero at 3.48 @xmath0 m or 3.35 @xmath0 m in the rest frame , where the optical depth should in fact be significant @xcite . we thus believe that the actual pah flux and equivalent width are lower than the values inferred using the curved continuum level ; we can use these latter values as stringent upper limits . considering the uncertainty of the continuum determination , we combine the values measured based on the two continuum levels and adopt f@xmath24 = 1.6@xmath280.3 @xmath12 10@xmath25 ergs s@xmath8 @xmath26 , ew@xmath24 = 0.035@xmath280.010 @xmath0 m , and @xmath27(observed ) = 0.65@xmath280.05 . the detection of 3.3 @xmath0 m pah emission indicates the presence of star - formation activity . however , its observed luminosity is 5.2@xmath281.0 @xmath12 10@xmath29 ergs s@xmath8 , which yields an observed 3.3 @xmath0 m pah to far - infrared luminosity ratio ( table 1 ) of @xmath141 @xmath12 10@xmath30 , an order of magnitude smaller than is found in star - formation - dominated galaxies ( @xmath141 @xmath12 10@xmath31 ; mouri et al . 1990 ) . the detected , weakly - obscured star - formation activity thus contributes little to the huge far - infrared luminosity of ugc 5101 and a dominant energy source must be located behind the dust . the presence of such a dust - obscured energy source is supported by the detection of strong 3.4 @xmath0 m carbonaceous dust absorption . if the 34 @xmath0 m continuum emission source behind the dust originated in star - formation activity , the rest - frame equivalent width of the 3.3 @xmath0 m pah emission should be similar to those of less obscured star - formation - dominated galaxies ( @xmath140.12 @xmath0 m ; imanishi & dudley 2000 ) , because both 34 @xmath0 m continuum and 3.3 @xmath0 m pah emission fluxes would be attenuated similarly by dust extinction . the observed rest - frame equivalent width is , however , smaller by more than a factor of three . consequently , the emission from behind the dust shows virtually no pah emission , suggesting that the source is an agn . this agn activity dominates the observed 34 @xmath0 m continuum flux , contributing @xmath1470 % of it , and is plausibly the energy source of ugc 5101 s far - infrared luminosity . if the dust obscuring such an energetically important agn had a torus - like geometry , we would expect ugc 5101 to be optically classified as a seyfert 2 , but in fact the non - seyfert optical classification of ugc 5101 @xcite implies that the agn is obscured by dust along all lines of sight ( that is , a buried agn ) . after subtracting the contribution from star - formation to the observed 34 @xmath0 m fluxes by using the spectral shape of the starburst galaxy ngc 253 in @xcite , we estimate the intrinsic optical depth of the 3.4 @xmath0 m carbonaceous dust absorption toward the buried agn to be @xmath27(intrinsic ) @xmath14 0.8 , which yields dust extinction of @xmath32 @xmath16 100 mag if a galactic extinction curve is assumed @xcite . assuming a@xmath33 @xmath14 0.05 @xmath12 a@xmath34 @xcite , we estimate the dereddened 34 @xmath0 m dust emission luminosity powered by the buried agn to be @xmath35l@xmath36 @xmath3710@xmath38 ergs s@xmath8 , which is comparable to the _ observed _ peak dust emission luminosity at 60 @xmath0 m and 100 @xmath0 m ( @xmath35l@xmath36 @xmath14 2 @xmath12 10@xmath38 ergs s@xmath8 , as calculated based on the _ iras _ fluxes shown in table 1 ) . in the case of a buried agn , dust radiative transfer controls the temperature of the dust shell ; the temperature of the dust decreases with increasing distance from the central agn . the entire luminosity is transferred at each temperature . the present data thus provide evidence for an inner @xmath14900 k , 34 @xmath0 m continuum emitting dust shell of a luminosity similar to that of the observed outer @xmath1440 k , 60100 @xmath0 m continuum emitting dust shell , as is expected from the buried agn scenario . for obscured agns , @xcite found that the column density of x - ray absorbing gas , parameterized by @xmath39 , relative to dust extinction towards the 34 @xmath0 m continuum emitting region ( @xmath32 ) is often higher by a large factor than the galactic @xmath39/@xmath32 ratio ( 1.8 @xmath12 10@xmath40 cm@xmath8 mag@xmath8 ; predehl & schmitt 1995 ) . dust obscuration toward the 34 @xmath0 m continuum emitting region around the buried agn in ugc 5101 is so high ( @xmath42 100 mag : this work ) that @xmath39 could easily exceed 10@xmath43 @xmath26 , in which case direct 210 kev x - ray emission from the buried agns would be very strongly attenuated . the non - detection of 210 kev x - rays from the putative buried agn with asca @xcite can thus be explained without difficulty . the radio to far - infrared flux ratio of ugc 5101 is a factor of two larger than the typical values for star - formation - dominated galaxies at 1.5 ghz @xcite , but inside their scatter at 151 mhz and 5 ghz @xcite . though the bulk of the radio emission is extended @xcite , the spatially - unresolved vlbi radio core , interpreted as an agn by @xcite , is luminous enough to account for the bolomotric luminosity of ugc 5101 with agn activity @xcite . besides ugc 5101 , studies in the thermal infrared have provided evidence for the presence of energetically dominant buried agns in two other optically non - seyfert ulirgs , iras 08572 + 3915 @xcite and iras f00183@xmath17111 @xcite . the far - infrared emission properties of these three ulirgs are summarized in table 1 . in the optical , all are classified as liners @xcite . for ulirgs classified optically as liners , which constitute @xmath1440% of ulirgs @xcite , two arguments based on _ iso _ studies have been made : ( 1 ) star - formation activity is energetically dominant , and ( 2 ) the liner - type optical line emission is due to superwind - driven shocks caused by star - formation activity @xcite . the first argument , however , is not applicable to the three ulirgs in table 1 . then what is the origin of the liner - type optical emission in these objects ? in buried agns , uv emission is blocked at the inner surface of the surrounding dust shell , but x - rays can penetrate deeply into the dust to produce x - ray dissociation regions ( xdrs ; maloney , holllenbach , & tielens 1996 ) . in xdrs , locally - generated uv photons , such as lyman - werner band h@xmath44 emission and ly@xmath22 emission , can ionize some species with ionization potential less than 1011 ev ( notably carbon and iron ) , but the emitting volume is dominated by largely neutral gas , which causes xdrs essentially always to have liner - type spectra ( maloney , in preparation ) . a buried agn can thus explain the observed liner - type optical emission . however , in practical cases , some star - formation activity may very well be present at the surface of galaxies even when buried agns are energetically dominant , as we infer for ugc 5101 . in such a case , due to the susceptibility of optical emission to dust extinction , the optical spectral diagnostics are likely to be dominated by this surface star - formation emission . the star - formation activity could be responsible for liner - type optical emission via shocks , or might even alter the optical emission to a hii region type . it is important to note that narrow - line emission from high - excitation clouds characteristic of seyfert galaxy spectra which are observed to be produced along the axes of nuclear dust tori would not contribute substantially at any wavelength in the case of buried agn . we have found evidence for the presence of an energetically dominant , buried agn in the ulirg ugc 5101 , which is characterized by a cool far - infrared color and a liner optical spectrum . this finding seems contrary to the currently established view , based on _ iso _ data , that such objects are powered by star formation . two other liner ulirgs also show evidence for energetically dominant buried agns , and we therefore conclude that at least some fraction of liner ulirgs are powered by buried agns . we are grateful to j. rayner and b. golisch for their support before and during the observing run . research in infrared astronomy at nrl is supported by the office of naval research . prm is supported by the nsf under grant ast-9900871 . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . alonso - 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we report on the results of 34 @xmath0 m spectroscopy of the ultra - luminous infrared galaxy ( ulirg ) ugc 5101 . it has a cool far - infrared color and a liner - type optical spectrum , and so , based on a view gaining some currency , would be regarded as dominated by star formation . however , we find that it has strong 3.4 @xmath0 m carbonaceous dust absorption , low - equivalent - width 3.3 @xmath0 m polycyclic aromatic hydrocarbon ( pah ) emission , and a small 3.3 @xmath0 m pah to far - infrared luminosity ratio . this favors an alternative scenario , in which an energetically dominant agn is present behind obscuring dust . the agn is plausibly obscured along all lines of sight ( a ` buried agn ' ) , rather than merely obscured along our particular line of sight . such buried agns have previously been found in thermal infrared studies of the ulirgs iras 08572 + 3915 and iras f00183@xmath17111 , both classified optically as liners . we argue that buried agns can produce liner - type optical spectra , and that at least some fraction of liner - type ulirgs are predominantly powered by buried agns .

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Proceed to summarize the following text: eit is a non - invasive real - time functional imaging modality for the continuous monitoring of physiological functions such as lung ventilation and perfusion . the image contrast represents the time change of the electrical conductivity distribution inside the human body . using an array of surface electrodes around a chosen imaging slice , the imaging device probes the internal conductivity distribution by injecting electrical currents at tens or hundreds of khz . the injected currents ( at safe levels ) produce distributions of electric potentials that are non - invasively measured from the attached surface electrodes . a portable eit system can provide functional images with an excellent temporal resolution of tens of frames per second . eit was introduced in the late 1970s@xcite , likely motivated by the success of x - ray ct . numerous image reconstruction methods and experimental validations have demonstrated its feasibility@xcite and clinical trials have begun especially in lung ventilation imaging and pulmonary function testing@xcite . however , eit images often suffer from measurement noise and artifacts especially in clinical environments and there still exist needs for new image reconstruction algorithms to achieve both high image quality and robustness . the volume conduction or lead field theory indicates that a local perturbation of the internal conductivity distribution alters the measured current voltage or trans - impedance data , which provide core information for conductivity image reconstructions . the sensitivity of the data to conductivity changes varies significantly depending on the distance between the electrodes and the conductivity perturbation : the measured data are highly sensitive to conductivity changes near the current - injection electrodes , whereas the sensitivity drops rapidly as the distance increases@xcite in addition , the boundary geometry and the electrode configuration also significantly affect the measured boundary data . the sensitivity map between the measured data and the internal conductivity perturbation is the basis of the conductivity image reconstruction . a discretization of the imaging domain results in a sensitivity or jacobian matrix , which is inverted to produce a conductivity image . the major difficulty arises from this inversion process , because the matrix is severely ill - conditioned . conductivity image reconstruction in eit is , therefore , known to be a fundamentally ill - posed inverse problem . uncertainties in the body shape , body movements , and electrode positions are unavoidable in practice , and result in significant amounts of forward and inverse modeling errors . measurement noise and these modeling errors , therefore , may deteriorate the quality of reconstructed images . numerous image reconstruction algorithms have been developed to tackle this ill - posed inverse problem with data uncertainties@xcite . in the early 1980s , an eit version of the x - ray ct backprojection algorithm@xcite was developed based on a careful understanding of the ct idea . the one - step gauss - newton method@xcite is one of the widely used methods and often called the linearized sensitivity method . several direct methods were also developed : the layer stripping method@xcite recovered the conductivity distribution layer by layer , the d - bar method@xcite was motivated from the constructive uniqueness proof for the inverse conductivity problem@xcite , and the factorization method@xcite was originated as a shape reconstruction method in the inverse scattering problem@xcite . recently , the discrete cosine transform was adopted to reduce the number of unknowns of the inverse conductivity problem@xcite . there is an open source software package , called eidors , for forward and inverse modelings of eit@xcite . there are also novel theoretical results showing a unique identification of the conductivity distribution under the ideal model of eit@xcite . most common eit image reconstruction methods are based on some form of least - squares inversion , minimizing the difference between the measured data and computed data provided by a forward model . various regularization methods are adopted for the stable inversion of the ill - conditioned jacobian matrix . these regularized least - squares data - fitting approaches adjust the degree of regularization by using a parameter , controlling the trade - off between data fidelity and stability of reconstruction . their performances , therefore , depend on the choice of the parameter . in this paper , we propose a new regularization method , which is designed to achieve satisfactory performances in terms of both fidelity and stability regardless of the choice of the regularization parameter . investigating the correlations among the column vectors of the jacobian matrix , we developed a new regularization method in which the structure of data fidelity is incorporated . we also developed a motion artifact removal filter , that can be applied to the data before image reconstructions , by using a sub - matrix of the jacobian matrix . after explaining the developed methods , we will describe experimental results showing that the proposed fidelity - embedded regularization ( fer ) method combined with the motion artifact removal filter provides stable image reconstructions with satisfactory image quality even for very large regularization parameter values , thereby making the method irrelevant to the choice of the parameter value . to explain the eit image reconstruction problem clearly and effectively , we restrict our description to the case of a 16-channel eit system for real - time time - difference imaging applications . the sixteen electrodes ( @xmath0 ) are attached around a chosen imaging slice , denoted by @xmath1 . we adopt the neighboring data collection scheme , where the device injects current between a neighboring electrode pair @xmath2 and simultaneously measures the induced voltages between all neighboring pairs of electrodes @xmath3 for @xmath4 . here , we denote @xmath5 . let @xmath6 be the electrical conductivity distribution of @xmath1 , and let @xmath7 denote the boundary surface of @xmath1 . the electrical potential distribution corresponding to the @xmath8th injection current , denoted by @xmath9 , is governed by the following equations : @xmath10 where @xmath11 is the outward unit normal vector to @xmath12 , @xmath13 is the electrode contact impedance of the @xmath14th electrode @xmath15 , @xmath16 is the potential on @xmath15 subject to the @xmath8th injection current , and @xmath17 is the amplitude of the injection current between @xmath18 and @xmath19 . assuming that @xmath20 , the measured voltage between the electrode pair @xmath3 subject to the @xmath8th injection current at time @xmath21 is expressed as : @xmath22 the voltage data @xmath23 in a clinical environment are seriously affected by the following unwanted factors@xcite : * unknown and varying contact impedances making the data @xmath24 for the @xmath8th injection current unreliable and * inaccuracies in the moving boundary geometry and electrode positions . here , we set @xmath25 and @xmath26 . among those sixteen voltage data for each injection current , thirteen are measured between electrode pairs where no current is injected , that is , the normal components of the current density are zero . for these data , it is reasonable to assume @xmath27 to get @xmath28 in . discarding the remaining three voltage data , which are sensitive to changes in the contact impedances , we obtain the following voltage data at each time : @xmath29^t\ ] ] the total number of measured voltage data is @xmath30 at each time . the data vector @xmath31 reflects the conductivity distribution @xmath6 , body geometry @xmath1 , electrode positions @xmath32 , and data collection protocol . inevitable discrepancies exist between the forward model output @xmath63 and the measured data @xmath51 due to modeling errors and measurement noise : the real background conductivity is not homogeneous , the boundary shape and electrode positions change with body movements and there always exist electronic noise and interferences . motion artifacts are inevitable in practice to produce errors in measured voltage data and deteriorate the quality of reconstructed images@xcite . to investigate how the motion artifacts influence the measured voltage data , we take time - derivative to both sides of assuming the domain @xmath1 varies with time . it follows from the reynolds transport theorem that @xmath64 where @xmath65 is the outward - normal directional velocity of @xmath7 . note that the last term of is the voltage change due to the boundary movement . using the chain rule , the first term of the right - hand side of is expressed as @xmath66 it follows from the integration by parts and that @xmath67 here , we neglected the contact impedances underneath the voltage - sensing electrodes and approximated @xmath68 . from - , can be expressed as : @xmath69 note that becomes when the boundary does not vary with time ( @xmath70 ) . we linearize by replacing @xmath44 with the computed potential @xmath71 in : @xmath72 after discretization , can be expressed as : @xmath73 where @xmath74 is given by @xmath75^t\ ] ] with @xmath76 . compared to , has the additional term @xmath74 that is the ( linearized ) error caused by the boundary movement . this term multiplied by the strong sensitivities on the boundary @xmath77 becomes a serious troublemaker , and can not be neglected in because the vectors @xmath78^t$ ] for @xmath79 have large magnitudes . to filter out the uncertain data @xmath74 related with motion artifacts from @xmath80 , we introduce the boundary sensitive jacobian matrix @xmath81 , which is a sub - matrix of @xmath59 consisting of all columns corresponding to the triangular elements located on the boundary . the boundary movement errors in the measured data @xmath80 are , then , assumed to be in the column space of @xmath81 . the boundary errors are extracted by @xmath82 where @xmath83 is a regularization parameter and @xmath84 is the identity matrix . then , the motion artifact is filtered out from data by subtraction @xmath85 . the proposed motion artifact removal is performed before image reconstruction using any image reconstruction method . in this paper , the filtered data @xmath86 were used in places of @xmath80 for all image reconstructions . severe instability arises in practice from the ill - conditioned structure of @xmath59 when some form of its inversion is tried . to deal with this fundamental difficulty , the regularized least - squares data - fitting approach is commonly adopted to compute @xmath87 with a suitably chosen regularization parameter @xmath88 and regularization operator @xmath89 . such image reconstructions rely on the choice of @xmath88 ( often empirically determined ) and @xmath89 using _ a priori _ information , suffering from over- or under - regularization . we propose the fidelity - embedded regularization ( fer ) method : @xmath90 where the regularization operator @xmath91 is the diagonal matrix such that @xmath92 to explain the fer method , we closely examine the correlations among column vectors of the sensitivity matrix @xmath59 , described in fig . [ figs : sensitivitymatrix ] . the correlation between @xmath62 and @xmath93 can be expressed as @xmath94 where @xmath95 denotes the standard inner product . here , @xmath96 is a solution of @xmath97 where @xmath98 is the characteristic function having @xmath99 on @xmath55 and @xmath100 otherwise . the identity follows from @xmath101 for @xmath102@xcite . this shows that the column vector @xmath62 is like an eeg ( electroencephalography ) data induced by dipole sources with directions @xmath103 at locations @xmath55 . given that two dipole sources at distant locations produce mutually independent data , the correlation between @xmath104 and @xmath105 decreases with the distance between @xmath55 and @xmath106 . [ figs : sensitivitymatrix ] shows a few images of the correlation @xmath107 as a function of @xmath108 for four different positions @xmath55 . the correlation decreases rapidly as the distance increases . in the green regions where the correlation is almost zero , @xmath93 is nearly orthogonal to @xmath109 . [ figs : sensitivitymatrix ] shows that if @xmath55 and @xmath106 are far from each other , the corresponding columns of the sensitivity matrix are nearly orthogonal . this somewhat orthogonal structure of the sensitivity matrix motivates an algebraic formula that directly computes the local ensemble average of conductivity changes at each point using the inner product between changes in the data and a scaled sensitivity vector at that point : @xmath110 where @xmath111 is the weighted average conductivity at the @xmath56th element @xmath55 and the weight is expressed in terms of the correlations between columns of @xmath59 . it turns out that this simple formula shows a remarkable performance in terms of robustness , but requires a slight compromise in spatial resolution . substituting @xmath112 into , the relation between @xmath113 and @xmath114 can be expressed as the following convolution form : @xmath115 where @xmath116 . the non - zero scaling factor @xmath117 is designed for normalization . the kernel @xmath118 satisfies the following : * @xmath119 for each @xmath56 , due to the non - zero scaling factor . * @xmath118 decreases as the distance between @xmath55 and @xmath106 increases ( except near boundary where strong sensitivity arises ) . hence , @xmath118 roughly behaves like a blurred version of the dirac delta function . this is the reason why the formula directly computes the local ensemble average of conductivity changes at each point . the algebraic formula can be seen as a regularized least - squares data - fitting method when the regularization operator is @xmath91 . then , the formula can be expressed using @xmath91 in as @xmath120 this can be formulated similarly as for an extremely large value of @xmath88 ( @xmath121 ) when @xmath122 : @xmath123 here , @xmath124 is a devised scaling term to prevent the reconstructed image from becoming zero when @xmath88 goes to infinity . the fer method in was proposed based on and . when the regularization parameter @xmath88 is small ( @xmath125 ) , the fer method is equivalent to the regularized least - squares data - fitting method . when @xmath88 is large ( @xmath126 ) , it converges to the algebraic formula and directly recovers the weighted average conductivity @xmath113 . the regularization operation @xmath91 fully exploits the somewhat orthogonal structure of the sensitivity matrix , thereby embedding data fidelity in the regularization process . adopting this carefully designed @xmath91 , the fer method provides stable conductivity image reconstructions with high fidelity even for very large regularization parameter values . we applied the fer method to experimental data to show its performance . we acquired the boundary geometry and electrode positions as accurate as possible to reduce forward modeling uncertainties@xcite . a handheld 3d scanner was used to capture the boundary shape of the thorax and electrode positions ( fig . [ figs : eit_system ] ) . then , we set the electrode plane as the horizontal cross - section of the 3d - scanned thorax containing the attached electrodes ( fig . [ figs : eit_system ] ) . the finite element method was employed to compute the sensitivity matrix @xmath59 by discretizing the imaging slice . here , we used a mesh with 12,001 nodes and 23,320 triangular elements for subject a and a different mesh with 13,146 nodes and 25,610 triangular elements for subject b. figs . [ fig : subjecta ] and [ fig : subjectb ] compare the performance of the proposed fer method in with the standard regularized least - squares method ( when @xmath127 is the identity matrix ) . the regularization parameter of the standard method was heuristically chosen for its best performance , and the parameter of the fer method was set to be one of three different values @xmath128 . the injection current was 1 ma@xmath129 at 100 khz , and the frame rate was 9 frames per second . the reference frame at @xmath130 was obtained from the maximum expiration state . the measured data , @xmath131 , represent the voltage differences between each time @xmath132 and @xmath130 . the blue regions , which denote where conductivity decreased by inhaled air , increased during inspiration and decreased during expiration . the fer method with @xmath121 was clearly more robust than the standard method that produced more artifacts originated from the inversion process . ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath142 & @xmath143 & @xmath144&@xmath145&@xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath142 & @xmath143 & @xmath144&@xmath145&@xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath142 & @xmath143 & @xmath144&@xmath145&@xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath142 & @xmath143 & @xmath144&@xmath145&@xmath146&@xmath147&@xmath148&@xmath149&@xmath150&@xmath151 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath152 & @xmath137&@xmath153&@xmath142&@xmath154&@xmath147&@xmath155&@xmath156&@xmath157 & + & & & & & & & & & & & + & @xmath158 & @xmath159 & @xmath160&@xmath161&@xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath152 & @xmath137&@xmath153&@xmath142&@xmath154&@xmath147&@xmath155&@xmath156&@xmath157 & + & & & & & & & & & & & + & @xmath158 & @xmath159 & @xmath160&@xmath161&@xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath158 & @xmath159 & @xmath160&@xmath161&@xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167 & + & & & & & & & & & & & + ' '' '' & @xmath130 & @xmath133 & @xmath134&@xmath135&@xmath136&@xmath137&@xmath138&@xmath139&@xmath140&@xmath141 & + & & & & & & & & & & & + & @xmath158 & @xmath159 & @xmath160&@xmath161&@xmath162&@xmath163&@xmath164&@xmath165&@xmath166&@xmath167 & + & & & & & & & & & & & + note that the degree of orthogonality of the columns of the sensitivity matrix depends on the current injection pattern . this makes the performance of the fer method depend on the current injection pattern since it incorporates the structure of the sensitivity matrix in the regularization process . for example , if we inject currents between diagonal pairs of electrodes , the corresponding sensitivity matrix becomes less orthogonal compared with that using the neighboring current injection protocol , thus producing more blurred images . in contrast , a more narrower injection angle , for example , using a 32-channel eit system with the adjacent injection pattern , can enhance the orthogonality of the corresponding sensitivity matrix . however , the narrower injection angle results in poor distinguishability@xcite and may deteriorate the image quality . to maximize the performance of the fer method , balancing between orthogonality of the sensitivity matrix and distinguishability should be considered when designing a data collection protocol . the direct algebraic formula or for @xmath121 can be expressed as a transpose of a scaled sensitivity matrix . this type of direct approach was suggested in the late 1980s by kotre@xcite , but was soon abandoned owing to poor performance and lack of theoretical grounding@xcite . since then , regularized inversion of the sensitivity matrix has been the main approach for eit image reconstruction . kotre s method using the normalized transpose of @xmath168 was regarded as an extreme version of the backprojection algorithm in eit@xcite , in the sense of the radon transform in ct ; in the case when @xmath168 is the radon transform of ct , its adjoint is known as the backprojector . with this interpretation , it seems that kotre s method is very sensitive to forward modeling uncertainties . in the fer method , @xmath118 is a scaled version of the adjoint of the jacobian , which can be viewed as a blurred version of the dirac delta function . the fer method with @xmath121 becomes a direct method for robust conductivity image reconstructions without inverting the sensitivity matrix @xmath59 . since the 1980s , many eit image reconstruction methods have been developed to overcome difficulties in achieving robust and consistent images from patients in clinical environments . recent clinical trials of applying eit to mechanically ventilated patients have shown its feasibility as a new real - time bedside imaging modality . they also request , however , more robust image reconstructions from patients data contaminated by noise and artifacts . the proposed fer method achieves both robustness and fidelity by incorporating the structure of the sensitivity matrix in the regularization process . unlike most other algorithms , the fer method also offers direct image reconstructions without matrix inversion . this has a practical advantage especially in clinical environments since the direct method does not require any adjustment of regularization parameters . in addition to time - difference conductivity imaging , the fer method enables robust spectroscopic admittivity imaging of both conductivity and permittivity , and can employ frequency - difference approaches as well . although we showed that @xmath113 in provides a satisfactory approximation to @xmath114 , it is difficult to estimate its accuracy . this is related to the kernel @xmath118 and the structure of the sensitivity matrix @xmath59 in which the current injection pattern is incorporated . it is a challenging issue to mathematically characterize how precisely the kernel @xmath118 approximates the dirac delta function with respect to the current inject pattern . k.l . and j.k.s . were supported by the national research foundation of korea ( nrf ) grant 2015r1a5a1009350 . e.j.w . was supported by a grant of the korean health technology r&d project ( hi14c0743 ) . t. meier et al . , assessment of regional lung recruitment and derecruitment during a peep trial based on electrical impedance tomography " , _ intensive care med . _ , 543 - 550 , 2008 . i. frerichs et al . , chest electrical impedance tomography examination , data analysis , terminology , clinical use and recommendations : concesus statement of the translational eit development study group " , _ thorax _ at press , 2016 . d. isaacson , distinguishability of conductivities by electric current computed tomography " , _ ieee trans . med . imaging _ , mi-5 , nol . 2 , pp . 91 - 95 , 1986 . b. h. brown , d. c. barber , and a. d. seagar , applied potential tomography : possible clinical applications " , _ clin . 109 - 121 , 1985 f. santosa and m. vogelius , a back - projection algorithm for electrical impedance imaging " , _ siam j. appl . 216 - 243 , 1990 m. cheney , d. isaacson , j. newell , s. simske , and j. goble , noser : an algorithm for solving the inverse conductivity problem " , _ internat . j. imaging systems and technology _ , vol . 66 - 75 , 1990 . s. siltanen , j. mueller , and d. isaacson , an implementation of the reconstruction algorithm of a nachman for the 2d inverse conductivity problem " , _ inverse problems _ , vol . 681 - 699 , 2000 . j. l. mueller , s. siltanen , d. isaacson , a direct reconstruction algorithm for electrical impedance tomography " , _ ieee trans . med . 21 , no . 6 , pp . 555 - 559 , 2002 . m. k. choi , b. bastian , and j. k. seo , regularizing a linearized eit reconstruction method using a sensitivity - based factorization method " , _ inverse probl . 22 , no . 7 , pp . 1029 - 1044 , 2014 . v. kolehmainen , m. vauhkonen , p. a. karjalainen , j. p. kaipio , assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns " , _ physiol . 4 , pp . 289 - 303 , 1997 .

electrical impedance tomography ( eit ) provides functional images of an electrical conductivity distribution inside the human body . since the 1980s , many potential clinical applications have arisen using inexpensive portable eit devices . eit acquires multiple trans - impedance measurements across the body from an array of surface electrodes around a chosen imaging slice . the conductivity image reconstruction from the measured data is a fundamentally ill - posed inverse problem notoriously vulnerable to measurement noise and artifacts . most available methods invert the ill - conditioned sensitivity or jacobian matrix using a regularized least - squares data - fitting technique . their performances rely on the regularization parameter , which controls the trade - off between fidelity and robustness . for clinical applications of eit , it would be desirable to develop a method achieving consistent performance over various uncertain data , regardless of the choice of the regularization parameter . based on the analysis of the structure of the jacobian matrix , we propose a fidelity - embedded regularization ( fer ) method and a motion artifact removal filter . incorporating the jacobian matrix in the regularization process , the new fer method with the motion artifact removal filter offers stable reconstructions of high - fidelity images from noisy data by taking a very large regularization parameter value . the proposed method showed practical merits in experimental studies of chest eit imaging .

You are an expert at summarizing long articles. Proceed to summarize the following text: hera deeply inelastic scattering ( dis ) results on structure functions demonstrate a rapid bremsstrahlung growth of the gluon density at small x. when interpreted in the same framework as the parton model , this growth is predicted to saturate because the gluon occupation number in hadron wave functions saturate at a value maximally of order @xmath1 ; dynamically , nonlinear effects such as gluon recombination and screening by other gluons deplete the growth of the gluon distribution@xcite . gluon modes with @xmath2 are maximally occupied , where @xmath3 is a dynamically generated semi - hard scale called the saturation scale . for small @xmath4 , @xmath5 is large enough that high occupancy states can be described by weak coupling classical effective theory@xcite . this color glass condensate description of high energy hadrons and nuclei is universal and has been tested in both dis and hadronic collisions . in particular , saturation based phenomenological predictions successfully describe recent lhc p+p data @xcite and predict possible geometrical scaling of transverse momentum distribution@xcite similar to the geometrical scaling observed previously in dis . the object common to dis and hadronic collisions is the dipole cross section @xmath6 . in the cgc framework , the dipole cross section can be expressed in terms of expectation values of correlators of wilson lines representing the color fields of the target . the energy dependence of this quantity comes from renormalization group evolution but to get the realistic impact parameter dependence one has to rely on models involving parametrizations constrained by experimental data . in the large @xmath7 limit , the dipole cross section is related to the un - integrated gluon distribution inside hadron / nucleus as @xmath8^{2}. \label{eq : unint - gluon}\ ] ] for hadron - hadron collisions , the inclusive gluon distribution which is @xmath9-factorizable into the products of un - integrated gluon distributions in the target and projectile is expressed as @xmath10 two models of the dipole cross - section that have been extensively compared to hera data are the ip - sat @xcite and the b - cgc @xcite models . in the former the impact parameter dependence is introduced through a normalized gaussian profile function @xmath11 and in the latter through a scale @xmath12 . for a detailed discussion of the parameters involved in these models and their values from fits to hera data , see ref . @xcite . the saturation scale in the fundamental representation for both the models can be calculated self consistently solving @xmath13=2(1-e^{-1/2})$ ] . the corresponding adjoint saturation scale @xmath14 , relevant for hadronic collisions , is obtained by multiplying @xmath15 by 9/4 . in the range @xmath16-@xmath17 , the behaviour of @xmath14 ( see fig.[fig : satscale ] left ) at @xmath18 can be approximated by a function of the form @xmath19 with @xmath20 for the b - cgc model and @xmath21 for the ip - sat model . [ fig : multdist ] multiparticle production in high energy hadronic collisions can be treated self consistently in the cgc approach . the glasma flux tube picture @xcite predicts @xcite that the n - particle correlation is generated by the negative binomial distribution @xmath22 . it is characterized by two parameters , the mean multiplicity @xmath23 and @xmath24 . at a given impact parameter of the collision , the mean multiplicity @xmath25 is obtained by integrating eq . [ eq : ktfact1 ] over @xmath26 . in the glasma picture , the parameter @xmath27 with @xmath28 @xcite . the quantity @xmath29 shown in fig.[fig : satscale ] ( right ) is the number of flux tubes in the overlap area @xmath30 of two hadrons . convolving @xmath31 with the probability distribution @xmath32 for an inelastic collision at @xmath33-fig . [ fig : multdist ] ( left)-one obtains @xcite the n - particle inclusive multiplicity distribution as shown in fig . [ fig : multdist ] ( right ) . various kinematic variables exhibit scaling with the saturation scale@xcite . the mid - rapidity multiplicity density scales with functional forms like @xmath34 and @xmath35 whereas a linear functional form seem to provide very good fit to the energy dependence of @xmath36 as shown in fig.[fig : scaling][left ] . these results are suggestive that @xmath37 is the only scale that controls the bulk particle multiplicity . in ref . @xcite it has been shown that @xmath26 spectra in @xmath38 collisions exhibit geometric scaling assuming a simple form of @xmath37 . in our case we use a scaling variable @xmath39 , where @xmath37 is directly calculated in the ip - sat model . as shown in fig.[fig : scaling][right ] , an approximate scaling below @xmath40 is observed for transverse momentum distribution in @xmath38 collision energy @xmath41 gev . going to lower energies we observe systematic deviations from the universal curve . + in summary , our description of multiplicity distribution successfully describes bulk lhc p+p data . in particular , we observe that the dominant contribution to multiplicity fluctuations is due to the intrinsic fluctuations of gluon produced from multiple glasma flux tubes rather than from the fluctuations in the sizes and distributions of hotspots . the @xmath26-spectra in p+p at high energies exhibits universal scaling as a function of @xmath39 . the observed scaling indicates that particle production in this regime is dominantly from saturated gluonic matter characterized by one universal scale @xmath37 . ridge like two particle correlation structures in @xmath42 in high multiplicity p+p collisions may provide more detailed insight into its properties @xcite . v. khachatryan _ et al . _ [ cms collaboration ] , phys . lett . * 105 * , 022002 ( 2010 ) . k. aamodt _ et al . _ [ alice collaboration ] , eur . j. c * 68 * , 345 ( 2010 ) . a. dumitru , k. dusling , f. gelis , j. jalilian - marian , t. lappi , r. venugopalan , arxiv:1009.5295 [ hep - ph ] .

dipole models based on various saturation scenarios provide reasonable fits to small - x dis inclusive , diffractive and exclusive data from hera . proton un - integrated gluon distributions extracted from such fits are employed in a @xmath0-factorization framework to calculate inclusive gluon distributions at various energies . the n - particle multiplicity distribution predicted in the glasma flux tube approach shows good agreement with data over a wide range of energies . hadron inclusive transverse momentum distributions expressed in terms of the saturation scale demonstrate universal behavior over a wider kinematic range systematically with increasing center of mass energies . saturation ; lhc p + p collision ; cgc ; deep inelastic scattering

You are an expert at summarizing long articles. Proceed to summarize the following text: it is well known that the properties of biological vesicles and membrane cells strongly depend on the their constituent blocks , usually composed of phospholipid bilayers with embedded proteins . these molecules are in general anisotropic ( rod or plate - like ) and the demixed states usually have liquid - crystal symmetries , such as isotropic ( i ) , nematic ( n ) or biaxial nematic ( b ) symmetries . for certain conditions these complex mixtures of biomolecules phase separate , creating regions rich in different species and consequently changing the membrane curvature . there is much experimental evidence of demixing transitions in monolayers and bilayers of mixed anisotropic biomolecules @xcite . the adsorption of a large variety of mixtures of rod - like molecules in langmuir monolayers has been extensively studied both experimentally and theoretically . many of these works focus on the chemical and thermodynamic conditions for which the monolayers become spatially heterogeneous , i.e. when the mixture demixes in different phases usually possessing liquid - crystal ordering @xcite . finally many experiments on the adsorption of rod - like colloidal particles at the interfaces separating two immiscible fluids showed the propensity of these particles to self - assemble into clusters of different geometries @xcite . the degree of adsorption of these particles at the interface and their relative orientation with respect to it strongly depend on their chemical compositions . this in turn can modify their wetting properties and consequently the effective capillary forces acting between particles . the resulting effect is the existence of anisotropy in the pair interaction potential which forces the particles to self - assemble into clusters @xcite . when colloids with very different chemical properties are adsorbed at the interface they usually phase separate into phases with different composition of species . a recent experimental work showed how demixing of adsorbed colloids strongly modifies their self - assembling properties @xcite . all the systems discussed above share the following properties : ( i ) they are mixtures of anisotropic particles , ( ii ) the degrees of freedom of their centres of mass are strongly restricted , usually resulting in an effective two - dimensional fluid , and ( iii ) the particle axes can rotate in 3d but with certain restrictions which depend on the degree of particle adsorption on the monolayer , bilayer or interface . the main motivation of the present work is the formulation of a very simple model for a binary mixture of anisotropic particles ( specifically a mixture of rods and plates ) which allows a detailed study of the conditions ( particle aspect ratios , degree of adsorption ) under which these mixtures demix into two different phases . to this purpose we choose particles to have a board - like shape and interact through a hard - core repulsion . also , for simplicity , we use the zwanzig approximation to account for the orientational degrees of freedom , which are restricted to be three . finally , we will use a mean - field density functional ( df ) based on the fundamental - measure theory ( fmt ) , derived for the present model in the late 90 s @xcite and more recently implemented to calculate the phase diagrams of rod and plate board - like particles @xcite . monolayers of one - component rods or plates were recently studied within this theory , considering uniaxial @xcite and biaxial @xcite particle geometries . in the latter work , phase diagrams were calculated as a function of a geometric parameter @xmath3 $ ] that measured particle shape , with @xmath4 for uniaxial rod and plate geometries , respectively . one - component monolayers of prolate or oblate freely - rotating ellipsoids were also recently studied via the parsons - lee df and molecular - dynamics simulations @xcite . in particular , the effect of orientational restriction of ellipsoids on the orientational properties of monolayers was studied . the ground states of monolayers of hard ellipsoids interacting through a quadrupole pair - potential were recently found @xcite . apart from the t - like configurations , three more particle orientations were predicted to be stable . here we extend our previous model @xcite to a binary mixture of uniaxial rods and plates . board - like - shaped rods and plates are taken to be symmetric : although they have different eccentricities ( prolate and oblate ) , their volumes and their aspect ratios ( ratio between major and minor particle edge - lengths ) are taken to be the same . this choice of shape geometries is motivated by their extensive use in studies of the biaxial - nematic ( b ) phases stability with respect to nematic - nematic ( n - n ) phase separation in binary @xcite and polydisperse @xcite mixtures of rods and plates . we are interested in the effect of particle adsorption on the phase behaviour of the mixture , in an effort to elucidate ( i ) the propensity of the system to phase separate into different phases , ( ii ) the nature ( second vs. first order ) of the n - b transition when the adsorption strengths are changed , ( iii ) the relative stability of the b phase with respect to the n or other non - uniform phases , and ( iv ) the representative phase diagrams of the system ( calculated for certain selected values of model parameters ) . in general we found a rich phase behaviour with the presence of two disconnected demixed n - b phase transitions , one located at low pressures , with a b phase rich in rods , and the other at very high pressures , with a b phase rich in plates ( although b - b demixing also occurs in a narrow range of pressures ) . when plates are strongly adsorbed , the mixture exhibits a strong first - order phase transition . different demixing scenarios depend on the relative values of plate and rod adsorption coefficients . when adsorption of rods is large as compared to that of plates , the b - phase stability is greatly enhanced , the n - b transition is always of second order , and no demixing occurs . the article is organized as follows . in sec . [ model ] we introduce the model and the theoretical tools used to perform the calculations . in sec . [ results ] we summarize all the results obtained , with subsections presenting different mixtures with various relative adsorption strengths and particle aspect ratios . finally some conclusions are drawn in sec . [ conclusions ] . we use the zwanzig model for a binary mixture of prolate ( rods ) and oblate ( plates ) board - like particles with centres of mass lying on the @xmath5 plane and with main axes pointing along the @xmath6 directions . the edge lengths of species @xmath7 ( @xmath8 for rods and @xmath9 for plates ) are represented through the tensor @xmath10 with @xmath11 the kronecker delta symbol , while @xmath12 and @xmath13 are the particle sizes parallel and perpendicular to the main particle axis , respectively . therefore , particles are uniaxial parallelepipeds with a square section of area @xmath14 . only symmetric mixtures will be studied , namely those composed of particles with the same volume ( which is set to unity , @xmath15 ) , and with aspect ratios @xmath16 and @xmath17 related by @xmath18 . thus the edge lengths of the different species are calculated as @xmath19 , @xmath20 . [ fig1 ] for a schematic representation of our model . the theory used in the present calculations is the uniform limit of the fmt density functional obtained by applying the dimensional crossover property . this feature allows to correctly transform the functional from 3d to 2d by assuming a 3d density profile , for species @xmath7 and orientation along the @xmath21-axis , of the form @xmath22 , i.e. imposing that the particle centres of mass are constrained to the flat surface perpendicular to @xmath23 . when this density is substituted into the 3d version of the excess free - energy functional , the resulting functional depends on the ( constant ) 2d number densities @xmath24 where @xmath25 is the total number density , @xmath26 is the molar fraction of species @xmath7 , while @xmath27 is the fraction of species @xmath21 with main axes pointing along the @xmath21-direction . obviously these quantities fulfill the following constraints : @xmath28 and @xmath29 or alternatively @xmath30 and @xmath31 , with @xmath32 the number density of species @xmath7 . the resulting excess free - energy density in reduced thermal units @xmath33 depends on the following weighted densities : @xmath34 and has the explicit form @xmath35 where @xmath36 is the boltzmann factor and @xmath37 the total area of the system . we note that the weighted density @xmath38 is just the total packing fraction , @xmath39 , of the binary mixture . the ideal part , @xmath40 is , as usual @xmath41 and the effective interaction between species @xmath7 , with projected area on the @xmath5 plane @xmath42 , and the surface , is accounted for by an external potential contribution to the free - energy density : @xmath43 note that this contribution is proportional to the projected particle areas , and that we allow for the possibility that the adsorption strengths @xmath44 be dependent on species @xmath7 . to find the equilibrium orientational properties of the fluid we minimize the total free - energy density @xmath45 with respect to the fractions @xmath27 . these can be related to the uniaxial nematic order parameters , @xmath46 which measure the order about the direction perpendicular to the surface , and to the biaxial nematic order parameters , @xmath47 which measure the degree of biaxial order . note that the factor @xmath48 in the definition of @xmath49 is necessary to take account of the orthogonality of the plate and rod main axes when their projections have the same orientations . we have calculated the phase diagram by searching for possible demixing transitions through ( i ) the equality between chemical potentials of species @xmath7 , @xmath50 , and ( ii ) the equality between the pressures of the demixed phases . the latter can be calculated as @xmath51 ( in reduced thermal units ) . these calculations are equivalent to finding the coexisting molar fractions of the demixed phases through the double - tangent construction of the gibbs free - energy density , @xmath52 as a function of the molar fraction @xmath53 at constant value of @xmath54 . this constraint allows us to find the total number density @xmath55 as a function of @xmath56 at fixed @xmath54 , while all the quantities @xmath27 should be calculated for the same values @xmath57 from the set of equations @xmath58 . to find the location of second - order phase transitions to biaxial phases we use a bifurcation analysis of the total free - energy density @xmath59 with respect to the ( small ) order parameters @xmath49 . for details of these calculations see appendix [ append ] . in the rest of the manuscript we use dimensionless densities @xmath60 and pressures @xmath61 . this section is devoted to the study of the phase diagram topologies as a function of the adsorption strengths @xmath62 and the aspect ratio @xmath0 of the mixture . we firstly studied the monolayer of rods and plates with zero adsorption to the surface by setting @xmath63 . the phase diagram obtained for a mixture with @xmath1 is shown in fig . [ fig2](a ) in the pressure - composition plane . the dashed line , which departs at its lowest pressure from the left vertical axis ( @xmath64 ) , represents a continuous n - b phase transition . from low to high pressures , but below the n - b spinodal , the configuration of plates changes from a nearly equimolar composition of their three species to that in which the species with the largest ( square ) projected area , equal to @xmath65 , has the lowest composition while the other two species , with rectangular projected shapes of aspect ratio @xmath0 and surface area @xmath66 , have equal compositions . the plate axes of the latter species are parallel to the surface but their rectangular sections are yet randomly oriented in 2d . this phase is a planar n with a negative uniaxial order parameter which decreases with pressure . the configuration of rods at low pressures always exhibits a preferential alignment perpendicular to the monolayer , resulting in a higher proportion of projected ( small ) squares of surface area @xmath67 . the other two species of rods , having rectangular shapes , aspect ratio @xmath0 , and surface area @xmath68 , have again the same composition . thus the uniaxial order parameter of rods is always positive and increases with pressure . the authors of the recent work @xcite showed that the uniaxial order parameter @xmath69 of one - component rods on a surface increases linearly from zero as a function of @xmath70 . as the pressure increases , the fraction of plates with their axes oriented parallel to the surface and that of rods oriented perpendicular to it become larger , and the uniaxial order parameters tend to @xmath71 and @xmath72 , respectively . at a certain pressure ( which depends on molar fraction ) , the @xmath5 orientational symmetry is broken in a continuous fashion , and the projected rectangular species for both , rods and plates , begin to align along a preferential direction , say the @xmath73 direction . from this pressure the b phase becomes stable and the b order parameters continuously increase from zero . see fig . [ fig0 ] for a sketch of projected shape configurations for the different phases . [ fig2 ] indicates that biaxial ordering is promoted by the plates : the n - b spinodal , departing from the left vertical axis ( @xmath74 ) , is a monotonically increasing function of @xmath56 and possesses an asymptote at @xmath75 . the latter means that one - component monolayers of rods do not exhibit b ordering . from now on we refer to this spinodal as the _ plate _ n - b _ spinodal_. at very high pressures , most of the plate axes align parallel to the monolayer , while those of rods are perpendicular to it . thus the system can be approximated by a 2d mixture of zwanzig hard rectangles ( of area @xmath66 and aspect ratio @xmath0 ) and parallel hard squares ( of area @xmath67 ) . this 2d mixture demixes into two phases , each one rich in one of the species , as shown in fig . [ fig2 ] . the critical point of the demixing transition is above the n - b spinodal , meaning that b - b coexistence takes place in some pressure interval . for pressures above the crossing point between the right demixing binodal and the n - b spinodal the system phase separates into a b phase , rich in plates , and a n phase , rich in rods . finally the dotted line in fig . [ fig2 ] represents the spinodal instability of the uniform phases with respect to density modulations , which corresponds to the presence of stable non - uniform phases ( see appendix [ append1 ] for details on these calculations ) . we can see that the n / b - b demixing is , except for a small interval of pressures , metastable with respect to transitions to non - uniform phases . fig . [ fig2 ] ( b ) shows the phase diagram when the adsorption strengths are still relatively small : @xmath76 . we can observe that the phase diagram topology is similar to that of the preceding case , except that now there appears a region , close to @xmath75 and bounded by a dashed line , where the b phase becomes stable . from now on this spinodal will be called the _ rod _ n - b _ spinodal_. the total free - energy is lowered when the fraction of rods with main axes parallel the monolayer increases , since this is proportional to the projected areas . in turn , these rods exhibit two continuous n - b and b - n transitions : b ordering increases with pressure , reaches a maximum , then decreases and finally disappears altogether . this reentrant behaviour of the b phase with pressure can be explained as follows . when the loss in free - energy given by a preferential adsorption and further alignment of rods with projected rectangular shapes can not compensate the free - energy increase due to the large excluded volumes between rectangular projected species , as compared to those of small squared species , the most favored configuration of particles is that of rods pointing perpendicular to the monolayer . as the total amount of rods lying on the surface becomes small a b - n transition takes place . this behaviour was already found in monolayers of one - component zwanzig rods with @xmath77 and zero adsorption . the values of @xmath78 were found to be 21.3 and 12 from the spatially continuous @xcite and discrete lattice models @xcite . when the continuous orientational degrees of freedom are restored , this transition disappears @xcite . however if particles can rotate freely except for a small solid angle with respect to the surface normal , the b phase again becomes stable @xcite . thus in real situations when the surface / interface promotes a preferential adsorption of rods with their axes parallel to the surface the b phase will certainly become stable . now we proceed to describing the phase behaviour of monolayers of rods and plates with relatively high adsorption strengths , specifically those with @xmath79 . the phase diagram in the pressure - composition and total density - composition planes are shown in fig . [ fig3](a ) and ( b ) , respectively . the most salient features that can be observed from the figure are : ( i ) the presence of strong n - b demixing at pressures located between two tricritical points , both lying on the rod - n - b spinodal which ends at @xmath75 , and ( ii ) the existence of a strong first order n - b transition departing from @xmath80 and ending in a tricritical point located at the plate - n - b spinodal . panel ( b ) shows the density of the coexisting phases along all these binodals and spinodals . we first describe the n - b demixing . [ fig3](b ) shows that the values of the coexisting densities in the demixed phases are similar , with b being the densest phase , rich in rods ( two different pairs of coexisting densities are shown with circles and squares ) . however the packing fraction has the opposite behaviour ( see the black line of fig . [ fig4 ] ) : the phase with the highest packing fraction is the n phase , rich in plates . this behaviour ( nearly the same coexisting densities but very different composition ) is typical in entropy - driven demixing . note that plates in both coexisting n and b phases have always a positive uniaxial order parameter along the binodals [ see fig . [ fig5](b ) ] , so the fraction of projected large squares ( cross - section of plates ) is relatively high . the projected rectangles corresponding to rods lying on the surface also have a relatively high fraction ( see the negative values of @xmath69 in ( a ) along the binodals , except for a region close to the upper tricritical point ) , as compared to that of the square projected areas ( when rods point perpendicular to the monolayer ) . as usually occurs in entropy - driven demixing , the total excluded area between the rod - projected rectangles and the plate - projected squares is lowered if the demixed phases are rich in one of the species . the high proportion of large squares is the reason behind the high packing fraction values of the coexisting n phase , as compared to that of the b phase . it is interesting to note the highly non - monotonic behaviour of the packing fraction along the demixing binodals ( see fig . [ fig4 ] ) , a direct consequence of the dependence of @xmath39 not only on @xmath70 and @xmath56 but also on the order parameters @xmath81 . finally fig . [ fig6 ] shows the evolution of the biaxial order parameters along the demixing binodals . the biaxial order of both species along the coexisting b phase rapidly increases and saturates to its highest value ( unity ) as the mixture gets away from the lower tricritical point . at some point they invert their monotonicity and decrease to zero at the upper tricritical point . from the other side of the phase diagram ( @xmath80 ) , and for intermediate values of pressure , a strong first order n - b phase transition takes place , as can be inferred from the large coexisting density gap in fig . [ fig3](b ) . this transition ends in a tricritical point located at the plate - n - b spinodal , and is driven by the reorientation of plates . note the positive and negative values of @xmath82 , corresponding to the coexisting n and b phases , respectively [ see fig . [ fig5](b ) ] . the rods are now mainly oriented perpendicular to the monolayer ( perfectly oriented in the n phase , and with a small degree of orientation in the b phase ) . it is interesting to note that the coexisting molar fractions are now similar ( with the b phase slightly rich in plates ) , while the coexisting densities are very dissimilar ( b being the densest phase ) . this transition is driven by a differential change in free - energy from a n phase with a high fraction of adsorbed larges squares ( plate axes perpendicular to the monolayer ) to a b phase with a high fraction of projected rectangles ( corresponding to plates with their axes lying on the monolayer and pointing along @xmath73 ) . when we follow a constant pressure path from the n to the b coexisting phases , the free - energy contribution corresponding to the external potential increases , while that coming from the entropic interaction part is lowered ( because the particle excluded areas decrease ) . the differential change in the total adsorbed area of particles is huge , so the transition becomes strongly first order . again the coexisting packing fractions are inverted : that of the b phase is lower ( see the gray solid curve in fig . [ fig4 ] ) . at very high pressures the same n / b - b demixing transition ending in a critical point takes place [ similar to the cases @xmath63 and @xmath83 . finally we calculated the instability of uniform phases with respect to non - uniform density modulations . the pressures and densities at which these instabilities occur are plotted as a function of @xmath56 in fig . [ fig3](a ) and ( b ) , respectively . we can see that the lower tricritical point is located above this curve , suggesting that all demixing transitions are metastable with respect to transitions or demixing between non - uniform phases . when the molar fraction of plates with axes perpendicular to the surface is high due to their large surface adsorption , their square cross - sections may crystallize in a simple square lattice at a certain pressure . if pressure is increased beyond this value , plates will reorient their axes parallel to the monolayer and consequently crystal ordering could be destabilized with respect to a uniform or nonuniform phase exhibiting b ordering . on the other side of the phase diagram , where the molar fraction of rods is high and the b phase is stable , the most likely scenario is that of b - smectic or b - columnar phases , where the projected rectangles exhibit two - dimensional smectic or columnar arrangements . these open questions should be settled out by performing df minimisation with respect to non - uniform density profiles , @xmath84 , and search for possible coexistences , a formidable task that we leave for future studies . in this section we study the effect of adsorption asymmetry on the phase behaviour of monolayers of rods and plates . to this purpose we have chosen the adsorption strengths as @xmath85 , i.e. rods are more strongly adsorbed on the surface than plates . to better compare the results obtained with those described in the preceding sections , we again set the aspect ratio to @xmath1 . the phase diagram for this mixture is plotted in fig . [ fig7 ] in the ( a ) pressure - composition and ( b ) density - composition variables . the main features we can extract from these results are : ( i ) when the rods are strongly adsorbed on the surface , the lower part of the rod - n - b spinodal meets the plate - n - b spinodal at intermediate compositions , creating a monotonic , fully connected spinodal curve over the whole composition interval . ( ii ) there is a lower tricritical point located on this curve , above which a demixing transition occurs . ( iii ) the demixing transition coalesces with the strong first - order transition driven by plates at higher pressures ( the one ending at @xmath80 ) . ( iv ) the upper part of the rod - n - b spinodal joins the right part of the plate - n - b spinodal , creating an island of n phase stability . ( v ) the entropic n / b - b demixing at high pressure remains invariant , which confirms the fact that the system behaves like a 2d mixture of squares and rectangles . [ fig8 ] shows the strong non - monotonic behaviour of the packing fraction along the coexistence binodals , with the presence of a large loop . ( vi ) the spinodal for the uniform phase instability with respect to density modulations is located again below the tricritical point . also , packing fraction inversion ( with respect to density ) does occur , with n being the densest phase . figs . [ fig9](a ) and ( b ) show the uniaxial order parameters along the binodals , with a b phase of rods and plates having axes lying on the monolayer , and a n phase with plate axes pointing perpendicular to it . interestingly , the rods in n phase are oriented parallel to the surface , although to a lesser degree . the insets show the completely saturated ordering of particles ( perpendicular and parallel to the surface for rods and plates , respectively ) along the boundaries limiting the island of n phase stability . finally fig . [ fig10 ] shows the biaxial order parameters along the coexisting binodals , which have the usual behaviour : a rapid increase as the system gets away from the tricritical point , and then saturation to perfect biaxial ordering . [ fig11](a ) shows the phase diagram of a mixture with @xmath1 , but with larger asymmetry in their adsorption strengths , @xmath86 . now the plates are slightly adsorbed on the surface , while rods are strongly adsorbed . we can see that : ( i ) the demixing and first - order n - b transitions ( present in the @xmath87-mixtures ) are substituted by continuous transitions , with the n - b spinodal now being a monotonically decreasing function of @xmath56 . thus the region of b stability is greatly enhanced . ( ii ) the island of n - phase stability and the n / b - b demixing at high pressure remain as before . [ fig11](b ) shows the phase diagram of the mixture @xmath88 , i.e. plates and rods are strongly and slightly adsorbed , respectively . note the presence of a strong first - order n - b transition driven by the desorption of plates at intermediate pressures . as before , this transition ends in a tricritical point located on the plate - n - b spinodal . the b phase , rich in rods , is again stable inside a island bounded by the rod - n - b spinodal . no demixing was found in this part of the phase diagram , with the n - b transition being of second order . the n / b - non - uniform - phase spinodals are now discontinuous and located above the b phase of rods . the last study concerns the phase behaviour of mixtures with higher aspect ratios , in particular those with @xmath90 and 40 . as shown in fig . [ fig12 ] the phase - diagram topologies are similar , but there is an important difference : now the lower tricritical point is always located below the spinodal instability to non - uniform phases . thus there is always a range of pressures , which increases with @xmath0 , for which demixing into a n phase and a b phase rich in rods is stable . in figs . [ fig12 ] ( a ) and ( b ) the phase diagrams for @xmath90 and @xmath91 , panel ( a ) , and @xmath92 , panel ( b ) , are shown . note that the former mixture is symmetric with respect to adsorption . the upper boundary of the rod - n - b spinodal meets the n - b - plate spinodal at intermediate compositions . the phase behaviour includes : n - b demixing between two tricritical points ( the lower one departing from the rod - n - b spinodal ) , and a strongly first - order n - b transition driven by the desorption of plates , beginning at @xmath80 and ending in a tricritical point located at the plate - n - b spinodal . this point and the upper critical point of the demixing transition are now very close to each other . an island of n - phase stability exists at high pressure as a consequence of the coalescence between the right part of the plate - n - b spinodal and the upper part of the rod - n - b spinodal . finally , n / b - b demixing of the effective two - dimensional mixture of squares and rectangles at high pressure is also present . the phase diagram topology for the second mixture studied , that with @xmath93 , is very similar to the for @xmath1 described before . again there is an important difference , namely the existence of stable n - b demixing in some pressure range . also the b phase , rich in rods , is stable in a rather large region of the phase diagram . we end this section by showing the phase diagram of a mixture with @xmath94 and @xmath76 , in figs . [ fig12](c ) and ( d ) . we concentrate on some details of the phase diagram topology not found before : ( i ) the presence of a lower critical point ( at low pressures and close to the lower n - b tricritical point ) above which b - b demixing takes place in a rather small range of pressures [ see inset of panel ( d ) ] . ( ii ) the two distinct tricritical points , located close to each other in previous cases [ e.g. the case @xmath90 and @xmath91 ] now coalesce into a single azeotropic point [ see panel ( d ) ] . we have systematically studied the phase behaviour of mixtures of rods and plates adsorbed on a monolayer . in our model , the particle centres of mass are taken to freely move on the surface , while particles can rotate in 3d within the restricted - orientation , zwanzig approximation . adsorption of the particle surfaces on the monolayer is mimicked through an attractive external potential proportional to the area of the particle surface contact , while the strengths of this interaction , @xmath62 , depend on the species type . rods and plates were taken to be symmetric , i.e. with the same volume and same aspect ratio @xmath0 . a fmt - based df , adapted to the present constrained geometry , was minimised , and phase diagrams of mixtures with @xmath1 , 20 and 40 and different values of @xmath62 were calculated . the main results can be summarized as follows . ( i ) when both adsorption strengths @xmath96 are zero or small , rods and plates orient perpendicular and parallel to the surface , respectively . the degree of orientation continuously increases with pressure , and at some value ( which depends on the composition ) a second - order n - b transition occurs , at which the plate axes orient along a director lying on the surface . although rods also exhibit biaxial ordering , this transition is governed by the orientational symmetry breaking of plate axes , and consequently the plate - n - b spinodal is a monotonically increasing function of @xmath56 . ( ii ) at some value of @xmath97 and , starting at the @xmath75 vertical axis at low pressures , there appears an island of b phase stability enclosed by a rod - n - b spinodal which is disconnected from the plate - n - b spinodal . the n - b transition is of second order and is governed by the alignment of rods with axes on the monolayer . as pressure increases rods prefer to align perpendicular to the surface , and biaxial ordering disappears . ( iii ) when rods and plates are symmetrically adsorbed on the monolayer ( @xmath98 ) and the strengths are relatively large , two tricritical points appear on the rod - n - b spinodal ; between these two points n - b demixing takes place , with demixed phases rich in one of the components . the b and n coexisting phases are mostly populated by rods and plates , respectively . also , there is a strongly first - order n - b phase transition with a large density gap starting at the @xmath80 vertical axis . this transition ends in a tricritical point located on the plate - n - b spinodal , and is driven by the desorption of the largest cross - section of the plates ( corresponding to axes perpendicular to the monolayer ) . finally at very high pressures , when the degree of order is high , the system effectively becomes a two - dimensional mixture of squares ( the smallest projected section of rods ) and rectangles ( the smallest projected sections of plates ) , which demix into a b phase , rich in plates , and a n phase , rich in rods . this demixing transition ends in a critical point above which there exists a rather narrow b - b demixing region . ( iv ) when the adsorption of particles is very asymmetric and @xmath99 , the lowest boundary of the rod - n - b spinodal connects with the left part of the plate - n - b spinodal , forming a monotonically - decreasing spinodal over the whole range of compositions . the n - b transition is always of second order . also the upper boundary of the rod - n - b spinodal connects with the right part of the plate - n - b - plate spinodal , forming an island of n stability . the highest pressure n / b - b demixing remains invariant . ( v ) when the adsorption of particles is very asymmetric and @xmath100 , the strongly first - order n - b transition governed by desorption of plates and their b ordering is present up to high molar fractions , while the n - b transition , governed by orientation of rods , is of second order and the n - b demixing disappear . the region of stability of the b phase ( enclosed by the rod - n - b spinodal ) is reduced as @xmath97 becomes smaller . we also calculated the spinodal instability of uniform phases with respect to density modulations with different symmetries ( smectic , columnar or crystalline ) . we found that a b phase rich in rods is stable over a relatively large interval of pressures , while the strong n - b phase transition is always metastable . for @xmath90 and 40 there exists a range of pressures for which n - b demixing is stable . of course n / b - b demixing at high pressure , ending in a lower critical point , is also metastable . we note that demixing regions could be made wider if we chose shape - asymmetric mixtures ( different volumes and/or different aspect ratios ) , and consequently the regions of n and b phase stability could be modified . even the orientational symmetries of the demixed phases could be different for asymmetric mixtures as shown in theoretical calculations of freely - rotating rod - plate onsager mixtures @xcite . we are confident that the results presented in this work will be qualitatively similar if we remove the restricted orientation approximation and consider the free rotation of particle axes . computer simulations of binary mixtures adsorbed on a flat monolayer could confirm this conclusion . mc simulations of 2d mixtures of rods on a lattice show an interesting phase behavior @xcite . when the aspect ratio of the longer rods is 7 there exist two i - n and n - i transitions as the density of longer rods is increased while that of the shorter rods is fixed bellow a certain critical density . this behavior resembles that of the present rod - plate mixture for which two n - b and b - n transitions take place at fixed composition as the pressure is increased and the adsorption strengths of rods is high enough . thus it would be interesting to perform dft calculations on mixtures of adsorbed rods to find the differences and similarities between monolayers and strictly 2d hard rod mixtures . also interesting are the similarities that the present system shares with the phase behaviour of monolayers of biaxial particles studied in @xcite . in that case the phase diagrams , in the density - biaxial parameter plane , present a n - b spinodal , completely analogous to that of the present system , if we replace the biaxial parameter by molar fraction . moreover , by increasing the aspect ratio of biaxial rods in the previous study , one obtains an island of b phase stability ; a similar effect is found in the present study when adsorption strength is increased . above a particular value for the largest aspect ratio of the biaxial particles , this island coalesces with the n - b spinodal , again a behaviour similar to the one found in the present study if the adsorption strength is increased beyond some critical value . additionally , we may ask ourselves how particle biaxiality would affect the present phase behaviour . if particles are biaxial , we would expect the n - b spinodal to shift to higher pressures , favouring the stabilisation of non - uniform phases . on the other hand , biaxiality would also reduce the island of b stability in the rod - rich part of the phase diagram . finally it would also be possible that particle biaxiality reduced demixing gaps , because particle projections become similar with biaxiality . the constrained minimisation of the free - energy density with respect to the variables @xmath27 gives the following set of equations : @xmath101 where @xmath102 using the definition of biaxial order parameter @xmath49 from ( [ biaxial ] ) , we arrive at @xmath103 expanding ( [ itera1 ] ) with respect to @xmath49 ( obviously the functions @xmath104 depend on @xmath105 ) up to first order gives a system of two equations which has a nontrivial solution only if the number density is such that @xmath106,\end{aligned}\ ] ] where @xmath107 is the solution of ( [ itera ] ) for @xmath108 , and the functions @xmath104 are calculated from ( [ uno])-([tres ] ) , with all the weighted densities depending only on @xmath107 , as @xmath109 and @xmath110,\nonumber\\&&\\ & & n_2=\eta=\rho\sum_s x_s\left[\kappa^{s/3}+\gamma_{sz}\left(\kappa^{-2s/3}-\kappa^{s/3}\right)\right ] \end{aligned}\ ] ] thus we need to solve a system of two non - linear equations to find the equilibrium values of @xmath107 , and consequently the number density @xmath25 at which a bifurcation from the n to the b phase occurs . we have calculated the spinodal instability of uniform phases with respect to spatial inhomogeneities through the divergence of the structure factor . the fourier transform of the direct correlation functions , calculated through the second functional derivative of the functional , reads @xmath111 where @xmath112 implies symmetrisation with respect to the pair of indexes @xmath113 , and the weighted densities @xmath114 correspond to the stable uniform n or b phases , with orientational order parameters calculated from the minimisation of the corresponding free energies . the fourier transforms of the weighting functions are @xmath115 where @xmath116 and @xmath117 . we define the @xmath118 structure factor matrix @xmath119 where @xmath120 and @xmath121 if @xmath9 , and @xmath122 and @xmath123 if @xmath8 ( with the corresponding relabelling @xmath124 and @xmath125 for @xmath56 , @xmath73 and @xmath23 , respectively ) . evaluating the determinant of this matrix @xmath126 $ ] at the wave vectors @xmath127 or @xmath128 ( corresponding to inhomogeneities along or perpendicular to the nematic director , respectively ) , we found the corresponding values at the spinodal instabilities , @xmath129 and @xmath130 , as the values for @xmath131 and @xmath70 where the absolute minimum of @xmath132 as a function of @xmath131 becomes zero for the first time . this is equivalent to solving the pair of equations @xmath133 100 p. sharma , a. ward , t. gibaud , m. f. hagan , and z. dogic , nature * 513 * , 77 ( 2014 ) . p. a. rice and h. m. mcconnell , proc . usa * 86 * , 6445 ( 1989 ) . j. p. hagen and h. m. mcconnell , bba - 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we theoretically study the phase behaviour of monolayers of hard rod - plate mixtures using a fundamental - measure density functional in the restricted - orientation ( zwanzig ) approximation . particles can rotate in 3d but their centres of mass are constrained to be on a flat surface . in addition , we consider both species to be subject to an attractive potential proportional to the particle contact area on the surface and with adsorption strengths that depend on the species type . particles have board - like shape , with sizes chosen using a symmetry criterion : same volume and same aspect ratio @xmath0 . phase diagrams were calculated for @xmath1 , 20 and 40 and different values of adsorption strengths . for small adsorption strengths the mixtures exhibit a second - order uniaxial nematic - biaxial nematic transition for molar fraction of rods @xmath2 . in the uniaxial nematic phase the particle axes of rods and plates are aligned perpendicular and parallel to the monolayer , respectively . at the transition , the orientational symmetry of the plate axes is broken , and they orient parallel to a director lying on the surface . for large and equal adsorption strengths the mixture demixes at low pressures into a uniaxial nematic phase , rich in plates , and a biaxial nematic phase , rich in rods . the demixing transition is located between two tricritical points . also , at higher pressures and in the plate - rich part of the phase diagram , the system exhibits a strong first - order uniaxial nematic - biaxial nematic phase transition with a large density coexistence gap . when rod adsorption is considerably large while that of plates is small , the transition to the biaxial nematic phase is always of second order , and its region of stability in the phase diagram considerably widens . at very high pressures the mixture can effectively be identified as a two - dimensional mixture of squares and rectangles which again demixes above a certain critical point . we also studied the relative stability of uniform phases with respect to density modulations of smectic , columnar and crystalline symmetry .

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Proceed to summarize the following text: it is postulated in quantum physics that quantities we observe are eigenvalues of operators representing the dynamics of the quantities . therefore , the energy spectra , i.e. the eigenvalues , are required to be real and bounded from below so that the system has a stable lowest - energy state . to satisfy such requirements , it was conjectured that the operators must be hermitian . non - hermitian hamiltonians have been commonly associated with complex eigenvalues and therefore decay of the quantities . however , it turned out that hermiticity is not necessarily required by a hamiltonian system to satisfy the postulate @xcite . of particular examples have been systems exhibiting the so - called parity - time ( @xmath0 ) symmetry , suggested by bender and co - workers @xcite . a necessary condition for a hamiltonian to be @xmath0symmetric is that its potential @xmath1 should satisfy the condition @xmath2 . optical analogues of such systems were proposed in @xcite using two coupled waveguides with gain and loss . note that such couplers were already studied in @xcite . the following successful experiments @xcite have stimulated extensive studies on @xmath0symmetric dimers , which are a finite - dimensional reduction of schrdinger equations modelling , e.g. , bose - einstein condensates with @xmath0symmetric double - well potentials @xcite . nontrivial characteristics of the systems allow them to be exploited , e.g. , for all - optical switching in the nonlinear regime , lowering the switching power and attaining sharper switching transition @xcite as well as a unidirectional optical valve @xcite . @xmath0symmetric analogues in coupled oscillators have also been proposed theoretically and experimentally recently @xcite . note that coupled oscillators with gain and loss have already been considered in @xcite . in this paper , we consider the following equations of motion @xcite @xmath3 where the dot represents differentiation with respect to the evolution variable , which is the propagation direction @xmath4 for nonlinear optics or the physical time @xmath5 for bose - einstein condensates , @xmath6 is the nonlinearity coefficient and @xmath7 is the gain - loss parameter . here , we consider two cases , i.e. when @xmath8 and @xmath9 . for the former case , one can scale the coefficient such that @xmath10 . it was shown numerically in @xcite that the nonlinearity suppresses periodic time reversals of optical power exchanges between the sites , leading to the symmetry breaking and a sharp beam switching to the waveguide with gain . when @xmath11 , eq . has two conserved quantities @xmath12 which are commonly referred to as the power and the hamiltonian / energy , respectively . using the liouville - arnold theorem ( or liouville - mineur - arnold theorem ) @xcite , is integrable since the degree of freedom is equal to the number of conserved quantities . by defining the site - occupation probability difference @xmath13 kenkre and campbell @xcite showed that @xmath14 satisfies a @xmath15-equation , which explains the presence of josephson tunneling and self - trapped states , with the latter corresponding to @xmath14 being sign - definite , as well as the transition between them . it was later shown @xcite that @xmath14 also satisfies the pendulum equation . when @xmath16 , eq . is still integrable @xcite . system is actually a special case of a notably integrable dimer derived in ( see also a brief review of integrable oligomers in @xcite ) . the conserved quantities in that case are @xmath17 it was reported that the general system could be reduced to a first - order differential equation with polynomial nonlinearity and it possesses blow - up solutions that was observed numerically . in this paper , we consider and show that it can be reduced to a pendulum equation with a linear potential and a constant drive . the same equation has been obtained recently , parallel to and independently from this work , by kevrekidis , pelinovsky and tyugin @xcite and by barashenkov , jackson and flach @xcite through a different formalism . the linear potential and constant drive explain the presence of unbounded solutions . we exploit the strong relation between the problem and the geometry of circles . we also discuss the qualitative pictures of all solutions of the governing equations . in section [ sec2 ] , we rewrite the governing equations in terms of power , population imbalance , and phase difference between the wavefields in the channels . in the section , we also derive a constant of motion . in section [ sec3 ] , we analyse the characteristics of the fixed points , which are the time - independent solutions of the system . in section [ sec4 ] , we reduce our system in section [ sec2 ] further into one equation . here , we show that the problem is described by a pendulum equation with a linear potential and a constant drive . in the section , we analyse the pendulum equation qualitatively through its phase - portrait . in section [ sec5 ] , we discuss the phase - portrait of the system that is composed of trajectories with the same value of a constant of motion . the constant corresponds to power , that is a conserved quantity when @xmath11 . the case of linear systems is discussed in section [ sec6 ] . finally , we conclude our work in section [ conc ] . writing @xmath18 and @xmath19 in polar form @xmath20 and defining the variable of phase difference between @xmath18 and @xmath19 @xmath21 the equations of motion can then be expressed in terms of , and as ( see the appendix ) @xmath22 we can limit the phase difference to be in the interval @xmath23 . note that the argument angle @xmath24 ( and hence @xmath25 ) is undefined when @xmath26 ( or @xmath27 ) . taking @xmath28 , the conditions for equilibrium points are @xmath29 this shows that no equilibrium points can exist with @xmath30 , therefore demonstrating the non - existence of the self trapped state that is observed for the case when @xmath11 . when @xmath31 it follows that no equilibrium points exist . this is a threshold of total @xmath0-symmetry breaking where no periodic solutions can exist . however , unbounded trajectories will be shown to always exist for any @xmath7 . after some manipulations ( see the appendix for the details ) , it is possible to find a constant of motion of , , and that is given by @xmath32 where @xmath33 is a constant . we can impose the constant to be non - negative with no loss of generality . it is important to note that one can draw similarities of to the cosine rule for triangles with three edge length : @xmath34 , @xmath33 and @xmath35 ( see fig [ sk2 ] ) . \(a ) ( b ) ( c ) by varying @xmath25 , trajectories of form circles with radius @xmath33 centred on a point a unit distance from the origin . three different sketches corresponding to different values of @xmath33 are given in fig . [ sk2 ] . equation ( [ eq : c ] ) can also be rearranged to give @xmath36 from which a few things can be said about the constant @xmath33 and how this affects what values @xmath25 can take . firstly @xmath33 must be real to give a real solution . if @xmath37 then @xmath38 . if @xmath39 then @xmath25 is bounded such that @xmath40 . the @xmath41 solution in and can exist only for @xmath33 within this range . only when @xmath42 , @xmath43 can equal @xmath44 , in which case @xmath45 . if @xmath46 then @xmath25 is unbounded it is always natural to first analyse the behaviour of the time - independent solutions . using the conditions for the equilibrium points eq . ( [ eq : eqpt ] ) alongside ( [ conmo ] ) , we can find the value of the power , @xmath47 , at equilibrium points for different values of @xmath33 , i.e.@xmath48 applying the inequality @xmath49 to eq . , eq . is also the minimum power that a ( generally time - dependent ) solution can attain . the value @xmath37 is particular as explicit solutions can be found . in this case @xmath50 , @xmath51 and @xmath52 , where @xmath53 is a constant determined by the initial conditions . this gives one example of where @xmath0-symmetry is broken for any non - zero @xmath54 . if @xmath55 , no equilibrium points can exist as the power is complex - valued . hence , all solutions are unbounded . when @xmath56 , an equilibrium point lies on the boundary of @xmath25 with @xmath57 . for trajectories with @xmath58 , there are two equilibrium points at @xmath57 . this is shown in fig . [ sk2 ] as the intersections of the circles and the dashed lines given by eq . ( [ eq : eqpt ] ) . the first of these is a centre and the second is a saddle node with the corresponding power @xmath59 and @xmath60 , respectively . exploiting the geometry of the triangles and circles in fig . [ sk2 ] , one can obtain that @xmath61 for the case of @xmath42 , the trivial equilibrium point at @xmath62 is found to represent a centre for these trajectories . there is also an equilibrium point located at @xmath63 and @xmath64 , that can be straightforwardly shown to be a saddle node . solutions with @xmath46 also have two equilibrium points located at @xmath63 and @xmath65 . the former represents a saddle node with @xmath66 and the latter is a centre with @xmath67 . from the results above , we obtain that generally speaking the larger @xmath33 is , the larger the power of stable solutions can be . in order to analyse the general solutions of the governing equations , it is useful to define a new variable @xmath68 , i.e. the arc angle , that parametrises the circles as indicated in fig . the variable can be expressed by @xmath69 where @xmath70 and @xmath71 . these equations can be rearranged to yield @xmath72 the relationships are plotted in fig . [ sk4 ] . using the remark following , note that is a lower bound to the power of solutions . after carrying out some rearranging and differentiation ( see the appendix ) , we find that @xmath73 where @xmath74 is , technically , a constant of motion . we use the word `` technically '' because care must be taken in how @xmath68 is defined . because we are choosing to define @xmath75 we must remember that as a trajectory crosses this boundary , the value of @xmath74 changes by an amount @xmath76 to ensure that the power , @xmath47 , remains continuous . if @xmath68 should be defined unbounded then @xmath74 would indeed remain constant for all time . it so happens that for @xmath28 , @xmath74 has similarities to @xmath47 with @xmath11 , where it is well known that the power remains constant for all trajectories . \(a ) ( b ) the true value of defining the variable @xmath68 and the constant @xmath33 is made apparent when we substitute @xmath77 with equations ( [ eq : cpsi1 ] ) and ( [ eq : k ] ) into equation ( [ eq : deltadot ] ) . the equations of motion then reduce to a second order differential equation @xmath78 this is a main result of this paper . when @xmath79 , we indeed obtain the equation for a pendulum similarly to @xcite that was derived through a different approach . the presence of @xmath16 introduces a linear potential and a constant drive into the pendulum equation . it may be possible to solve eq . ( [ eq : oscillator1 ] ) in terms of jacobi elliptic functions . asymptotic solutions of the equation were derived in @xcite . however , instead we will study the qualitative behaviours of the solutions in the ( @xmath68,@xmath80)-phase plane . the first integral of eq . ( [ eq : oscillator1 ] ) can be obtained by using eq . ( [ eq : cpsi3 ] ) , i.e.@xmath81 the phase - portrait can then be plotted rather easily and is demonstrated in fig . [ sk5 ] for @xmath82 . the phase plane can have two topologically different structures , one with and one without a stable region . [ padd1 ] shows how this phase plane , combined with fig . [ sk4]a ( cf . ) can be plotted in a three - dimensional graph . this then gives an idea of how solutions appear in the @xmath83 phase plane . when @xmath84 there are boundaries on @xmath25 which correspond to @xmath85 . these are represented by vertical dashed lines in fig . trajectories that cross these boundaries will behave differently , in the @xmath86 plane , from those trajectories which do not . this results in a more diverse set of solutions when @xmath84 than for otherwise . one can note that when @xmath42 these vertical dashed lines merge at @xmath87 to correspond to the point where @xmath25 is in fact undefined . it is useful to know the value of the constant @xmath74 for the trajectories that touch the boundary at @xmath88 . such values are found by using eq . ( [ eq : k ] ) and eq . ( [ eq : cpsi3 ] ) to be @xmath89 how a trajectory behaves depends on how its @xmath74-value compares to the four main values of @xmath74 , namely @xmath90 , @xmath91 , @xmath92 and @xmath93 , where @xmath94 with @xmath95 and @xmath96 being the values of @xmath68 at the equilibrium points , i.e.@xmath97 which can be obtained using - ( or similarly exploiting the geometric pictures in fig . [ sk2 ] ) . by plotting out these values of @xmath74 against @xmath33 for a fixed @xmath54 one sees that there is a point where @xmath90 and @xmath93 cross . three types of solution exist for all @xmath84 . the first of these is a stable trajectory that never touches a boundary for @xmath25 . this corresponds to @xmath98 . second are trajectories with @xmath99 . these cross one boundary of @xmath25 . the final type is an unstable trajectory with @xmath100 . in addition to the trajectories above , there are two types of solutions that can not coexist for the same @xmath33-value . this is because one solution exists when @xmath101 and the other when @xmath102 , which is obvious that both conditions can not be met at the same time . when @xmath101 , there is additionally another type of stable trajectories that crosses both boundaries of @xmath25 this is demonstrated in fig . however , if @xmath102 then instead there is another type of unstable trajectory , as shown in in fig . all of the different types of trajectory are sketched in fig . [ sk3 ] . \(a ) ( b ) \(a ) ( b ) \(a ) ( b ) + ( c ) ( d ) in the conservative case @xmath11 , the power @xmath47 is independent of time , see . in that case , plots in figs . [ padd1 ] and [ sk3 ] correspond to varying power , i.e. each trajectory has different value of @xmath47 , and a fixed energy , i.e. the same value of first integral . however , it is not the common practice as usually one plots trajectories corresponding to the same value of power . in the general case when @xmath16 , @xmath47 is no longer constant . it is therefore not possible to compose a similar phase portrait consisting of trajectories with constant @xmath47 . however , we find that one could instead plot phase - portraits with the same value of @xmath74 , which interestingly runs parallel with the different solutions observed at the same values of @xmath47 for @xmath11 , see . when comparing periodic trajectories that all have the same value of power @xmath74 , it is useful to study the ( @xmath68,@xmath47)-phase plane , which qualitatively can be obtained by plotting and analysing ( see fig . [ sk4]b ) and . by doing this , we find that different values of @xmath74 can give different results in the ( @xmath25,@xmath14)-phase plane . there are three important @xmath74 values to consider . the first is the minimum value @xmath74 can take . the second is the point where the equilibrium point at @xmath63 bifurcates from a centre to a saddle node . this is found from substituting @xmath56 into eq . ( [ eq : ksaddle ] ) . the third important value of @xmath74 is when the separatrix passing through the saddle point has a @xmath33-value of one , found from substituting @xmath42 into eq . ( [ eq : ksaddle ] ) . these values of @xmath74 are given respectively by @xmath103 substituting @xmath11 into the above expressions gives the important values of @xmath74 as 0 , 1 and 2 . realising that here @xmath104 makes clear that these are generalisations of the critical values found for @xmath47 when @xmath11 , where @xmath105 corresponds respectively to the presence of only trivial solution , the emergence of a pair of fixed points @xmath106 , and the emergence of rotational ( or running ) states @xcite . when @xmath107 all three of these values are equal , which is the threshold of total @xmath0-symmetry breaking , i.e. there are no periodic , bounded solutions for @xmath31 . note that for any @xmath108 , there are two equilibrium points . the @xmath33-values of the equilibria give the upper and lower bounds the periodic paths can have for a certain @xmath74 . \(a ) ( b ) ( c ) in fig . [ sk6 ] , we sketch the possible phase - portraits of the system for @xmath16 within three different intervals determined by the critical values of @xmath74 above . note that the figures are actually composed of trajectories taken from the phase portraits with different values of @xmath33 sketched in fig . they can also be seen as the dynamical regimes for arbitrary initial conditions , showing the regions of periodic and unbounded solutions , that we refer to here as stability and instability regions . similar figures obtained using comprehensive numerical computations were presented in @xcite . in panel ( a ) , all the solutions are periodic . as @xmath74 increases , there will be a critical value above which the only equilibrium point becomes unstable . when @xmath11 , this is where a pair of fixed points with nonzero @xmath14 emerges . for @xmath16 , instead we have unbounded solutions . the instability region is nevertheless contained within the region of stable , periodic solutions . when @xmath74 is increased further passing , the stability region that contains the instability region vanishes completely . sukhorukov , xu , and kivshar @xcite also presented the dependence of the minimal input intensity on the gain / loss coefficient and the corresponding phase difference required for nonlinear switching , i.e. the minimum intensity above which the solutions would be unbounded . according to the analytical results presented herein , those would correspond to the separatrices of ( see fig . [ sk5 ] ) . analytical expressions should be derivable , that we leave for the interested reader . after analysing the nonlinear dimer with @xmath8 , finally we discuss the linear equations described by with @xmath9 . this is particularly interesting because the system has been realised experimentally in @xcite . transforming the equations into the polar forms by similarly defining @xmath47 , @xmath14 and @xmath25 , we obtain the same equations except ( [ eq : thetadot ] ) , which is now given by @xmath109 following the similar reduction , we obtain that instead of the constant of motion is given by @xmath110 this , instead of circles , is an equation for a straight line passing a distance @xmath33 from the origin . by parametrising the line with the new variable @xmath68 defined by @xmath111 we find that ( cf . and ) @xmath112 where @xmath74 is also a constant of integration . using the above equations combined with the identity @xmath113 ( obtained from and ) in equation ( [ eq : deltadot ] ) gives @xmath114 this is a forced , simple harmonic equation and therefore can be solved explicitly to give @xmath68 in terms of @xmath5 . due to the second equation in , we therefore can conclude that the oscillating power reported in , e.g. , @xcite has internal frequency @xmath115 . we have studied analytically linear and nonlinear @xmath0-symmetric dimers , where we described the whole dynamics of the system . the effect of nonlinearity that induces @xmath0-symmetry breaking for gain / loss parameters below that of the linear system , which in a previous work was referred to as nonlinear suppression of time - reversals @xcite , has been analysed as well . our analytical study may offer a new insight into the global dynamics of directional waveguide couplers with balanced gain and loss or bose - einstein condensates in a double - well potential with a balanced sink and source of atoms . in additions to our qualitative analysis , one could extend the study here to the analytical expression of the solutions of the nonlinear @xmath0-symmetric dimer , that may be expressed in terms of jacobi elliptic functions . in that case , the oscillation frequency of the power in the nonlinear system would be obtained . here , we only derive the frequency of the linear system . differentiating the expression gives @xmath116 which by substituting into the governing equations yields @xmath117 from the real parts and @xmath118 using the product rule for differentiation we can also obtain the following four equations @xmath119 using the above equations and noting that @xmath120 eqs . , , and can be immediately obtained . next , one can compare and to obtain that @xmath121 therefore , we can define a constant @xmath33 such that @xmath122 which is nothing else but eq . . after parametrisation of the constant of motion by @xmath68 , i.e. eqs . - , one can differentiate to obtain that @xmath123 now combining it with yields @xmath124 using , we see that @xmath125 which can be integrated to give . a. ruschhaupt , f. delgado , and j. g. muga , j. phys . a 38 , l171 ( 2005 ) . r. el - ganainy , k. g. makris , d. n. christodoulides , and z. h. musslimani , opt . 32 , 2632 ( 2007 ) . s. klaiman , u. guenther , and n. moiseyev , phys . lett . 101 , 080402 ( 2008 ) . y. j. chen , a. w. snyder , and d. n. payne , ieee j. quantum electron . 28 , 239 ( 1992 ) . a. guo , g. j. salamo , d. duchesne , r. morandotti , m. volatier - ravat , v. aimez , g. a. siviloglou , and d. n. christodoulides , phys . 103 , 093902 ( 2009 ) . c. e. ruter , k. g. makris , r. el - ganainy , d. n. christodoulides , m. segev , and d. kip , nature physics 6 , 192 ( 2010 ) . miroshnichenko , b.a . malomed , and yu.s . kivshar , phys . a 84 , 012123 ( 2011 ) . h. cartarius and g. wunner , phys . a 86 , 013612 ( 2012 ) . graefe , j. phys . 45 , 444015 ( 2012 ) . j. schindler , a. li , m.c . zheng , f. m. ellis , and t. kottos , phys . a 84 , 040101(r ) ( 2011 ) . h. ramezani , j. schindler , f. m. ellis , u. gnther , and t. kottos , phys . a 85 , 062122 ( 2012 ) . z. lin , j. schindler , f. m. ellis , and t. kottos , phys . a 85 , 050101(r ) ( 2012 ) . j. schindler , z. lin , j.m . lee , h. ramezani , f.m . ellis and t. kottos , j. phys . a : math . theor . 45 , 444029 ( 2012 ) .

the coupled discrete linear and kerr nonlinear schrdinger equations with gain and loss describing transport on dimers with parity - time ( @xmath0 ) symmetric potentials are considered . the model is relevant among others to experiments in optical couplers and proposals on bose - einstein condensates in @xmath0symmetric double - well potentials . it is known that the models are integrable . here , the integrability is exploited further to construct the phase - portraits of the system . a pendulum equation with a linear potential and a constant force for the phase - difference between the fields is obtained , which explains the presence of unbounded solutions above a critical threshold parameter . the behaviour of all solutions of the system , including changes in the topological structure of the phase - plane , is then discussed .

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Proceed to summarize the following text: electronic - structure calculations on periodic systems are conventionally done using the so - called bloch orbital based approach which consists of assuming an itinerant form for the single - electron wave functions . this approach has the merit of incorporating the translational invariance of the system under consideration , as well as its infinite character , in an elegant and transparent manner . an alternative approach to electronic - structure calculations on periodic systems was proposed by wannier @xcite . in this approach , instead of describing the electrons in terms of itinerant bloch orbitals , one describes them in terms of mutually orthogonal orbitals localized on individual atoms or bonds constituting the infinite solid . since then such orbitals have come to be known as wannier functions . it can be shown that the two approaches of description of an infinite solid are completely equivalent and that the two types of orbitals are related by a unitary transformation @xcite . therefore , the two approaches differ only in terms of their practical implementation . however , the description of metallic systems in terms of wannier functions frequently runs into problems as it is found that for such systems the decay of the orbitals away from the individual atomic sites is of power law type and not of exponential type . in other words , the wannier functions for such systems are not well localized @xcite . this behavior is to be expected on intuitive grounds as electrons in metals are indeed quite delocalized . on the other hand , for the situations involving surfaces , impurity states , semiconductors and insulators , where the atomic character of electrons is of importance , wannier functions offer a natural description . recent years have seen an increased amount of activity in the area of solid - state calculations based on localized orbitals @xcite , of which wannier functions are a subclass . most of these approaches have been proposed with the aim of developing efficient order - n methods for electronic structure calculations on solids within the framework of density functional theory . with a different focus , nunes and vanderbilt @xcite have developed an entirely wannier - function based approach to electronic - structure calculations on solids in the presence of electric fields , a case for which the eigenstates of the hamiltonian are no longer bloch states . however , we believe that there is one potential area of application for wannier orbitals which remains largely unexplored , namely in the _ ab initio _ treatment of electron - correlation effects in solids using the conventional quantum - chemical methods @xcite . it is intuitively obvious that an _ ab initio _ treatment of electron correlations on large systems will converge much faster with localized orbitals as compared to delocalized orbitals because the coulomb repulsion between two electrons will decay rapidly with the increasing distance between the electrons . in the quantum - chemistry community the importance of localized orbitals in treating the correlation effects in large systems was recognized early on and various procedures aimed at obtaining localized orbitals were developed @xcite . some of the localized - orbital approaches were also carried over to solids chiefly by kunz and collaborators @xcite at the hartree - fock level . this approach has been applied to a variety of systems @xcite . kunz , meng and vail @xcite have gone beyond the hartree - fock level and also included the influence of electron correlations for solids using many - body perturbation theory . the scheme of kunz et al . is based upon nonorthogonal orbitals which , in general , are better localized than their orthogonal counterparts . however , the subsequent treatment of electron correlations with nonorthogonal orbitals is generally much more complicated than the one based upon true wannier functions . in our group electron correlation effects on solids have been studied using the incremental scheme of stoll @xcite which works with localized orbitals . in such studies the infinite solid is modeled as a large enough cluster and then correlation effects are calculated by incrementally correlating the hartree - fock reference state of the cluster expressed in terms of localized orbitals @xcite . however , a possible drawback of this procedure is that there will always be finite size effects and no _ a priori _ knowledge is available as to the difference in results when compared with the infinite - solid limit . in order to be able to study electron - correlation effects in the infinite - solid limit using conventional quantum - chemical approaches , one first has to obtain a hartree - fock representation of the system in terms of wannier functions . this task is rather complicated because , in addition to the localization requirement , one also imposes the constraint upon the wannier functions that they be obtained by the hartree - fock minimization of the total energy of the infinite solid . in an earlier paper @xcite henceforth referred to as i we had outlined precisely such a procedure which obtained the wannier functions of an infinite insulator within a hartree - fock approach and reported its preliminary applications to the lithium hydride crystal . in the present paper we describe all theoretical and computational details of the approach and report applications to larger systems namely lithium fluoride and lithium chloride . unlike i , where we only reported results on the total energy per unit cell of the system , here we also use the hartree - fock wannier functions to compute the x - ray structure factors and compton profiles . additionally , we also discuss the localization characteristics of the wannier functions in detail . all the physical quantities computed with our procedure are found to be in excellent agreement with those computed using the crystal program @xcite which employs a bloch orbital based _ ab initio _ hartree - fock approach . in a future publication we will apply the present formalism to perform _ ab initio _ correlation calculations on an infinite insulator . the rest of this paper is organized as follows . in section [ theory ] we develop the theoretical formalism at the hartree - fock level by minimizing the corresponding energy functional , coupled with the requirement of translational symmetry , and demonstrate that the resulting hf equations correspond to the hf equations for a unit cell of the solid embedded in the field of identical unit cells constituting the rest of the infinite solid . thus an embedded - cluster picture for the infinite solid emerges rigorously from this derivation . subsequently a localizing potential is introduced in the hf equations by means of projection operators leading to our working equations for the hartree - fock wannier orbitals for an infinite solid . finally , these equations are cast in the matrix form using a linear combination of atomic orbitals approach which is used in the actual calculations . in section [ results ] we present the results of our calculations performed using the aforementioned formalism on lif and licl crystals . finally , in section [ conclusion ] we present our conclusions . various aspects related to the computer implementation of the present approach are discussed in the appendix . we consider the case of a perfect solid without the presence of any impurities or lattice deformations such as phonons . we also ignore the effects of relativity completely so that the spin - orbit coupling is also excluded . in such a case , in atomic units @xcite , the nonrelativistic hamiltonian of the system consisting of the kinetic energy of electrons , electron - nucleus interaction , electron - electron repulsion and nucleus - nucleus interaction is given by @xmath0 where in the equation above @xmath1 denotes the position coordinates of the @xmath2-th electron while @xmath3 and @xmath4 respectively denote the position and the charge of the @xmath5-th nucleus of the lattice . for a given geometry of the solid the last term representing the nucleus - nucleus interaction will make a constant contribution to the energy and will not affect the dynamics of the electrons . to develop the theory further we make the assumptions that the solid under consideration is a closed - shell system and that a single slater determinant represents a reasonable approximation to its ground state . moreover , we assume that the same spatial orbitals represent both the spin projections of a given shell , i.e. , we confine ourselves to restricted hartree - fock ( rhf ) theory . with the preceding assumptions , the total energy of the solid can be written as @xmath6 where @xmath7 and @xmath8 denote the occupied spatial orbitals assumed to form an orthonormal set , @xmath9 denotes the kinetic energy operator , @xmath10 denotes the electron - nucleus potential energy , @xmath11 denotes the nucleus - nucleus interaction energy and @xmath12 etc . represent the two - electron integrals involving the electron repulsion . the equation above is completely independent of the spin degree of freedom which , in the absence of spin - orbit coupling , can be summed away leading to familiar factors of two in front of different terms . clearly the terms involving @xmath13 , @xmath12 , and @xmath11 contain infinite lattice sums and are convergent only when combined together . so far the energy expression of eq.([eq - esolid ] ) does not incorporate any assumptions regarding the translational symmetry of a perfect solid . in keeping with our desire to introduce translational symmetry in the real space , without having to invoke the * k*-space as is usually done in the bloch orbital based theories , we make the following observation . a crystalline solid , in its ground state , is composed of identical unit cells and the orbitals belonging to a given unit cell are identical to the corresponding orbitals belonging to any other unit cell and are related to one another by a simple translation operation . assuming that the number of orbitals in a unit cell is @xmath14 and if we denote the @xmath15-th orbital of a unit cell located at the position given by the vector @xmath16 of the lattice by @xmath17 then clearly the set @xmath18 denotes all the orbitals of the solid . in the previous expression @xmath19 is the total number of unit cells in the solid which , of course , is infinite . henceforth , greek labels @xmath20 will always denote the orbitals of a unit cell . the translational symmetry condition expressed in the real space can be stated simply as @xmath21 where @xmath22 is an operator which represents a translation by vector @xmath23 . using this , one can rewrite the energy expression of eq.([eq - esolid ] ) as @xmath24 where @xmath25 denotes an orbital centered in the reference unit cell , @xmath26 involves the interaction energy of the nuclei of the reference cell with those of the rest of the solid ( @xmath27 ) , and we have removed the subscript @xmath28 from the energy . the preceding equation also assumes the important fact that the orbitals obtained by translation operation of eq.([eq - trsym ] ) are orthogonal to each other . we shall elaborate this point later in this section . an important simplification to be noted here is that by assuming the translational invariance in real space as embodied in eq.([eq - trsym ] ) , we have managed to express the total hartree - fock energy of the infinite solid in terms of a finite number of orbitals , namely the orbitals of a unit cell @xmath14 . if we require that the energy of eq.([eq - esolidf ] ) be stationary with respect to the first - order variations in the orbitals , subject to the orthogonality constraint , we are led to the hartree - fock operator @xmath29 where j and k the conventional coulomb and exchange operators , respectively are defined as @xmath30 any summation over greek indices @xmath20 will imply summation over all the @xmath14 orbitals of a unit cell unless otherwise specified . as mentioned earlier , the terms @xmath10 , @xmath31 and @xmath32 involve infinite lattice sums and their practical evaluation will be discussed in the next section . the eigenvectors of the hartree - fock operator of eq.([eq - hff ] ) will be orthogonal to each other , of course . however , in general , these solutions would neither be localized , nor would they be orthogonal to the orbitals of any other unit cell . this is because the orbitals centered in any other unit cell are obtained from those of the reference cell using a simple translation operation as defined in eq.([eq - trsym ] ) , which does not impose any orthogonality or localization constraint upon them . since our aim is to obtain the wannier functions of the infinite solid , i.e. , all the orbitals of the solid must be localized and orthogonal to each other , we will have to impose these requirements explicitly upon the eigenspace of ( [ eq - hff ] ) . this can most simply be accomplished by including in ( [ eq - hff ] ) the projection operators corresponding to the orbitals centered in the unit cells in a ( sufficiently large ) neighborhood of the reference cell @xmath33 where @xmath34 stands for @xmath25 , an orbital centered in the reference unit cell , @xmath35 , @xmath36 , and @xmath37 collectively denotes the unit cells in the aforementioned neighborhood . clearly the choice of @xmath37 will be dictated by the system under consideration the more delocalized electrons of the system are , the larger will @xmath37 need to be . in our calculations we have typically chosen @xmath37 to include up to third nearest - neighbor unit cells of the reference cell . in the equation above @xmath38 s are the shift parameters associated with the correponding orbitals of @xmath37 . for perfect orthogonality and localization , their values should be infinitely high . by setting the shift parameters @xmath38 s to infinity we in effect raise the orbitals localized in the environment unit cells ( region @xmath37 ) to very high energies compared to those localized in the reference cell . thus the lowest energy solutions of eq.([eq - hff1 ] ) will be the ones which are localized in the reference unit cell and are orthogonal to the orbitals of the environment cells . of course , in practice , it suffices to choose a rather large value for these parameters , and the issue pertaining to this numerical choice is discussed further in section [ results ] . eq.([eq - hff1 ] ) will generally be solved iteratively as described in the next section . if the initial guesses for the orbitals of the unit cell @xmath39 are localized , subsequent orthogonalization by means of projection operators will not destroy that property @xcite and the final solutions of the problem will be localized orthogonal orbitals . therefore , projection operators along with the shift parameters , simply play the role of a localizing potential @xcite as it is clear that upon convergence their contribution to the hartree - fock equation vanishes . the orbitals contained in unit cells located farther than those in @xmath37 should be automatically orthogonal to the reference cell orbitals by virtue of the large distance between them . it is clear that the orthogonalization of the orbitals to each other will introduce oscillations in these orbitals which are also referred to as the orthogonalization tails . combining the orthogonality of the neighboring orbitals to the reference cell orbitals with the translation symmetry of the infinite solid , it is easy to see that the orbitals of any unit cell are orthogonal to all the orbitals of the rest of the unit cells . therefore , orbitals thus obtained are essentially wannier functions . after solving for the hf equations presented above one can obtain the electronic part of the energy per unit cell simply by dividing the total energy of eq.([eq - esolidf ] ) by @xmath19 , which , unlike the total energy , is a finite quantity . in paper i we arrived at exactly the same hf equations as above , although we had followed a more intuitive path utilizing the so - called `` embedded - cluster '' philosophy , whereby we minimized only that portion of the total energy of eq.([eq - esolid ] ) which corresponds to the `` cluster - environment '' interaction . the fact that the derivations reported in paper i , and here , both lead to the same final equations has to do with the translation invariance which allows the total energy to be expressed in the form of eq.([eq - esolidf ] ) . therefore , we emphasize that the equations derived above are exact and do not involve any approximation other than the hartree - fock approximation itself . thus results of all the computations utilizing this approach should be in complete agreement with the equivalent computations performed using the traditional bloch orbital based approach as is implemented , e.g , in the program crystal @xcite . by inspection of eq.([eq - hff1 ] ) it is clear that it is of the embedded - cluster form in the sense that if one calls the reference unit cell the `` central cluster '' , it describes the dynamics of the electrons of this central cluster embedded in the field of identical unit cells of its environment ( rest of the infinite solid ) . we have performed a computer implementation of the formalism presented in the previous section within a linear combination of atomic orbital ( lcao ) approach , whereby we transform the differential equations of eq.([eq - hff1 ] ) into a set of linear equations solvable by matrix methods . atomic units were used throughout the numerical work . we proceed by expanding the orbitals localized in the reference cell as @xmath40 where @xmath41 has been used to denote the reference cell , @xmath16 represents the location of the @xmath42th unit cell ( located in @xmath41 or @xmath37 ) and @xmath43 represents a basis function centred in the @xmath42th unit cell . in order to account for the orthogonalization tails of the reference cell orbitals , it is necessary to include the basis functions centred in @xmath37 as well . clearly , the translational symmetry of the crystal as expressed in eq.([eq - trsym ] ) demands that the orbitals localized in two different unit cells have the same expansion coefficients @xmath44 , and differ only in the location of the centers of the basis functions . the lcao formalism implemented in most of the quantum - chemistry molecular programs , as also in the crystal code@xcite , expresses the basis functions @xmath43 of eq.([eq - lcao ] ) as linear combinations of cartesian gaussian type basis functions ( cgtfs ) of the form @xmath45 where @xmath46 . in the previous equation , @xmath47 denotes the exponent and the vector @xmath48 represents the center of the basis function . the centers of the basis functions @xmath48 are normally taken to be at the locations of the appropriate atoms of the system . cgtfs with @xmath49 are called respectively @xmath50 type basis functions@xcite . the individual basis functions of the form of eq.([eq - cgto ] ) are called _ primitive _ functions while the linear combinations of them are called the _ contracted _ functions . the formalism is totally independent of the type of basis functions , but for the sake of computational simplicity , we have programmed our approach using gaussian lobe - type functions@xcite . in this approach one approximates the @xmath51 and higher angular momentum cgtfs as linear combinations of @xmath52-type basis functions displaced by a small amount from the location of the atom concerned . for example , in the present study a primitive @xmath51 type cgtf centered at the origin was approximated as @xmath53 where , @xmath54 is the normalization constant and @xmath55 . in the present study the value of 0.1 atomic units ( a.u . ) was employed for @xmath56 . for approximating the @xmath57 , @xmath58 and @xmath59 types of basis functions , the displacement vectors @xmath60 are chosen to be along the positive @xmath61 , @xmath62 and @xmath63 directions , respectively . by substituting eq.([eq - lcao ] ) in eq.([eq - hff1 ] ) we obtain the hf equations in the lcao matrix form @xmath64 the fock matrix @xmath65 occuring in the equation above is defined as @xmath66 where the contribution of all the operators appearing in eq.([eq - hff1 ] ) has been replaced by the corresponding matrices in the representation of the chosen basis set . above , unprimed functions @xmath67 and @xmath68 represent the basis functions corresponding to the orbitals of the reference unit cell while the primed functions @xmath69 and @xmath70 denote the basis functions corresponding to the orbitals of @xmath37 . in particular , the overlap matrix is given by @xmath71 and the coulomb and the exchange matrix elements are defined as @xmath72 and @xmath73 where @xmath74 denotes the elements of the density matrix @xmath75 of the orbitals of a unit cell evaluated as @xcite @xmath76 the matrix form of the hf equations ( [ eq - scf ] ) is a pseudo eigenvalue problem which can be solved iteratively to obtain the hf orbitals . the energy per unit cell can be computed by means of a simple matrix - trace operation @xmath77 where above @xmath9 , @xmath10 , @xmath31 and @xmath32 and @xmath75 denote the matrices of the corresponding operators in the representation of the chosen basis set , and @xmath26 was defined after eq.([eq - esolidf ] ) . in practice one proceeds according to the following algorithm : 1 . start with some localized initial guess for the orbitals of the reference cell . for ionic systems considered here we chose these to be the orbitals of the individual ions centered on the corresponding atomic sites . for covalent systems , it would be reasonable to use suitable bonding combinations of atomic orbitals . 2 . use these orbitals to construct the fock matrix as defined in eq.([eq - fock ] ) . 3 . diagonalize the fock matrix to obtain a new set of orbitals of the reference cell . 4 . compute the energy per unit cell by using eq.([eq - ecell ] ) . go to step 2 . iterate until the energy per unit cell has converged . various mathematical formulas and computational aspects related to the evaluation of different contributions to the fock matrix are discussed in the appendix . in this section we describe the evaluation of the x - ray structure factors and compton profiles from the hartree - fock wannier functions obtained from the formalism of the previous section . both these properties can be obtained from the first - order density matrix of the system defined for the present case as @xmath78 where @xmath79 is the @xmath15-th hf orbital of the unit cell located at position @xmath23 . the factor of two above is a consequence of spin summation . by measuring the x - ray structure factors experimentally one can obtain useful information on the charge density of the constituent electrons . theoretically , the x - ray structure factor @xmath80 can be obtained by taking the fourier transform of the diagonal part of the first - order density matrix @xmath81 by means of compton scattering based experiments , one can extract the information on the momentum distribution of the electrons of the solid . in the present study we compute the compton profile in the impulse approximation as developed by eisenberger and platzman @xcite . under the impulse approximation the compton profile for the momentum transfer @xmath82 is defined as @xcite @xmath83 where @xmath84 and @xmath85 are , respectively , the changes in momentum and the frequency of the incoming x- or @xmath86-ray due to scattering , @xmath87 is the compton electron momentum , @xmath88 is the projection of @xmath87 in the direction of @xmath84 , the delta function imposes the energy conservation and @xmath89 denotes the electron momentum distribution obtained from the diagonal part of the full fourier transform of the first - order density matrix @xmath90 by choosing the @xmath63-axis of the coordinate system defining @xmath87 along the direction of @xmath84 , one can perform the @xmath59 integral in eq.([eq - cp1 ] ) to yield @xmath91 integrals contained in the expressions for the x - ray structure factor and the compton profile ( eqs . ( [ eq - xfac ] ) and ( [ eq - cp ] ) , respectively ) can be performed analytically when the density matrix is represented in terms of gaussian lobe - type basis functions . these analytic expressions are used to evaluate the quantities of interest in our computer code , once the hartree - fock density matrix has been determined . in this section we present the results of the calculations performed on crystalline lif and licl . prencipe et al . @xcite studied these compounds , along with several other alkali halides , using the crystal program @xcite . crystal , as mentioned earlier , is a bloch orbital based _ ab initio _ hartree - fock program set up within an lcao scheme , utilizing cgtfs as basis functions . in their study , prencipe et al . employed a very large basis set and , therefore , their results are believed to be very close to the hartree - fock limit . in the present work our intention is not to repeat the extensive calculations of prencipe et al . @xcite , but rather to demonstrate that at the hartree - fock level one can obtain the same physical insights by applying the wannier function based approach as one would by utilizing the bloch orbital based approach . moreover , because of the use of lobe functions as basis functions , we run into problems related to numerical instability when very diffuse @xmath92type ( and beyond ) basis functions are employed . in future we intend to incorporate true cgtfs as basis functions in our program , which should make the code numerically much more stable . therefore , we have performed these calculations with modest sized basis sets . we reserve the use of large basis sets for the future calculations , when we intend to go beyond the hartree - fock level to utilize these wannier functions to do correlated calculations . the reason we have chosen to compare our results to those obtained using the crystal program is because not only is crystal based upon an lcao formalism employing gaussian type of basis functions similar to our case , but also it is a well - tested program and widely believed to be the state of the art in crystalline hartree - fock calculations @xcite . all the calculations to be presented below assume the observed face - centered cubic ( fcc ) structure for the compounds . the reference unit cell @xmath41 was taken to be the primitive cell containing an anion at the @xmath93 position and the cation at @xmath94 , where @xmath95 is the lattice constant . the calculations were performed with different values of the lattice constants to be indicated later . the basis sets used for lithium , fluorine and chlorine are shown in tables [ tab - basli ] , [ tab - basf ] and [ tab - bascl ] , respectively . for lithium we adopted the basis set of dovesi et al . used in their lithium hydride study @xcite , while for fluorine and chlorine basis sets originally published by huzinaga and collaborators @xcite were used . the values of the level - shift parameters @xmath38 s of eq.([eq - fock ] ) should be high enough to guarantee sufficient orthogonality while still allowing for numerical stability . thus this choice leaves sufficient room for experimentation . we found the values in the range @xmath96 suitable for our work . we verified by explicit calculations that our results had indeed converged with respect to the values of the shift parameters . in the course of the evaluation of integrals needed to construct the fock matrix , all the integrals whose magnitudes were smaller than @xmath97 were discarded both in our calculations as well as in the crystal calculations . the comparison of our ground - state energies per unit cell with those obtained using the identical basis sets by the crystal program @xcite is illustrated in tables [ tab - enlif ] and [ tab - enlicl ] for different values of lattice constants . the biggest disagreement between the two types of calculations is 0.7 millihartree . a possible source of this disagreement is our use of lobe functions to approximate the @xmath51-type cgtfs . however , since the typical accuracy of a crystal calculation is also 1 millihartree @xcite , we consider this disagreement to be insignificant . such excellent agreement between the total energies obtained using two different approaches gives us confidence as to the essential correctness of our approach . from the results it is also obvious that the basis set used in these calculations is inadequate to predict the lattice constant and the bulk modulus correctly . to be able to do so accurately , one will have to employ a much larger basis set such as the one used by prencipe et al . @xcite . since hartree - fock lattice constants generally are much larger than the experimental value , we reserve the large - scale hartree - fock calculations for future studies in which we will also go beyond the hartree - fock level to include the influence of electron correlations . valence wannier functions for lif and licl are plotted along different crystal directions in figs . [ fig - lif2p-001 ] , [ fig - lif2p-110 ] , [ fig - licl3p-001 ] and [ fig - licl3p-110 ] . lattice constants for these calculations were assigned their experimental values @xcite of 3.99 @xmath98 and 5.07 @xmath98 , for lif and licl , respectively . although core orbitals were also obtained from the same set of calculations , we have not plotted them here because they are trivially localized . the @xmath51-character of the wannier functions is evident from the antisymmetric nature of the plots under reflection . the additional nodes introduced in the orbitals due to their orthogonalization to orbitals centered on the atoms of region @xmath37 are also evident . the localized nature of these orbitals is obvious from the fact that the orbitals decay rapidly as one moves away from the atom under consideration . the orthogonality of the orbitals of the reference cell to those of the neighborhood ( region @xmath37 ) was always better than @xmath99 . now we discuss the data for x - ray structure factors . these quantities were also evaluated at experimental lattice constants mentioned above . the x - ray structure factors obtained by our method are compared to values calculated with the crystal program , and experimental data , in tables [ tab - lifxfac ] and [ tab - liclxfac ] for lif and licl , respectively . for the case of lif we directly compare the theoretical values with the experimental data of merisalo et al . @xcite , extrapolated to zero temperature by euwema et al . @xcite . for licl it was not possible for us to extrapolate the experimental data of inkinen et al . @xcite , measured at @xmath100 , to the corresponding zero temperature values . therefore , to compare our licl calculations to the experiment , we correct our theoretical values for thermal motion using the debye - waller factors of @xmath101 and @xmath102 , measured also by inkinen et al . @xcite the debye - waller corrections were applied to the individual form factors of li@xmath103 and cl@xmath104 ions . from both the tables it is obvious that our results are in almost exact agreement with those of crystal . this implies that our wannier function hf approach based description of the charge density of systems considered here , is identical to a bloch orbital based hf description as formulated in crystal @xcite . for the case of lif the agreement between our results and the experiment is also quite good , maximum errors being @xmath105 5% . for the case of licl , our corrected values of x - ray structure factors deviate from the experimental values at most by approximately 3% . perhaps by using a larger basis set one can obtain even better agreement with experiments . finally we turn to the discussion of compton profiles . directional and isotropic compton profiles , computed using our approach and the crystal program , are compared to the isotropic compton profiles measured by paakkari et al . @xcite , in tables [ tab - lifcp ] and [ tab - liclcp ] . we obtain the isotropic compton profiles from our directional profiles by performing a directional average of the profiles along the three crystal directions according to the formula @xmath106 valid for an fcc lattice @xcite . while experimental data for directional compton profiles exist in the case of lif @xcite , no such measurements have been performed for licl , to the best of our knowledge . for lif there is close agreement between our results and the ones calculated using the crystal program . for licl our results disagree with the crystal results somewhat for small values of momentum transfer , although relatively speaking the disagreement is quite small the maximum deviation being @xmath105 0.3% for @xmath107 and the @xmath108 $ ] direction . the possible source of the disagreement may be that to get the values of compton profiles for all the desired values of momentum transfer , we had to use the option of crystal @xcite where the compton profiles are obtained by using the real - space density matrix rather than its more accurate * k*-space counterpart . however , as is clear from the tables , even for those worst cases , there is no significant difference between the averaged out isotropic compton profiles obtained in our computations and those obtained from crystal . at the larger values of momentum transfer , our results are virtually identical to the crystal results . the close agreement with crystal clearly implies that our wannier function based description of the momentum distribution of the electrons in the solid is identical to the one based upon bloch orbitals . considering the fact that we have used a rather modest basis set , it is quite surprising that the values of isotropic compton profiles obtained by us are in close agreement with the corresponding experimental values @xcite . an inspection of tables [ tab - lifcp ] and [ tab - liclcp ] reveals that the calculated values always agree with the experimental ones to within 6% . however , ours as well as the crystal calculations presented here are not able to describe the observed anisotropies in the directional compton profiles @xcite for lif which is also the reason that we have not compared the theoretical anisotropies to the experimental ones . for small values of momentum transfer the calculated values are even in qualitative disagreement with the experimental results , although for large momentum transfer the qualititative agreement is restored . this result is not surprising , however , because , as berggren et al . have argued @xcite in their detailed study , the proper description of the compton anisotropy mandates a good description of the long - range tails of the crystal orbitals . to be able to do so with the gaussian - type of basis functions used here , one will unlike the present study have to include basis functions with quite diffuse exponents . in conclusion , an _ ab initio _ hartree - fock approach for an infinite insulating crystal which yields orbitals in a localized representation has been discussed in detail . it was applied to compute the total energies per unit cell , x - ray structure factors and directional compton profiles of two halides of lithium , lif and licl . the close agreement between the results obtained using the present approach , and the ones obtained using the conventional bloch orbital based hf approach , demonstrates that the two approaches are entirely equivalent . the advantage of our approach is that by considering local perturbations to the hartree - fock reference state by conventional quantum - chemical methods , one can go beyond the mean - field level and study the influence of electron correlations on an infinite solid in an entirely _ ab initio _ manner . presently projects along this direction are at progress in our group , and in a future publication we will study the influence of electron correlations on the ground state of a solid . one of us ( a.s . ) gratefully acknowledges useful discussions with prof . roberto dovesi , and his help regarding the use of the crystal program . in this section we discuss the calculation of various terms in the fock matrix . since the kinetic - energy matrix elements @xmath109 and the overlap - matrix elements @xmath110 have simple mathematical expressions and are essentially unchanged from molecular calculations , we will not discuss them in detail . however , we will consider the evaluation of the rest of the contributions to the fock matrix at some length . the electron - nucleus attraction term of the fock matrix contains the infinite lattice sums involving the attractive interaction acting on the electrons of the reference cell due to the infinite number of nuclei in the solid . when treated individually , this term is divergent . however , when combined with the coulombic part of the electron repulsion to be discussed in the next section , convergence is achieved because the divergences inherent in both sums cancel each other owing to the opposite signs . this fact is a consequence of the charge neutrality of the unit cell and is used in the ewald - summation technique @xcite to make the individual contibutions also convergent by subtracting , from the corresponding potential a shadow potential emerging from a ficitious homogeneous charge distribution of opposite sign . in addition , in the ewald method , one splits the lattice potential into a short - range part whose contribution is rapidly convergent in the @xmath111-space and a long - range part which converges fast in @xmath84-space . therefore , in the ewald - summation technique one replaces the electron - nucleus interaction potential due to a lattice composed of nuclei of charge @xmath112 , by the effective potential @xcite @xmath113 where @xmath23 represents the positions of the nuclei on the lattice , @xmath114 are the vectors of the reciprocal lattice , @xmath85 is the volume of the unit cell , @xmath115 is a convergence parameter to be discussed later and erfc represents the complement of the error function . matrix elements of the ewald potential of eq.([eq - ewaldp ] ) with respect to primitive @xmath52-type basis functions were derived by stoll @xcite to be @xmath116 above @xmath51 and @xmath82 label the primitive basis functions , @xmath117 and @xmath118 represent the positions of the unit cells in which they are located and @xmath119 represents the overlap matrix element between the two primitives given by @xmath120 the vectors @xmath121 and @xmath122 above specify the centers of the two basis functions relative to the origin of the unit cell , @xmath47 and @xmath123 represent the exponents of the two gaussians , @xmath124 and @xmath125 with @xmath126 , @xmath127 and where the parameter @xmath129 takes over the role of the convergence parameter @xmath115 of eq.([eq - ewaldp ] ) . the remaining quantities are the the same as those in eq.([eq - ewaldp ] ) . it is clear that the function @xmath130 involves lattice sums both in the direct space and the reciprocal space . although the final value of the function will be independent of the choice of the convergence parameter @xmath129 , both these sums can be made to converge optimally by making a judicious choice of it . large values of @xmath129 lead to faster convergence in the real space but to slower one in the reciprocal space and with smaller values of @xmath129 the situation is just the opposite . therefore , for optimal performance , the choice of @xmath129 is made dependent on the value of @xmath15 . in the present work we make the choice so that if @xmath131 , @xmath132 and if @xmath133 , @xmath134 . in the former case the sum is both , in the real and the reciprocal space while in the latter case the sum is entirely in the reciprocal space . although we have written an efficient computer code to evaluate the function @xmath130 , it remains the most computer intensive part of our program . the computational effort involved in the computation of these integrals can be reduced by utilizing the translational symmetry . one can verify that as a consequence of translation symmetry @xmath135 where @xmath136 is also a vector of the direct lattice , @xmath137 represents the reference unit cell and the last term is a compact notation for the second term . since the number of unique @xmath138 vectors is much smaller than the number of pairs @xmath139 , the use of eq.([eq - usym ] ) reduces the computational effort considerably . to further reduce the computational effort we also use the interchange symmetry @xmath140 additional savings are achieved if one realizes that matrix elements @xmath141 become smaller as larger the distance @xmath142 between the interacting charge distributions becomes . as is clear from eq.([eq - upq ] ) , a good estimate of the magnitude of an integral is the overlap element @xmath143 @xcite . therefore , we compute only those integrals whose overlap elements @xmath143 are larger than some threshold @xmath144 . in the present calculations we chose @xmath145 . to calculate the coulomb contribution to the fock matrix , one needs to evaluate the two - electron integrals with infinite lattice sum @xmath146 where @xmath147 and @xmath52 represent the primitive basis functions and @xmath148 and @xmath149 represent the unit cells in which they are centered . this integral , treated on its own is divergent , as discussed in the previous section . however , using the ewald - summation technique , one can make this series conditionally convergent with the implicit assumption that its divergence will cancel the corresponding divergence of the electron nucleus interaction . since the details of the ewald - summation technique for the coulomb part of electron repulsion are essentially identical to the case of electron - nucleus interaction , we will just state the final results @xcite @xmath150 where @xmath151 and @xmath152 all the notations used in the equations above were defined in the previous section . the expression @xmath153 used in eq.([eq - jmate ] ) , as against @xmath154 of eq.([eq - jmat ] ) , is meant to remind us that the matrix elements stated in eq.([eq - jmate ] ) are those of the two - electron ewald potential and not those of the ordinary coulomb potential . like in the case of electron - nucleus attraction , one can utilize the translational symmetry for the present case to reduce the computational effort significantly . the corresponding relations in the present case are @xmath155 where as before @xmath137 represents the reference unit cell , @xmath136 , @xmath156 and the last term in eq.([eq - jsym ] ) is a compact notation for the second term . since the number of pairs @xmath157 is much smaller than the number of quartets @xmath158 , use of eq.([eq - jsym ] ) results in considerable savings of computer time and memory . in addition , we also use the four interchange relations of the form of eq.([eq - usym2 ] ) to further reduce the number of nonredundant integrals . additionally , these integrals also satisfy the interchange relation @xmath159 to keep the programming simple , however , at present we do not utilize this symmetry . in future , we do intend to incorporate this symmetry in the code . similar to the case of electron - nucleus integrals , here also we use the magnitude of the product @xmath160 to estimate the size of the integral to be computed and proceed with its calculation only if it is greater than a threshold @xmath161 , taken to be @xmath162 in this study . in order to compute the exchange contribution to the fock matrix , one has to compute the following two - electron integrals involving infinite lattice sum @xmath163 where the notation is identical to the previous two cases . by using the translational symmetry arguments one can show even for the exchange case that @xmath164 where the last term in eq.([eq - ksym ] ) above is a compact notation for the second term . as in the previous two cases , the use of translational symmetry results in considerable savings of computer time and storage . explicitly @xmath165 although eq.([eq - kfin ] ) contains an infinite sum over lattice vectors @xmath166 , the contributions of each of the terms decreases rapidly with the increasing distances @xmath167 and @xmath168 between the interacting charge distributions . a good estimate of the contribution of the individual terms is provided by the product of overlap matrix elements between the interacting charge distributions namely , @xmath169 and @xmath170 @xcite . therefore , in the computer implementation , we arrange the vectors @xmath166 so that the corresponding overlaps are in the descending order and the loop involving the sum over @xmath166 in eq.([eq - kfin ] ) is terminated once the individual overlap matrix elements or their product are less than a specified threshold @xmath171 . the computer code for evaluating these integrals is a modified version of the program written originally by ahlrichs @xcite . the value of the threshold @xmath171 used in these calculations was @xmath162 . the exchange integrals also satisfy interchange symmetries similar to those of eqs.([eq - usym2 ] ) and ( [ eq - jsymn ] ) , which are not used in the present version of the code for the ease of programming . in future , however , we plan to use them as well . as described above , to minimize the need of computer time and storage , we have made extensive use of translational symmetry . however , the integral evaluation can be further optimized considerably by making use of point group symmetry as is done in the crystal program @xcite . implementation of point group symmetry , as well as the use of cgtos instead of lobe - type functions , is planned for future improvements of the present code .

an _ ab initio _ hartree - fock approach aimed at directly obtaining the localized orthogonal orbitals ( wannier functions ) of a crystalline insulator is described in detail . the method is used to perform all - electron calculations on the ground states of crystalline lithium fluoride and lithium chloride , without the use of any pseudo or model potentials . quantities such as total energy , x - ray structure factors and compton profiles obtained using the localized hartree - fock orbitals are shown to be in excellent agreement with the corresponding quantities calculated using the conventional bloch - orbital based hartree - fock approach . localization characteristics of these orbitals are also discussed in detail .

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Proceed to summarize the following text: when the first galaxies emerged @xmath4 million years after the big bang , their starlight reionized and heated the intergalactic hydrogen that had existed since cosmological recombination . much is currently unknown about this process , including what spatial structure it had , when it started and completed , and even which sources drove it . the ly@xmath0 forest provides one of the only robust constraints on this process , showing that it was at least largely complete by @xmath5 , when the universe was one billion years old @xcite . this letter argues that there exists another , potentially groundbreaking signature of reionization in the ly@xmath0 forest data . the amount of absorption in the ly@xmath0 forest can be quantified by the effective optical depth , @xmath6 , where @xmath7 $ ] is the transmitted fraction of a quasar s flux , @xmath8 indicates an average over a segment of the forest of length @xmath9 , and @xmath10 is the optical depth in ly@xmath0 at location @xmath11 along a sightline . the optical depth , @xmath10 , scales approximately as the h i number density , which after reionization scales as @xmath12 . here , @xmath13 is temperature , @xmath14 is the gas density in units of the cosmic mean , and @xmath15 is the h i photoionization rate , which scales with the amplitude of the local ionizing radiation background . observations of high-@xmath16 quasars show a steep increase in the dispersion of @xmath17 among coeval forest segments around @xmath18 @xcite . in the limit of a uniform ionizing background , the well - understood fluctuations in @xmath19 fall well short of producing the observed dispersion at @xmath20 , as shown recently by @xcite ( hereafter b2015 ) . previous studies have attempted to explain this excess with spatial fluctuations in the ionizing background . the properties of spatial fluctuations in the background depend on the number density of sources and the mean free path of photons , @xmath21 . while @xmath21 is well constrained at @xmath22 , being too large to yield significant background fluctuations for standard source models @xcite , b2015 showed that the excess @xmath17 dispersion at @xmath23 could be matched in a model where @xmath21 decreases by a factor of @xmath24 between @xmath25 and @xmath23 a time scale of just @xmath26 million years . however , such rapid evolution in @xmath21 is inconsistent with extrapolations based on measurements at lower redshifts @xcite , and would imply that the emissivity of ionizing sources , in turn , increases by an unnatural factor of @xmath24 over the same cosmologically short time interval . because of these issues , b2015 speculated that the excess dispersion was evidence for large spatial variations in the mean free path . alternatively , fluctuations in the ionizing background could have been enhanced if the sources of ionizing photons were rarer than the observed population of galaxies . however , current models require half of the background to arise from bright sources with an extremely low space density of @xmath27@xcite . this scenario is a possibility of current debate ( e.g. * ? ? ? * but see daloisio et al . in prep . ) . in this letter , we explore a source of dispersion in @xmath17 that has so far been neglected and that , unlike ionizing background fluctuations , has straightforward implications for the reionization process itself . in addition to @xmath14 and @xmath15 , the ly@xmath0 opacity depends on @xmath28 , mainly because the amount of neutral hydrogen after reionization is proportional to the recombination rate , which scales as @xmath29 . previous attempts to model ly@xmath0 opacity fluctuations had not included the residual temperature fluctuations that must have been present if reionization were patchy and temporally extended . as ionization fronts propagated supersonically through the igm , the gas behind them was heated to tens of thousands of degrees kelvin by photoionizations of h i and he i. after reionization , the gas cooled mainly through adiabatic expansion and through inverse compton scattering with cosmic microwave background ( cmb ) photons @xcite . since different regions in the igm were reionized at different times , these heating and cooling processes imprinted an inhomogeneous distribution of intergalactic temperatures that persisted after reionization @xcite . we will show that these residual temperature variations likely account for much of the observed dispersion in @xmath17 at @xmath30 , and may even account for all of it a scenario that would yield new information on the timing , duration , and patchiness of reionization . the remainder of this letter is organized as follows . in [ sec : methods ] and [ sec : toymodel ] , we describe our simulations and methodology . in [ sec : results ] , we present our main results . in [ sec : conclusion ] , we offer concluding remarks . we use comoving units for distances and physical units for number densities . to model the impact of relic temperature fluctuations from reionization on the distribution of @xmath17 , we ran a suite of 20 cosmological hydrodynamics simulations using a modified version of the code of @xcite . the simulations were initialized at @xmath31 from a common cosmological initial density field . we used a matter power spectrum generated by camb @xcite assuming a flat @xmath32cdm model with @xmath33 , @xmath34 , @xmath35 , @xmath36 , @xmath37 , and @xmath38 , consistent with recent measurements @xcite . our production runs use a cubical box with side length @xmath39 , with @xmath40 dark matter particles and @xmath41 gas cells . in each simulation , reionization was modeled in a simplistic manner by instantaneously ionizing and heating the gas to a temperature @xmath42 at a redshift of @xmath43 . subsequently , ionization was maintained with a homogeneous background with spectral index @xmath44 , consistent with recent post - reionization background models @xcite . utilizing the periodic boundary conditions of our simulations , we trace skewers of length @xmath45 ( following the convention of the b2015 @xmath17 measurements ) at random angles through all of the hydro simulation snapshots . each skewer is divided into @xmath46 equally spaced velocity bins of size @xmath47 ( where @xmath48 is the cosmological scale factor ) , and ly@xmath0 optical depths are computed using the method of @xcite . although reionization occurs instantaneously at @xmath43 within each simulation box , we piece together skewer segments from simulations with different @xmath43 to model the effect of an inhomogeneous reionization process , as we describe further in the next section . the post - reionization temperatures in the simulations are relatively insensitive to the spectrum of the ionizing background , but they are sensitive to the amount of heating that is assumed to occur at the time a gas parcel is reionized . previous calculations @xcite have bracketed the range of possible reionization temperatures to @xmath49 k. ( we note that previous large - scale reionization simulations do not accurately capture @xmath42 , as they do not resolve the @xmath50 physical kpc ionization fronts . ) thus , we have run two sets of ten simulations one set with @xmath51 k and the other with @xmath52 k where each set contains instantaneous reionization redshifts of @xmath53 . this redshift range spans the likely duration of reionization . simulations with @xmath54 are driven to a common temperature by @xmath55 , so they are well approximated by the @xmath56 simulation . figure [ fig : physics ] shows the post - reionization thermal and associated hydrodynamic relaxation of intergalactic gas ( and its effect on the ly@xmath0 forest opacity ) in our simulations . the top - left panel shows the volume - weighted average gas temperature in the ten @xmath52 k simulations . the bottom - left panel shows this same average , but limited to gas cells with densities of @xmath57 , the deepest voids that dominate transmission in the highly - saturated @xmath58 forest . even at @xmath59 , the @xmath60 gas temperatures differ by up to a factor of five between the simulations that were reionized at different times . the right panels show the h i number densities ( top ) and the transmission ( bottom ) at @xmath61 for the same skewer through our @xmath56 ( blue / solid ) and @xmath62 ( red / dashed ) simulations . the transmission is nearly zero in the @xmath56 case , owing to colder temperature and hence enhanced h i densities , whereas there is significant transmission in the @xmath62 case . reionization is a process that is inhomogeneous and temporally extended , unlike in our individual hydro simulations . modeling the thermal imprint of patchy reionization on the ly@xmath0 forest thus requires an additional ingredient : a model for the redshifts at which points along our skewers are reionized . in this section , we present a simplified toy model to illustrate how we piece together skewer segments from our hydro simulations , and to provide insight into how the timing , duration and morphology of reionization affect the amplitude of ly@xmath0 opacity fluctuations . for illustrative purposes , let us assume that the ly@xmath0 forest is made up of segments of equal length , @xmath63 , where each segment is reionized at a single redshift . let us further assume that the reionization redshift of each segment is drawn from a uniform probability distribution over the interval @xmath64 $ ] , resulting in a global reionization history in which the mean ionized fraction , @xmath65 , is linear in redshift ( since @xmath65 is the cumulative probability distribution of the reionization redshift ) . assuming that the reionization redshift field is in the hubble flow , each segment of length @xmath63 spans @xmath66 velocity bins of our hydro simulation skewers . for the first segment , with reionization redshift @xmath67 , we take the initial @xmath68 spectrum velocity bins of a randomly drawn skewer that has @xmath43 closest to @xmath67 . for the next segment , with reionization redshift @xmath69 , we take the next @xmath68 velocity bins _ from the same skewer _ through the simulation that has @xmath43 closest to @xmath69 ( note that all simulations were started from the same initial density field ) . we repeat this procedure until an entire @xmath70 sightline is filled . in what follows , we compare these toy models against the b2015 measurements of the cumulative probability distribution function of @xmath17 , @xmath71 . the measurements considered here span @xmath72 in bins of width @xmath73 . we construct @xmath71 from 4000 randomly drawn sightlines at the central redshift of each bin . for each redshift , we rescale the nominal post - reionization photoionization rate of our simulations ( @xmath74 ) by a constant factor , such that our model @xmath71 is equal to the observed distribution at either @xmath75 or @xmath76 , depending on which value provides the better visual fit . we have performed extensive numerical convergence tests using simulations of varying resolution and box size . we found excellent convergence of @xmath71 for our production runs in both box size and resolution ( especially for our patchy reionization models ) . in our toy model ( colored curves ) compared to the b2015 measurements ( black histograms ) and to an instantaneous reionization model with @xmath77 ( black / dotted curves ) . * top row : * varying the coherence length , @xmath78 , assuming a uniform @xmath43 probability between @xmath79 . * bottom row : * varying the redshift interval over which the @xmath43 values are drawn , assuming @xmath80 . , width=340 ] figure [ fig : toymodel_cdfs ] shows the @xmath71 of these toy models for a range of @xmath81 , compared against the b2015 measurements in the two highest redshift bins ( black histograms ) . the curves with shorter reionization durations , or with smaller @xmath63 , fall closer to the black / dotted curves , which assume that reionization occurs instantaneously ( at @xmath77 , although it does not depend on this choice ) . figure [ fig : toymodel_cdfs ] affords three insights : ( 1 ) the larger the coherence length , @xmath63 , over which gas shares a similar reionization redshift , the larger the spread in @xmath17 ; ( 2 ) the width of the observed @xmath71 can be fully accounted for only if reionization ended at @xmath82 and was well underway by @xmath2 ; ( 3 ) the maximal width is achieved by a late - ending ( @xmath83 ) and extended reionization model in which large contiguous segments ( @xmath84 ) of the ly@xmath0 forest were reionized at @xmath85 . in the next section , we apply these insights to construct more realistic models of reionization that reproduce the observed width of @xmath71 . we generate physically - motivated reionization redshift fields using simulations based on the excursion - set model of reionization ( esmr ) @xcite , which has been shown to reproduce the ionization structure found in full radiative transfer simulations @xcite . in particular , our esmr uses a realization of the linear cosmological density field and a top hat in fourier space filter to generate a realization of reionization in a cubical box with side @xmath86 , sampled with @xmath87 cells . for details of the algorithm , see @xcite . we use the simplest formulation of the esmr , with two free parameters : ( 1 ) the minimum mass of halos that host galactic sources of ionizing photons , @xmath88 ; ( 2 ) the ionizing efficiency of the sources , @xmath89 . we tune @xmath89 to obtain a reionization history that is approximately linear in redshift , similar to those in our toy model . the esmr calculation yields a cube of reionization redshifts . we construct mock absorption spectra by first tracing @xmath70 skewers through this cube . we then piece together spectrum segments from our suite of hydro simulations , much like in our toy model , except here we match @xmath43 to the reionization redshifts along the esmr skewers . this process does not account for how @xmath43 correlates with density , an effect that will underestimate ( making our models conservative ) the width of @xmath71 , which we address shortly . the left panel of figure [ fig : zreionfield ] shows a 2d slice through our fiducial reionization redshift field in which reionization spans @xmath90 . in the right panels , the blue / solid curves show the fraction of transmitted flux along four different sightlines through this field , selected to span a range of @xmath17 . the red / dashed curves show the corresponding reionization redshifts along the sightlines . on average , regions that reionize at later times yield more transmission , while regions that reionize at @xmath85 result in dark gaps in the ly@xmath0 forest . the variation among these mock spectra is similar to the well - known variation seen in observations of the @xmath3 ly@xmath0 forest @xcite . in our interpretation , this variation reflects differences in the reionization redshifts and hence temperatures fluctuations . ] between segments of the ly@xmath0 forest . the leftmost four panels of figure [ fig : eps_cdfs ] show @xmath71 in three esmr models that take @xmath91 and @xmath92 , with reionization histories shown in the inset of the @xmath93 panel . the cmb electron scattering optical depths in these models are @xmath94 and @xmath95 , within uncertainties of the latest _ planck _ measurement of @xmath96 @xcite . these models are compared against the b2015 measurements ( black histograms ) and against the homogeneous reionization reference model with @xmath77 ( dotted curves ) . the blue / long - dashed and green / short - dashed curves correspond to scenarios in which reionization spans @xmath97 and @xmath98 , respectively . while these two models produce significantly more width in @xmath71 than the homogeneous reionization model , they fall short of producing the full range of @xmath17 required to match the observations . however , the fiducial @xmath90 reionization model ( red / solid curves ) generally provides a good match to the measurements . a significant success of the fiducial model is that the observed redshift evolution of the @xmath71 width is reproduced without any additional tuning of parameters . ( we have checked that this success also holds over @xmath99 , lower redshifts than those shown where the effect is reduced . ) indeed , there are not any parameters that can be tuned in our model to change the post - reionization evolution of the width . the pink shaded regions indicate the 90% confidence levels of our fiducial model , estimated from bootstrap realizations , showing consistency with all the data aside from a single high - opacity point at @xmath100 and at @xmath23 . the discrepancy at the highest opacities may arise because our method of constructing mock absorption spectra does not capture correlations between temperature and density that should be present , since denser regions are more likely reionized earlier . such correlations would act to increase the width of @xmath71 , as the denser regions around galaxies are ionized earlier in our models . one might naively think these correlations are small , because correlations between the density on the much larger scales of the h ii bubbles and the smaller scales of voids in the forest should be weak , but calculations show that they may not be negligible @xcite the right panels of figure [ fig : eps_cdfs ] show the effect of varying @xmath42 and @xmath88 . for @xmath101 , the distribution of @xmath17 is somewhat narrower than the case with @xmath91 . hotter temperatures are likely achieved towards the end of reionization , when h ii bubbles are larger and propagate at quicker speeds . a model with @xmath102 near the end of reionization , and smaller temperatures earlier on , would likely produce more width than a model with @xmath102 at all times . however , any conclusions about @xmath42 prior to modeling the density / reionization - redshift correlations are premature . the blue / long - dashed and green / short - dashed curves in the right panels show the effect of varying @xmath88 . for the former atomic cooling halos ( achs ) curve , @xmath103 is set to the minimum mass required to achieve a halo virial temperature of @xmath104 . for both curves , we tune @xmath89 to match the reionization history of our fiducial model ( red curve in the inset ) . we find that the effects of varying @xmath103 are minor . we have shown that residual temperature inhomogeneities from a patchy and extended reionization process likely account for much of the opacity fluctuations in the @xmath58 ly@xmath0 forest . inhomogeneities in the ionizing background may also contribute at a significant level , though current models in this vein have required very small mean free paths or extremely rare sources . we showed that residual temperature fluctuations alone could account for the entire spread of observed @xmath17 . a significant success of this interpretation is that it is able to reproduce the rapid growth of @xmath17 fluctuations with redshift , despite having very little freedom in its post - reionization evolution . in this scenario , the observations favor a late but extended reionization process that is roughly half complete by @xmath2 and that ends at @xmath3 . unlike ionizing background fluctuations , which do not necessarily signal the end of reionization , temperature fluctuations directly probe the timing , duration , and patchiness of this process . if most of the opacity variations owe to temperature , it would mean that , on average , the darkest @xmath105 segments of the @xmath106 ly@xmath0 forest were reionized earliest , and the brightest segments last a potentially powerful probe of cosmological reionization . + the authors acknowledge support from nsf grant ast1312724 . h.t . also acknowledges support from nasa grant atp - nnx14ab57 g . computations were performed with nsf xsede allocation tg - ast140087 . we thank paul la plante , jonathan pober , phoebe upton sanderbeck , george becker , adam lidz , and fred davies for helpful discussions .

recent observations of the ly@xmath0 forest show large - scale spatial variations in the intergalactic ly@xmath0 opacity that grow rapidly with redshift at @xmath1 , far in excess of expectations from empirically motivated models . previous studies have attempted to explain this excess with spatial fluctuations in the ionizing background , but found that this required either extremely rare sources or problematically low values for the mean free path of ionizing photons . here we report that much or potentially all of the observed excess likely arises from residual spatial variations in temperature that are an inevitable byproduct of a patchy and extended reionization process . the amplitude of opacity fluctuations generated in this way depends on the timing and duration of reionization . if the entire excess is due to temperature variations alone , the observed fluctuation amplitude favors a late - ending but extended reionization process that was roughly half complete by @xmath2 and that ended at @xmath3 . in this scenario , the highest opacities occur in regions that reionized earliest , since they have had the most time to cool , while the lowest opacities occur in the warmer regions that reionized most recently . this correspondence potentially opens a new observational window into patchy reionization .

You are an expert at summarizing long articles. Proceed to summarize the following text: ultracold atoms in optical lattices offer unprecedented possibilities of controlling quantum matter and mimicking the systems of condensed - matter and high - energy physics @xcite . particularly fascinating is the possibility to study ultracold atoms under the influence of strong artificial abelian and non - abelian magnetic " fields . the experimental realization of artificial abelian magnetic " fields , which reproduce the physics of electrons in strong magnetic fields , is currently achieved through diverse schemes : for atoms in a trap the simplest way is to rotate the trap @xcite , while for atoms in optical lattices this can be accomplished by combining laser - assisted tunneling and lattice acceleration methods @xcite , by the means of lattice rotations @xcite , or by the immersion of atoms in a lattice within a rotating bose - einstein condensate ( bec ) @xcite . several phenomena were predicted to occur in these arrangements such as the hofstadter butterfly " @xcite and the escher staircase " @xcite in single - particle spectra , vortex formation @xcite , quantum hall effects @xcite , as well as other quantum correlated liquids @xcite . as shown by one of us in ref . @xcite , it is simple to generalize the scheme of jaksch and zoller for generating artificial abelian magnetic " fields @xcite in order to mimic artificial non - abelian magnetic " fields . to this aim we have to consider atoms with more internal states ( flavors " ) . the gauge potentials that can be realized using standard manipulations , such as laser - assisted tunneling and lattice acceleration , can have practically arbitrary matrix form in the space of flavors " . in such non - abelian potentials , the single - particle spectrum generally depicts a complex structure termed by one of us hofstadter moth " @xcite , which is characterized by numerous extremely small gaps . the model of ref . @xcite has stimulated further investigations , including studies of nontrivial quantum transport properties @xcite , as well as studies of the integer quantum hall effect ( iqhe ) for cold atoms @xcite , spatial patterns in optical lattices @xcite , modifications of the landau levels @xcite , and quantum atom optics @xcite . one should note , however , that the @xmath0 gauge potentials proposed in ref . @xcite and used in most of the following works are characterized by _ non - constant _ wilson loops : atoms performing a loop around a plaquette undergo a unitary transformation which depends linearly on one of the spatial coordinates . although such gauge potentials are interesting _ per se _ , the features characterizing the hofstadter moth " result from this _ linear spatial dependence _ of the wilson loop , rather than from their non - abelian nature . indeed , the hofstadter moth "- like spectrum may actually be found in the standard abelian case with a wilson loop proportional to @xmath1 ( see fig . 1 ) . two of us have shown that cold fermionic atoms trapped in optical lattices and subjected to artificial magnetic " fields should exhibit an iqhe @xcite . if a static force is applied to atoms , for instance by accelerating the lattice , the transverse hall conductivity gives the relation between this external forcing and the transverse atomic current through the lattice . it has been shown that this transverse conductivity is quantized , @xmath2 , where @xmath3 is an integer and @xmath4 is planck s constant . note that this quantity can be easily measured from density profiles , as shown recently by umucalilar _ . _ @xcite . the quantization of @xmath5 occurs , however , only if the fermi energy of the system is located inside a gap of the single - particle spectrum . while the observation of the iqhe seems to be experimentally feasible in abelian magnetic " fields , it is hardly so in the deeply non - abelian regime in which the gaps of the moth " become very small @xcite . the question therefore arises whether the consideration of non - abelian gauge potentials characterized by a _ constant _ wilson loop could stabilize the spectral gaps and guarantee the robustness of the iqhe in ultracold fermionic gases and whether an anomalous iqhe , as observed in graphene , can exist in such systems . in this work we provide affirmative answers to these questions by considering the iqhe in a system that features a non - abelian gauge potential characterized by specific non - commutating constant components and by a _ constant _ wilson loop . we calculate the energy spectrum and we obtain a robust band structure with well developed gaps , which differs drastically from the case of the gauge potential of ref . @xcite . in particular , we note the existence of van hove singularities in the density of states and we obtain their analytical expression . we then evaluate the conductivity @xmath6 for neutral currents using topological methods : we express @xmath6 in terms of the topologically invariant chern numbers associated to each energy band @xcite . we eventually present a salient fractal phase diagram which represents the integer values of the transverse conductivity inside the infinitely many gaps of the spectrum . in this way , we show that the iqhe survives in the non - abelian regime , but undergoes strong modifications with striking similarity to the anomalous iqhe in graphene @xcite : the transverse conductivity suddenly changes sign due to the presence of van hove singularities and is , under certain conditions , anomalous because of conical energy spectra . we consider a system of non - interacting two - component fermionic atoms trapped in a 2d optical square lattice of unit length @xmath7 , with sites at @xmath8 , with @xmath9 integers . the non - interacting limit can be reached using feshbach resonances , or simply at low densities . the optical potential is strong , so that the tight - binding approximation holds . the schrdinger equation for a single particle subjected to an artificial gauge potential then reads @xmath10 where @xmath11 ( resp . @xmath12 ) is the tunneling operator and @xmath13 ( resp . @xmath14 ) is the tunneling amplitude in the @xmath1 ( resp . @xmath15 ) direction . in the following , we use @xmath7 as the length , and @xmath16 as the energy units , and set @xmath17 , except otherwise stated . the tunneling operators are related to the gauge potential according to @xmath18 . here we consider a general non - abelian gauge potential @xmath19 where @xmath20 and @xmath21 are parameters , @xmath22 , @xmath23 are pauli matrices and @xmath24 is the number of abelian magnetic flux quanta per unit cell . in order to realize such a potential we may consider the method of ref . however , the specific form of this gauge potential allows us to consider an even more practical scheme based on a generalization of the method currently developed by klein and jaksch @xcite . we may use @xmath25k atoms in @xmath26 or @xmath27 hyperfine manifolds , or @xmath28li with @xmath29 . for @xmath25k one should optically pump and restrict the atomic dynamics to the two lowest zeeman sublevels in each of the hyperfine manifolds . one can then employ different lattice tiltings in the @xmath1 and @xmath15 directions to perform laser ( raman assisted ) tunnelings that change the internal states of the atoms ; this allows to control the parameters @xmath20 and @xmath21 , and fixes the tunneling rate . finally , the immersion of the system in a rotating bec will allow to control @xmath24 @xcite . in experiments , one routinely reaches the values of ( laser assisted , or direct ) tunneling rates in the range of 5 - 10 khz ( @xmath30 0.5 @xmath31k ) , fermi temperatures of the same order , and temperatures @xmath32 0.2 @xmath33 50 - 100 nk ( see for instance @xcite ) . + the tunneling operators are @xmath34 unitary matrices , @xmath35 which act on the two - component wave function @xmath36 . the single - particle hamiltonian is invariant under translations defined by the operators @xmath37 and @xmath38 under the condition that @xmath39 , where @xmath40 and @xmath41 are integers . consequently , the system is restricted to a @xmath42 super - cell and one can express the wave function as @xmath43 , with @xmath44 a @xmath41-periodic function . the wave vector @xmath45 belongs to the first brillouin zone , a @xmath46-torus defined as @xmath47 $ ] and @xmath48 $ ] . the schrdinger equation then reduces to a generalized harper equation @xmath49 artificial gauge potentials generally induce the following non - trivial unitary transformation for atoms hopping around a plaquette of the lattice : u = u_x u_y ( m+1 ) u_x^ u_y^ ( m ) . in the presence of the gauge potential eq . , atoms performing a loop around a plaquette undergo the unitary transformation : u= e^i 2 ^2 + 2 ^2 + 2 2 & 2 ^ 2 - i ^2 2 + -2 ^ 2 - i ^2 2 & ^2 + 2 ^2 - 22 . if one sets @xmath50 or @xmath51 , where @xmath52 is an integer , the matrix @xmath53 is proportional to the identity and the system behaves similarly to the hofstadter model @xcite . when @xmath54 , where @xmath55 , one finds that @xmath56 and the system is equivalent to the @xmath57-flux model in which half a flux quanta is added in each plaquette @xcite . in these particular cases where @xmath58 , the system is in the abelian regime . for any other values of the parameters @xmath20 and @xmath21 , the matrix @xmath59 is a non - trivial @xmath0 matrix and the system is non - abelian . + a gauge invariant quantity which characterizes the system is given by the wilson loop @xmath60 it is straightforward to verify that the system is non - abelian when @xmath61 , and that @xmath62 . in fig . [ wilson ] , where we show the wilson loop s magnitude as a function of the parameters , @xmath63 , we can easily identify the regions corresponding to the abelian ( @xmath64 ) and to the non - abelian regimes ( @xmath61 ) . we note that the abelian @xmath57-flux regime is reached at a singular point , @xmath54 , where @xmath55 . we also point out that the statement according to which the non - abelian regime is reached when @xmath65 \ne 0 $ ] , and which can be found in previous works @xcite , is incorrect : for the situation where @xmath54 , one finds that @xmath65 = 2 i \ , e^{2 i m \pi \phi } \sigma_z$ ] , while the system is abelian because of its trivial wilson loop , @xmath66 . contrary to the non - abelian systems considered in previous works @xcite , we emphasize that the gauge potential eq . leads to a wilson loop which does not depend on the spatial coordinates . in the following section , we show that this feature leads to energy spectra and fractal structures which significantly differ from the hofstadter moth " @xcite . the energy spectrum can be obtained through direct diagonalization of eq . . in the abelian regime corresponding to @xmath67 or @xmath68 , where @xmath55 , one finds @xmath41 doubly - degenerated bands for @xmath69 . in this particular case , the representation of the spectrum as a function of the flux @xmath24 leads to the fractal hofstadter butterfly " @xcite . for the other abelian case @xmath70 , the system behaves according to the @xmath57-flux lattice : the spectrum @xmath71 depicts a hofstadter butterfly " which is contained between @xmath72 $ ] , i. e. shifted by @xmath73 with respect to the original butterfly " , and the system remarkably describes zero - mass dirac particles @xcite . in the non - abelian regime , which is reached for arbitrary values of the parameters @xmath74 , the spectrum is constituted of @xmath75 separated bands as illustrated in fig . [ bandfig ] . for these general situations , the representation of the spectrum as a function of the flux @xmath24 leads to new interesting features . as in the abelian case , one observes repetitions of similar structures at various scales . however , new patterns arise in the non - abelian case , as illustrated in fig . [ flux ] for @xmath76 and in fig . [ one ] for @xmath77 and @xmath78 . it is worth noticing that for arbitrary values of the parameters @xmath74 , the spectra show well - developed gaps contrasting with the hofstadter moth " which appears in the non - abelian system proposed in ref . we further notice that the spectrum is periodic with period @xmath79 and is symmetric with respect to @xmath80 and @xmath73 . in the non - abelian regime close to @xmath81 , one observes that conical intersections are preserved in the energy spectrum . as shown in the next section , the particles behave similarly to dirac particles in this non - abelian region and the system exhibits an anomalous quantum hall effect . we eventually note that when the flux @xmath82 , the density of states reveals several van hove singularities at the energies @xmath83 and @xmath84 , where @xmath85 . as the flux increases , these singularities evolve and generally merge . we evaluate the linear response of the system described by eq . to an external force ( lattice acceleration ) applied along the @xmath15 direction and we evaluate the transverse conductivity @xmath86 using kubo s formula . following the method of ref . @xcite , one can generalize the well - known tknn expression @xcite to the present non - abelian framework , yielding @xmath87 where @xmath88 is the @xmath89th component of the wave function corresponding to the band @xmath90 such that @xmath91 , and @xmath92 refers to the first brillouin zone of the system . the fermi energy @xmath93 is supposed to lie within a gap of the spectrum . the transverse conductivity is then given by the contribution of all the states filling the bands @xmath94 situated below this gap . . conceals a profound topological interpretation for the transverse conductivity based on the fibre bundle theory @xcite . in the present framework , such bundles are conceived as the product of the parameter space @xmath92 with the non - abelian gauge group @xmath0 . this product space , which is supposed to be locally trivial but is generally expected to twist globally , is characterized by the non - abelian berry s curvature =( _ k_x ^y-_k_y ^x + [ ^x , ^y ] ) dk_x dk_y , where @xmath95 is the berry s connection . the triviality of the fibre bundle is measured by the chern number c(e_)= _ ^2 tr , which is a topological invariant and is necessarily an integer . note that each band @xmath90 is associated to a specific fibre bundle , on which a chern number is defined . one eventually finds that the hall - like conductivity eq . is given by a sum of integer chern numbers , _ x y= - _ e _ < e _ f c(e _ ) . [ final ] as a consequence , the transverse hall - like conductivity of the system evolves by steps corresponding to integer multiples of the inverse of planck s constant and is robust against small perturbations . the evaluation of these topological invariants leads to a complete understanding of the iqhe which takes place in the present context . the aim is then to compute the chern number associated to each band @xmath96 of the spectrum . this computation can be achieved numerically thanks to an efficient method developed by fukui _ _ @xcite and which can be applied to our specific system . this method is summarized as follows : the brillouin zone @xmath92 , defined by @xmath47 $ ] and @xmath97 $ ] , is discretized into a lattice constituted by points denoted @xmath98 . on the lattice one defines a curvature @xmath99 expressed as _ 12 ( _ l)= u_1 ( _ l ) u_2 ( _ l + ) u_1 ( _ l + ) ^-1 u_2 ( _ l)^-1 , where the principal branch of the logarithm with @xmath100 is taken , @xmath101 is a unit vector in the direction @xmath31 , and u _ ( _ l)= _ j u_j ( _ l ) u_j ( _ l+ ) / _ ( _ l ) , defines a link variable with a normalization factor @xmath102 such that @xmath103 . the chern number associated to the band @xmath90 is then defined by c= _ l _ 12 ( _ l ) . this method ensures the integral character of the chern numbers and holds for non - overlapping bands . in the situations where the spectrum reveals band crossings , a more general definition of the link variables @xmath104 has been proposed in ref . @xcite . we first compute the chern numbers for a specific case , illustrated in fig . [ bandfig ] . for @xmath105 and @xmath106 , the chern numbers associated to the six bands are respectively @xmath107 . according to eq . , the transverse conductivity s values associated to the 5 gaps are @xmath108 as shown in fig . [ bandfig ] . the phase diagram describing the iqhe for our model can eventually be drawn . in this diagram we represent the quantized transverse conductivity as a function of the fermi energy @xmath93 and flux @xmath24 . here we illustrate a representative example of such a phase diagram which was obtained for @xmath109 , @xmath110 ( cf . [ one ] ) . this striking figure differs radically from the phase diagrams obtained by osadchy and avron in the abelian case @xcite since the chern numbers associated to the gaps are no longer satisfying a simple diophantine equation @xcite . consequently , the measurement of the transverse conductivity in this system should show a specific sequence of robust plateaus , heralding a new type of quantum hall effect . this new effect is comparable to the iqhe observed in si - mosfet or the anomalous iqhe observed in graphene in the low flux " regime @xmath111 corresponding to experimentally available magnetic fields . in this regime , the quantized conductivity evolves monotonically by steps of one between sudden changes of sign across the aforementioned van hove singularities ( see fig . [ one ] ) . moreover , in the vicinity of @xmath112 , the quantized conductivity increases by double integers because of dirac points in the energy spectrum , in close similarity with the anomalous iqhe observed in graphene . summarizing , we have proposed in this paper how to realize in cold atomic systems a textbook example of non - abelian gauge potential characterized by a _ constant _ wilson loop . our main result is that despite the coupling between the different flavor " components of the single - particle wave functions , the spectrum exhibits well - developed gaps of order of 0.1 - 0.2@xmath113 , i.e. about 50 - 100 nk . the iqhe survives in the deeply non - abelian regime and acquires a unique character specific to the non - abelian nature of the gauge fields . it is characterized by a particular sequence of robust plateaus corresponding to the quantized values of the transverse conductivity . moreover , the non - abelian coupling induces controllable van hove singularities as well as an anomalous hall effect , similar to the effect induced by the hexagonal geometry in graphene . experimental observation of this distinctive effect requires to achieve @xmath114 smaller than the gaps , i.e. of order of 10 - 50 nk , which is demanding but not impossible . the main experimental challenge consists here in combining several established methods into one experiment : laser assisted tunneling @xcite , bec immersion @xcite , and density profile measurements @xcite . we acknowledge support of the eu ip programme scala , esf - mec euroquam project fermix , spanish mec grants ( fis 2005 - 04627 , conslider ingenio 2010 qoit ) , the belgian federal government ( iap project nosy `` ) , and the ' ' communaut franaise de belgique `` ( contract ' ' arc `` no . 04/09 - 312 ) and the f.r.s .- fnrs belgium . m.l acknowledges erc adg ' ' quagatua " . n. g. thanks s. goldman for his comments , pierre de buyl for its support and icfo for its hospitality . a.k . acknowledges support of the polish government research grant for 2006 - 2009 and thanks j. korbicz and c. menotti for discussions . the authors thank d. jaksch , j. v. porto , and c. salomon for their valuable insights in various aspects of this work .

nowadays it is experimentally feasible to create artificial , and in particular , non - abelian gauge potentials for ultracold atoms trapped in optical lattices . motivated by this fact , we investigate the fundamental properties of an ultracold fermi gas in a non - abelian @xmath0 gauge potential characterized by a _ constant _ wilson loop . under this specific condition , the energy spectrum exhibits a robust band structure with large gaps and reveals a new fractal figure . the transverse conductivity is related to topological invariants and is shown to be quantized when the fermi energy lies inside a gap of the spectrum . we demonstrate that the analogue of the integer quantum hall effect for neutral atoms survives the non - abelian coupling and leads to a striking fractal phase diagram . moreover , this coupling induces an anomalous hall effect as observed in graphene .

You are an expert at summarizing long articles. Proceed to summarize the following text: in 2008 the paul scherrer institut ( psi ) celebrated its 20@xmath0 anniversary and many years of delivering high intensity muon beams . several upgrades made the 590 mev/51 mhz ring cyclotron to be up to today the most powerful proton accelerator of its kind in the world , which delivers several 10@xmath1 muons per second to experiments . the accelerator runs now routinely with 2.0 ma proton current and was already pushed to 2.15 ma for tests . in the near future running at 2.3 ma is foreseen , and an extensive program was launched to boost the operating proton current to 2.6 ma , by 2011 , and ultimately to 3.0 ma , envisaged for 2012 @xcite . precision experiments should benefit from a correspondingly increased muon intensity . charged lepton - flavor conservation has been empirically verified to a high precision , but is not a consequence of a known underlying symmetry . the decay @xmath3 is lepton - flavor violating and hence , excluding neutrino flavor mixing , forbidden within the standard model ( sm ) . neutrino masses and mixing , which is established now , introduce a contribution to this decay within the sm , however , on an unmeasurably small level of order @xmath410@xmath5 @xcite . on the other hand , there are several attractive theories beyond the sm , such as supersymmetry , which generally predict lepton - flavor - violating processes at a level within today s experimental reach . a corresponding experimental signal would be free of sm background and hence a clear indication for ` new physics ' . the goal of the meg experiment at psi @xcite is to reach a sensitivity of 10@xmath6 , improving the present limit @xcite by almost 2 orders of magnitude . consequently one needs a detector managing a challenging high muon stop rate up to @xmath7 muons / s . the experimental principle is based on the simultaneous detection of the back - to - back emitted mono - energetic decay positron and gamma . the positrons are detected in high rate drift - chambers located in a magnetic field for momentum determination and in scintillation counters for timing . the gammas are detected in the world s largest liquid xenon scintillation counter , as sketched in fig.[meg - apparatus ] . excellent timing , energy and spatial resolution for both reaction products are required to beat the main background caused by ordinary muon decay and pile - up . 2008 saw the first months of physics run of meg and the accumulated statistics looks promising to already improve the present limit on @xmath3 significantly . the fermi constant @xmath8 describes the strength of the charged - current weak interaction . along with the fine structure constant @xmath9 and the @xmath10-boson mass , it is one of the three pillars of the electroweak standard model and directly related to the electroweak gauge coupling @xcite . the most precise determination of @xmath8 is based on the mean lifetime of the positive muon , @xmath11 , and can be extracted from : @xmath12 with @xmath13 representing higher order qed and hadronic corrections as well as finite - lepton - mass phase space factors , which have only recently been computed to a sub - ppm level @xcite . a first computation of order @xmath14 using a finite electron mass shifted the value of @xmath13 by another 0.43ppm @xcite . hence , a comparably precise experimental determination of @xmath11 is highly desirable . the mulan experiment @xcite installed a muon beam kicker on the pie3 beamline at psi , which allows after directing positive muons onto a target for a selectable time period ( e.g. 5 @xmath15s ) , to steer away the beam for the following , for instance , 22 @xmath15s , the decay positrons are recorded in a soccer - ball shaped detector ( see fig.[mu - lifetime - plot]b ) made from 170 double - layer scintillator tiles , which are read out via custom - made 500 mhz fadc modules able to separate pulse pile - up events on the ns level . systematic issues , caused by positron detection differences in the counters , due to polarized muons precessing in the earth s magnetic field , are dealt with via measurements in different targets , which are in a homogeneous magnetic field and either fully maintain the muon polarization ( silver ) , depolarize the muons to a large extent ( sulphur ) , or cause a very fast muon precession due to an internal few tesla high magnetic field ( arnokrome@xmath16-iii ) . several 10@xmath17 muon decays were recorded for each target . the first mulan result , based on part of the data has set a new precision benchmark , as shown in fig.[mu - lifetime - plot]a . additionally , several dedicated systematic measurements are presently under analysis . the final precision goal on @xmath11 is 1ppm , which translates into a 0.5ppm precision on @xmath8 . the fast experiment @xcite relies on the detection of the full decay sequence @xmath18 and corresponding times in a fast imaging target made of 32 x 48 pixels , constructed from plastic scintillator bars in a homogeneous b field . this approach allows a good control of muon polarization effects . fast is scheduled to achieve a statistics of several @xmath19 in 2008/2009 . its goal is a 2ppm measurement of @xmath11 . as a by - product , fast can also measure the @xmath20 lifetime and improve the present world average . the determination of the proton s weak pseudoscalar coupling constant @xmath21 has been the driving force behind decades of muon capture measurements . the psi result on the muon capture rate in @xmath22he @xcite has set a precision landmark in this field . however , with 3 involved nucleons some questions still remained in the precise theoretical prediction . a specially exciting turn came with the precise triumf results from a measurement of radiative muon capture ( rmc ) in hydrogen @xcite , which disagreed with theory and results derived from ordinary muon capture ( omc ) measurements @xcite , as shown in fig.[gp - plot]a . after decades of worldwide experimental efforts , the mucap experiment has achieved the first unambiguous determination of the proton s pseudoscalar coupling @xmath21 @xcite and has solved a longstanding discrepancy . the result is in excellent agreement with recent calculations based on heavy baryon chiral perturbation theory ( hbchpt ) @xcite . experimental determinations of @xmath21 depend on the ortho - para transition rate @xmath23 in the @xmath24 molecule . the most precise previous measurement of ordinary muon capture ( omc ) @xcite and the rmc experiment @xcite both depend significantly on the value of @xmath23 , which itself is poorly known due to mutually inconsistent experimental @xcite and theoretical @xcite results . in contrast , the first mucap result for @xmath21@xcite is almost independent of molecular effects . sc/@xmath15pc - muon entrance counters , tpc - time projection chamber = target , epc , esc - electron tracking and timing counters.__,title="fig : " ] the mucap result was only possible with an enduring joint effort and a rigorous experimental technique @xcite . the setup is shown in fig.[gp - plot]b . the active target , a time projection chamber , was filled with 10 bar of ultra - pure ( high @xmath10 contamination in the few ppb range ) and isotopically pure hydrogen @xcite . muon stops and corresponding decay electron tracks were recorded in 3 dimensions , which allowed for very selective cuts and hence an unprecedented control and possible study of systematic effects . specifically , muon capture events on high @xmath10 elements are even on the ppb contamination level visible in the mucap detector . target conditions were selected in order to control effects due to muonic molecule formation and muon catalyzed fusion . the muon capture rate was finally determined by comparing the lifetime of negative muons in hydrogen with the positive muon lifetime from the more precise result @xcite . in order to extract @xmath21 , the singlet muon capture rate was compared to two recent calculations @xcite adding the newly calculated radiative correction @xcite . presently , the full data set of more than 10@xmath25 recorded @xmath26 stops in hydrogen is being analyzed in a blind analysis , with the final precision goal of 1@xmath27 on the singlet muon capture rate in hydrogen . 20 s@xmath28 to the total rate . b ) estimations on the axial two - body current term @xmath29 . all methods up to now include certain assumptions and approximations which might be questioned . all references are given in @xcite _ , title="fig : " ] knowing @xmath21 from mucap facilitates the interpretation of the doublet muon capture rate on the deuteron ( @xmath30 ) , the measurement goal of the recently started musun experiment @xcite which also aims at 1@xmath27 precision . such a result would allow a precise test of modern effective field theories and would represent the most stringent test of electro - weak interaction in a two - body system @xcite . moreover , it would allow determination of the axial two - body current term which scales with the low energy constant denoted @xmath29 ( or d@xmath31 ) @xcite . this parameter is of astrophysical interest , as it appears in the same form in the cross - section for i ) muon capture on the deuteron , ii ) @xmath32 fusion the main fusion reaction in the sun , and iii ) neutrino - deuteron reactions , which are the detection reactions used by the sudbury neutrino observatory @xcite . hence , via the absolute neutrino rates , the precise @xmath15d capture rate determination would ` calibrate ' the sun . existing experimental results on @xmath30 are not precise enough ( fig.[l1a - plot]a ) and also using other sources leaves the present experimental knowledge on @xmath29 rather sparse , as shown in fig.[l1a - plot]b . with precision neutrino physics on the horizon the precise knowledge of @xmath29 will be necessary , as it also influences the determination of @xmath33 and @xmath34@xcite . the experimental principle of musun will follow the successful mucap approach , but for control and optimization of muonic molecule formation and muon catalyzed fusion reactions in deuterium one has to use a high density cryo - target at @xmath430k @xcite . @xmath35 fusion reactions occur at much higher rates than muon capture and hence represent a severe background , but also allow to monitor the hyperfine populations of the muonic atom @xcite . high @xmath10 target purity will be even more critical as in mucap . in a first engineering run a new pad - based tpc was successfully tested with high purity deuterium in late 2008 . there are several other precision measurements of fundamental parameters ongoing or under discussion at psi , which either will test the standard model or search for new physics . * the muonic lamb - shift experiment @xcite is preparing for its first physics data taking in 2009 , and wants to precisely determine the proton charge radius via observation of the 2p-2s energy difference in muonic hydrogen . * a precise test of the electron - muon universality is being performed within the pen experiment and corresponding data are being analyzed @xcite . * the search for the lepton - flavor - violating process @xmath36 would be a sensitive search for new physics and complement the present meg activity . there is an ongoing discussion on 2 suggested experimental approaches , how to obtain a sensitivity which improves the present experimental limit by roughly 3 orders of magnitude @xcite . * a sensitive search for a cp violating muon electric dipole moment ( edm ) was suggested in @xcite using a compact storage ring , which could make use of psi s high muon intensity and reach a sensitivity of 5@xmath3710@xmath38 e@xmath39 cm . in this way it would test ` new physics ' and pave the way for higher sensitivity tests of muon and other charged particle edms . * the discussion about dark matter and dark energy has also put interest in particles decaying into mirror worlds , other dimensions or to other branes . hence the decay products would be invisible . a search for the invisible decay of muons was suggested in @xcite , and might be also searched for by using the mucap setup . * high brightness muon beams would also allow a first test of the gravitational interaction of antimatter of a purely leptonic system , which involves second generation particles , namely muonium ( @xmath40 ) @xcite . given these and more ideas , one can be sure that precision measurements using muons , at psi and other facilities in the world , will also in future contribute to a deeper understanding and testing of the standard model and provide a fair chance to first find ` new physics ' beyond our presently accepted theory . i would like to express my sincere gratitude for advice , support and many stimulating discussions to k. kirch , p. kammel , d. hertzog , s. ritt , c. casella , c. petitjean , m. seidel , and especially all members of the mucap , mulan and musun collaborations . p. kammel in : proceedings of the exa02 - international workshop on exotic atoms - future perspectives , vienna , austria , 2002 ; austrian academy of sciences press , vienna , 2003 ; arxiv : nucl - ex/0304019 ; b. lauss in : proceedings of the exa05 - intern . conference on exotic atoms , vienna , austria , 2005 , austrian academy of sciences press , vienna 2005 ; arxiv : nucl - ex/0401005 .

muons can serve as probes to precisely determine fundamental parameters of the standard model or search for ` new physics ' . the high intensity muon beams at the paul scherrer institut ( psi ) allow for precision measurements and searches for rare or forbidden processes . both types of experiments challenge the standard model in a way complementary to high energy physics . we give a short overview of recent results and ongoing experiments at psi , and of ideas for the future . muon physics , rare decay , muon lifetime , fermi constant , muon capture 14.60ef , 13.35bv , 12.15.-y , 21.45.bc , 25.30.mr , 13.15.+g

You are an expert at summarizing long articles. Proceed to summarize the following text: as theoretically predicted in @xcite and experimentally discovered in @xcite , two - band solid state systems in 3d can host topologically protected `` weyl semimetallic phases '' in which the quasiparticle excitations at the band crossings ( `` weyl points '' ) share some features with weyl fermions from relativistic quantum mechanics . the experimental signature , namely `` fermi arcs '' of surface states which connect the projected weyl points in the surface brillouin zone , is just as remarkable . based on this initial success , much effort has been put into the general study of topological semimetallic phases in the hope of predicting and eventually realising new exotic fermionic quasiparticles in condensed matter systems . most proposals have focused on _ local _ aspects in the sense of finding new types of topological obstructions to locally opening up gaps in semimetal band crossings . staying in the two - band case , there are generalisations of the basic weyl semimetal phase to `` type - ii '' ones @xcite , as well as `` quadratic '' ones @xcite . one can consider model hamiltonians with more than two bands and in a different number of spatial dimensions @xcite . following the example set by topological insulator theory , one may also introduce antiunitary symmetry constraints such as time - reversal and charge - conjugation @xcite . to circumvent certain difficulties in achieving non - trivial obstructions , point symmetries were also introduced into the game , and a host of possibilities arise @xcite . our results in this paper have a different focus , concentrating on ( 1 ) isolating in a conceptually simple but rigorous way the general mathematical mechanism allowing for local semimetallic charges , ( 2 ) providing the full _ global _ topological characterisation of semimetal band structures , fermi arcs , and the relation to topological insulator invariants , and ( 3 ) introducing a new family of semimetals whose topological invariants have a very different character to those commonly used in the literature . figure [ fig : flowchart ] summarises the physical and mathematical concepts which we introduce in this paper . ) which are dual to each other.,scaledwidth=100.0% ] with regards to ( 1 ) , we see , with much hindsight , that the local topological charges protecting basic types of semimetal crossings are completely inherited from the singular vector field which specifies the hamiltonian @xcite . the poincar hopf theorem @xcite then shows how the brillouin zone topology forces a global charge cancellation condition , and the prediction of fermi arcs becomes a corollary of the bulk - boundary correspondence . this brings us to point ( 2 ) . while the 3d weyl semimetal may be intuitively understood using a stokes theorem argument to predict `` jumps '' in 2d chern numbers across the weyl points , there are several subtleties involved , not the least of which is the fact that the local charges ( equal to the jumps ) fail to completely characterise the semimetal , and can not predict , a priori , the fermi arc topology @xcite . it is here that the finer topology of the brillouin torus comes into play : the fact that @xmath0 has non - trivial cycles means that `` the ( fermi ) arc joining two weyl points '' is an ambiguous notion , even up to deformations of the arc ( contrast with the simply - connected spheres ) . the resolution of this ambiguity requires additional global data which captures the full topology of semimetal band structures , and is closely related to the concept of euler structures introduced by turaev to resolve ambiguities in torsions of reidemeister type @xcite ; both hinge on the fact that the first homology of the underlying manifold may be non - zero . by considering a homological version of _ euler structures _ @xcite , we find that the language of _ euler chains _ very concisely represents all the _ topological _ features of a semimetal . the euler chain representation directly predicts the _ topology _ of the fermi arcs , and the latter are the experimental signature of a topological semimetal . furthermore , the notion of topological equivalence between semimetals becomes clear in this language . a semimetal hamiltonian determines a fermi arc , e.g. through a transfer matrix formalism @xcite , and this determination descends to the level of topological invariants in the sense that equivalent hamiltonians give rise to equivalent fermi arcs . having isolated the basic mathematical principles underlying the geometry and topology of semimetals , we move on to ( 3 ) : the generalisation of our theory for 2-frames or 2 vector fields along the lines of the atiyah dupont theorem @xcite which is the analogue of the poincar hopf theorem in this context , where the _ kervaire semicharacteristic _ plays the role of the euler characteristic . it is a topological invariant of a very different nature to those that had been previously utilised in the literature on topological phases ( e.g. chern classes and related characteristic classes , @xmath1-theory , winding numbers / degrees ) . this leads us to define some new mathematical notions _ kervaire structures _ and _ kervaire chains _ which are two mutually dual objects characterising a new class of @xmath2-topological semimetals introduced for the first time in this paper , experimentally predicting torsion fermi arcs . in section [ sec : physback ] , we give a quick overview of the use of cohomological characteristic classes to study topological insulating phases , and then connect it to semimetal phases via the mayer vietoris technique of `` patching together '' the non - singular portion ( avoiding the weyl points ) and topologically trivial portions ( small neighbourhoods of the weyl points ) . the basic `` semimetal mayer vietoris sequence '' is given a geometric interpretation as an extension problem , parallel to the physical problem of deforming a semimetal band structure into a globally insulating one , in section [ sec : extension ] . in section [ sec : euler ] , we introduce the concept of euler structures and the euler chain representation of a semimetal . we illustrate with diagrams the basic intuitive ideas , and apply the euler chain concept to fermi arc topology and their rewirings . in section [ sec : higherdimension ] , we study dirac - type hamiltonians parametrised by singular vector fields , emphasising the merits of an abstract coordinate - free analysis with regards to symmetry considerations . in particular , we justify the generality of dirac - type hamiltonians through natural symmetry constraints , in order to avoid introducing `` spurious topology '' through ad - hoc parametrisations of toy model hamiltonians . in section [ sec : semimetalgerbe ] , a geometric connection to gerbes is made in the 4d case , and in section [ sec:5dsemimetal ] , the connection to quaternionic valence bundles in 5d is made , and the geometric interpretation as an extension problem is in section [ sec : extension ] . in section [ sec : torsionsemimetal ] , we introduce a new class of semimetals modelled on hamiltonians which are bilinear in gamma matrices . the topology of such semimetals is characterised by the notion of kervaire structures and chains , and we predict an interesting phenomenon of @xmath2 surface fermi arcs . finally we suggest some directions for future work in section [ sec : outlook ] . * conventions . * in this paper , @xmath3 will always be a compact connected smooth @xmath4-dimensional manifold without boundary , playing the role of a general parameter manifold for a family of hamiltonians . in specific instances , it may have an orientation , metric , spin@xmath5 structure etc . , and may be the brillouin zone for a family of bloch hamiltonians , as additionally specified . all ( co)-homology groups have integer coefficients unless otherwise indicated . the real clifford algebra @xmath6 has @xmath7 generators squaring to @xmath8 and @xmath9 generators squaring to @xmath10 , corresponding to an orthonormal basis @xmath11 in @xmath12 for the bilinear form @xmath13 ; @xmath14 refers to @xmath15 , and the complex clifford algebra @xmath16 has @xmath17 anticommuting generators squaring to @xmath10 . 2d chern insulators are classified by a first chern number , equal to the integrated berry curvature 2-form of the valence bands over the brillouin zone @xmath18 . when the fermi energy @xmath19 lies in a spectral gap , the fermi projection onto the valence bundle @xmath20 may be defined . over @xmath21 the @xmath20 are classified by their rank and first chern class , with the latter generally considered to be the interesting invariant . since the first chern class of @xmath20 is that of the determinant line bundle , the classification problem of 2d chern insulators is formally equivalent to that of u(1 ) line bundles over the brillouin zone for which the _ integral cohomology group _ @xmath22 provides a complete answer . in terms of differential forms , @xmath23 measures the failure of the ( berry ) curvature 2-form @xmath24 of @xmath20 to be globally exact ( the berry connection 1-form @xmath25 only exists locally ) . the integral of @xmath24 over @xmath21 yields an integer - valued first chern number , which may be interpreted as a topological obstruction to globally defining a basis of valence bloch eigenstates for @xmath20 . a typical way to construct a valence line bundle with first chern class @xmath26 is to consider @xmath27 traceless . ] bloch hamiltonians @xmath28 , parametrised by a nowhere - vanishing smooth vector field @xmath29 on @xmath21 . here @xmath30 are the pauli matrices . the spectrum is @xmath31 , and the spectrally - flattened hamiltonian @xmath32 is parametrised by the unit vector map @xmath33 . note that @xmath34 and @xmath32 determine the same valence line subbundle @xmath20 of the bloch bundle @xmath35 . the map @xmath36 has an integer degree given by the formula @xmath37)=\text{deg}(\hat{{\bm{{h}}}})[s^2]$ ] , where @xmath38 , [ s^2]$ ] denote the fundamental classes . the degree is a homotopy invariant , and all degrees do occur . by identifying @xmath39 with @xmath40 ( the bloch sphere construction ) , we see that @xmath20 is given by the pullback of the tautological ( hopf ) line bundle over @xmath40 under @xmath36 . due to the low - dimensionality of @xmath21 , all maps into @xmath41 are approximated by maps into @xmath40 , and so the first chern class of @xmath20 is precisely the degree of @xmath36 . in higher dimensions , and also in the presence of additional symmetries ( e.g. time - reversal or point group symmetries ) constraining the form of the hamiltonians , topological band insulators are classified by more complicated invariants . in the mathematical physics literature , these have included higher degree cohomology invariants playing the role of characteristic classes for the valence bundles , e.g. higher chern / instanton numbers , symplectic bundle invariants @xcite , fu kane mele / fkmm - type invariants @xcite , or generalised cohomology invariants ( @xmath1-theory ) @xcite . we are primarily interested in @xmath42 or @xmath43 spatial dimensions , and in those dimensions , the relevant `` berry curvature form '' @xmath24 can be a higher - degree form @xcite , and so a higher degree cohomology group @xmath44 comes into play . cohomology groups with other coefficients also arise quite generally in obstruction theory , which can be used to study semimetal - insulator transitions . all these ( generalised ) cohomological invariants are also homotopy invariants , and are closely related to degree theory ( as we saw from the 2d chern insulator described above ) . although the prototypical topological semimetal is the weyl semimetal in @xmath45 , the general theory presented in this paper works in higher dimensions . this is not merely of theoretical interest . in the case of topological insulators , there are concrete proposals and experiments in which topological phases of physical systems in three or fewer dimensions formally realise those in @xmath46 . some examples are quasicrystalline systems @xcite , time - periodic ( floquet ) insulators @xcite and their photonic counterparts @xcite , and general `` virtual '' topological insulators @xcite . it is in this spirit that we embark on the mathematical study of semimetallic phases for general @xmath4 . in a semimetal , the fermi level does not lie in a spectral gap , but instead passes through band crossings at some weyl submanifold @xmath47 of the momentum space manifold @xmath3 . such crossings may arise as `` accidental degeneracies '' @xcite and , for `` dirac - type hamiltonians '' in @xmath4-dimensions occur generically at points . a quick way to see this is to consider dirac - type bloch hamiltonians of the form @xmath48 here each @xmath49 is a smooth assignment of @xmath4-component vectors , and @xmath50 is the vector of traceless hermitian @xmath51-matrices representing irreducibly and self - adjointly the @xmath4-generators of a clifford algebra ( for @xmath45 , these are the usual @xmath27 pauli matrices ) . by the clifford algebra relations @xmath52 , the square of @xmath53 is the scalar @xmath54 , so the eigenvalues of @xmath53 are @xmath55 , each with degeneracy equal to half the dimension of the @xmath51-matrices . this means that a band crossing at @xmath56 requires @xmath4 conditions @xmath57 , and we see that the locus @xmath47 of band crossings generically has codimension @xmath4 . when @xmath3 is a @xmath4-manifold , @xmath47 is generically a collection of isolated `` weyl points '' , the ( rather misleading ) terminology arising from the @xmath45 case in which the low - energy excitations around a band crossing are essentially described by the weyl equation @xcite . furthermore , each weyl point @xmath58 serves as a generalised _ monopole _ , and may be assigned a _ local topological charge _ via the unit - vector map restricted to a small enclosing @xmath59-sphere @xmath60 , @xmath61 and defining the local charge @xmath62 to be the degree of @xmath63 . the local charge information of all the crossings for the hamiltonian @xmath64 is concisely summarised by the 0-chain @xmath65 . the unit vector map @xmath36 on @xmath66 , and its restrictions to @xmath60 , can be used to pullback the hopf line bundle with connection when @xmath45 , the basic gerbe with connection when @xmath67 and the quaternionic hopf line bundle with connection when @xmath68 ( see appendix [ appendix ] for their definitions ) . as explained in section [ sec : extension ] , these geometrical structures arise naturally when analysing dirac - type hamiltonians , their band crossings , and the topology of their valence bundles . interesting phenomena arise when we study the band crossings _ globally_. for instance , if @xmath29 can be identified as a tangent vector field over @xmath3 , as is often the case in physical models , then the classical poincar hopf theorem guarantees that @xmath69 where @xmath70 is the euler characteristic of @xmath3 , which is zero for @xmath71 or @xmath4 odd . this global `` charge - cancellation '' condition is well - known in lattice gauge theory ( where @xmath71 ) as the nielsen ninomiya theorem @xcite and discussed in the semimetal context in @xcite . together with the bulk - boundary correspondence , it predicts the appearance of surface fermi arcs connecting the projected weyl points on a surface brillouin zone . these arcs are an experimental signature of topological non - triviality in a semimetal band structure , and have been discovered recently @xcite . more generally , @xmath29 could be a section of some oriented rank-@xmath4 vector bundle @xmath72 over a compact oriented @xmath4-manifold @xmath3 . the local topological charges at the zeros of @xmath29 are defined as before , as is the euler characteristic of @xmath72 ( evaluating the euler class on the fundamental class ) . when the euler characteristic @xmath73 vanishes ( and when @xmath74 following @xcite ) , a notion of _ euler structures _ can be defined , and they provides links between topological semimetal invariants , seiberg witten invariants and reidemeister torsions . more pertinently , euler structures are a torsor over @xmath75 , and they provide an elegant way to understand some global aspects of fermi arc topology , including certain ambiguities first studied in @xcite , as well as the `` rewiring '' of fermi arcs @xcite . this is already interesting when @xmath71 , for which there are non - trivial euler structures . we take the following as a working definition : [ defn : abstracthamiltonian ] let @xmath3 be a compact , oriented @xmath4-dimensional riemannian spin@xmath76 manifold with @xmath77 . abstract dirac - type hamiltonian _ is a smooth vector field @xmath29 over @xmath3 , which is _ insulating _ if @xmath29 is non - singular ( nowhere zero ) , and _ semimetallic _ if @xmath29 has a finite set @xmath47 of singularities . this definition is motivated by the construction of concrete bloch hamiltonians from such vector fields , generalising , given in section [ sec : higherdimension ] . in @xcite , we carried out an analysis of the global structure of semimetal band structures and fermi arcs . the mayer vietoris principle provided the key to understanding the connection between topological semimetals and insulators ( fig . [ fig : metal - or - insulator ] ) . in particular , we showed that the local charge information at weyl points needs to be supplemented with some global data to fully characterise semimetal band structure and fermi arcs . we summarise the key constructions in @xcite , then provide several different ways to understand them . * definition of @xmath78 . * let @xmath3 be a compact oriented @xmath4-manifold ( usually the brillouin torus @xmath0 in concrete models ) with @xmath77 , and consider a smooth family of ( concrete ) @xmath79 bloch hamiltonians @xmath80 . the spectrum comprises @xmath17 bands , and we assume that band crossings at the fermi level ( normalized to @xmath81 ) occur on a finite set @xmath82 of isolated `` weyl points '' . on the complement @xmath66 , the bloch hamiltonians are gapped , and there is no difficulty in defining the valence bundle @xmath20 over @xmath66 and its characteristic classes in @xmath83 . for each @xmath58 , choose ( mutually disjoint ) open balls @xmath84 containing @xmath85 , and @xmath86-spheres @xmath87 surrounding @xmath85 . let @xmath88 and @xmath89 be the respective disjoint unions . thus @xmath3 is covered by @xmath90 and @xmath66 , and the intersection of the covering sets is @xmath91 which is homotopy equivalent to @xmath92 ( figure [ fig : mv - subspaces ] illustrates the setting for two weyl points ) . used in the semimetal mv - sequence , for two weyl points @xmath93.,scaledwidth=80.0% ] the mv sequence links the cohomology groups of @xmath3 with those of the covering subspaces and the intersection @xmath92 , by an exact sequence @xmath94 the maps which increase cohomological degree are the mv connecting maps , while the others are ( differences of ) restriction maps . note that @xmath90 is contractible , so the @xmath95 term vanishes for @xmath96 . when a semimetallic bloch hamiltonian is specified abstractly by a vector field @xmath29 as in definition [ defn : abstracthamiltonian ] , the zero set of @xmath29 will correspond to the weyl points @xmath47 , and we have the subsets @xmath97 exactly as above . then @xmath29 defines a topological invariant for the semimetal which lives in @xmath98 ( section [ sec : euler ] ) . the most interesting part of the mv sequence is at @xmath99 , which we call the cohomological _ semimetal mv sequence _ , @xmath100 here @xmath101 is the direct sum of the local charge groups for each @xmath85 , and the last mv connecting map @xmath102 is the `` total charge operator '' which adds up the local charges @xcite . for integer coefficients , @xmath103 and the charges are @xmath104-valued , but we will also consider @xmath2 coefficients in section [ sec : torsionsemimetal ] . let us be more explicit in the basic 3d situation , with @xmath27 bloch hamiltonians parametrised by @xmath105 and @xmath106 . the mv - sequence reads @xmath107 consider local charges @xmath108 which cancel , then there exists ( by exactness ) a line bundle @xmath20 over @xmath109 whose restriction to @xmath110 has chern class given by the local charge at @xmath111 . away from @xmath112 ( with charges @xmath113 ) , @xmath114 has a first chern number @xmath115 on the 2d subtorus in the @xmath116-@xmath117 direction ( blue ) . consider the region @xmath118 between the two blue 2-tori , with a small ball around @xmath119 removed . this is a 3-manifold whose boundary @xmath120 consists of the two blue 2-tori ( oppositely oriented ) together with a small 2-sphere around @xmath121 . note that the top and bottom faces are identified , as are the front and back faces so they do not contribute to @xmath120 . everywhere on @xmath118 , the valence line bundle is well - defined , and its berry curvature 2-form @xmath24 is closed . by stokes theorem , @xmath122 , so the difference between the first chern numbers for the left 2-torus and the right 2-torus is precisely the first chern number on the small 2-sphere , which is @xmath123 . this argument is easily generalised to higher dimensions and higher curvature forms ( remark [ remark : higherjumps]).,scaledwidth=60.0% ] * intuitive idea behind relation local charges and first chern numbers of a semimetal . * suppose the vector field @xmath29 on @xmath124 vanishes at @xmath93 with local indices @xmath125 respectively . for simplicity , we first assume that @xmath111 have the same @xmath126-coordinates , as in figure [ fig : chern - jump ] . take a trivial bloch bundle @xmath127 , the pauli matrices @xmath128 with respect to a trivialisation , and let @xmath129 . then on @xmath109 , the valence line bundle @xmath20 is well - defined . at each @xmath130 away from the weyl points , @xmath20 has first chern numbers @xmath131 obtained by pairing @xmath132 with the fundamental class of the 2-torus at @xmath130 ( integrating the berry curvature over the 2-torus ) . the integer @xmath133 is sometimes called a `` weak invariant '' for `` weak '' topological insulators supported on lower - dimensional subtori of the brillouin torus . as a function of @xmath130 , @xmath133 remains constant unless a weyl point is traversed , whence it jumps by an amount equal to the local charge there . a possible function @xmath133 is given at the bottom of figure [ fig : chern - jump ] , and different choices of @xmath29 can give rise to different @xmath134 . note that even with @xmath134 and the local charges @xmath135 still do not completely specify the class of @xmath20 there are two other independent first chern numbers @xmath136 which are constant functions of @xmath137 and @xmath138 respectively ( since both weyl points are traversed simultaneously as either @xmath137 or @xmath138 is varied ) . in general , the @xmath111 need not have the same @xmath126-coordinates , and so there are similar jumps in the functions @xmath139 and @xmath140 , see figures [ fig : intersection]-[fig : intersection2 ] . there is a simple way to concisely and invariantly ( i.e. coordinate - free ) capture all the jumps in the various first chern numbers , through a poincar dual picture in terms of euler chains ( see section [ sec : semimetaleulerchain ] and remark [ remark : higherjumps ] ) . * all first chern numbers can appear in a weyl semimetal . * let us also consider @xmath141 which arise as a subbundle of a @xmath142 which is not necessarily trivial . such @xmath141 are easily obtained by tensoring @xmath143 with a line bundle @xmath144 with @xmath145 . the `` twisted '' hamiltonian @xmath146 acts on the `` twisted '' bloch bundle @xmath147 , and has fermi line bundle @xmath148 ( defined over @xmath109 ) . therefore @xmath149 and , in particular , the first chern numbers @xmath150 increases by @xmath151 and similarly for @xmath152 . in this way , all elements of @xmath153 can be realised as valence line bundles of some semimetallic bloch hamiltonian . in @xmath45 , the exact sequence has a direct interpretation as a solution to a geometrical extension problem , which we can interpret in terms of insulator - semimetal transitions . let us recall that @xmath154 classifies line bundles over @xmath3 ( e.g. determinant valence bundles ) . so for @xmath155 , the exactness of tells us the following : 1 . a collection of line bundles @xmath156 with chern classes @xmath157 may be extended to a line bundle over @xmath66 ( comes from a semimetal ) iff @xmath158 ( local charges cancel ) . a line bundle @xmath159 extends to all of @xmath3 iff each local charge vanishes . thus a semimetal ( whose valence bundle represents a class in @xmath160 ) can be gapped into an insulator iff all the local charges are zero . a _ topological semimetal _ with a topologically protected crossing at @xmath85 has non - zero local charge there . the insulator invariants in @xmath161 ( `` weak '' chern insulators ) appear faithfully in @xmath160 . thus two semimetals with the same local charge ( same image under @xmath162 ) can be _ globally inequivalent _ , differing by some element in ( the image under @xmath163 of ) @xmath161 . an important consequence of point 3 is that the local charge information is not sufficient to determine the global band structure of the semimetal . as explained in @xcite , this ambiguity has crucial consequences for the determination of the resulting surface fermi arcs . we will have more to say about this in section [ sec : bbc ] , see also figure [ fig : bulk2boundary ] . more generally , in @xmath77 , a dirac - type bloch hamiltonian in the sense of definition [ defn : abstracthamiltonian ] determines by restriction smooth maps @xmath164 , and the semimetal mv sequence can be interpreted at this level . in appendix [ appendix ] we describe the relevant geometric structures on @xmath165 , @xmath166 , and these can be pulled back to @xmath92 via all the @xmath63 . 1 . in @xmath45 , let @xmath144 be the hopf line bundle over @xmath39 . consider a collection of pullback line bundles @xmath167 over @xmath60 , such that @xmath168 , where @xmath169 denotes the degree of the map @xmath63 . although they individually may not extend to @xmath170 , together @xmath171 over @xmath92 does extend to @xmath66 by exactness of the mv sequence . see the appendix for more information . 2 . in @xmath67 , the @xmath172 part of the semimetal mayer vietoris sequence @xmath173 has the following analogous geometric interpretation . a collection of pullback basic gerbes @xmath174 with dixmier - douady numbers @xmath175 may be extended to a gerbe over @xmath66 ( comes from a gerbe semimetal ) iff @xmath176 ( local dd charges cancel ) . the pullback basic gerbe @xmath177 extends to all of @xmath3 iff each local dd charge vanishes . thus a gerbe semimetal ( with topological invariant in @xmath178 ) can be gapped into an insulator iff all the local dd charges are zero . a _ topological gerbe semimetal _ with a topologically protected crossing at @xmath85 has non - zero local dd charge there . the insulator invariants in @xmath179 ( `` weak '' gerbe insulators ) appear faithfully in @xmath178 . thus two gerbe semimetals with the same local dd charge ( same image under @xmath162 ) can be _ globally inequivalent _ , differing by some element in ( the image under @xmath163 of ) @xmath179 . conversely , the pullback gerbes @xmath180 over @xmath60 , where @xmath181 is the basic gerbe over @xmath182 , such that @xmath183 . although the @xmath180 may not individually extend to @xmath170 , the gerbe @xmath184 does extend to @xmath66 by exactness of the mv sequence . see section [ sec : semimetalgerbe ] and the appendix for more details . 3 . in @xmath68 , the analysis is analogous to the case of line bundles done above . one replaces the line bundle @xmath144 over @xmath40 by the quaternionic line bundle @xmath185 over @xmath186 and one uses the mv sequence in , where the first chern class is replaced by the 2nd chern class . see section [ sec:5dsemimetal ] and the appendix for more details . although there is a mv sequence analysis in the case of kervaire semimetals ( section [ sec : kervairestructures ] ) , we are unable to formulate a geometric extension problem in this context ( see section [ sec : outlook ] ) . in section [ sec:2dberry ] , we briefly alluded to the relation between chern numbers and berry connections associated to valence subbundles for a family of hamiltonians over @xmath3 . in general , a rank-@xmath17 eigen - subbundle has a @xmath187 adiabatic ( non - abelian berry ) connection , whose curvature gives the real ( de rham ) chern classes of the subbundle @xcite . for @xmath188 , the curvature 2-form is involved in computing first chern numbers . in the presence of time - reversal symmetry , the subbundle is complex even - dimensional ( @xmath189 ) and in fact a quaternionic symplectic bundle with quaternionic dimension @xmath134 ; the berry connection becomes a @xmath190 one , see @xcite and section [ sec : quaternionbundle ] . for @xmath191 , the 4-form obtained from squaring the curvature enters in the computation of second chern numbers . in a similar vein , the additional imposition of a chiral symmetry allows a `` semimetal gerbe '' to be associated to the family of hamiltonians , see section [ sec : semimetalgerbe ] . there is a `` berry connection 2-form '' with 3-form curvature , which computes the dixmier douady invariants of the gerbe . fermi arcs are more naturally analyzed in homology , and for this reason , we poincar - dualise @xmath192 with the physical meanings of the first two groups explained in section [ sec : euler ] . roughly speaking , the poincar dual of a semimetal invariant in @xmath160 is an euler chain in the _ relative _ homology group @xmath193 , whose boundary is the 0-chain of local charge data . equation can also be exhibited directly as a mayer vietoris sequence for _ borel moore _ homology @xmath194 @xcite . the groups @xmath195 may be defined for locally compact spaces @xmath118 using _ locally finite _ chains . for compact spaces , @xmath194 and the usual ( singular ) homology @xmath196 coincide . however , @xmath197 whereas @xmath198 , for @xmath84 an open ball containing @xmath85 . there are isomorphisms @xmath199 , @xmath200 , poincar duality @xmath201 , and restriction maps to @xmath194 of open subsets , see chapter 2.6 of @xcite . the poincar dual of written in terms of @xmath194 is @xmath202 which is precisely the mv - sequence for @xmath194 ( ix.2.3 of @xcite ) , with respect to the open cover @xmath203 of @xmath3 ; c.f . the `` localization '' long exact sequence , ix.2.1 of @xcite and 2.6.10 of @xcite . the sequences and are the same on account of @xmath204 . the language of borel moore homology has certain advantages , but for this paper we stay in the more elementary setting of relative homology so as to avoid having to introduce too much technical machinery . in @xcite , we explained how fermi arcs represent _ relative homology _ classes in @xmath205 , where @xmath206 projects out one torus direction , so @xmath207 is the surface brillouin zone . the euler chain representing the topological data of a semimetal is pushed forward to a surface fermi arc , and this is the poincar dual picture of the bulk - boundary correspondence , see section [ sec : bbc ] . since we parametrise dirac - type bloch hamiltonians by vector fields @xmath29 on @xmath3 , we would also like to analyse the semimetal mv - sequence in terms @xmath29 and its homotopies . in the insulating case , we might want to maintain the gap condition and so consider homotopies through non - singular ( i.e. nowhere vanishing ) vector fields . in the semimetal case , we want to at least maintain the gap condition everywhere except perhaps at a finite set of isolated weyl points . we may require the set of weyl points and their charges to be kept fixed , or we may allow them to move around and be created / annihilated in pairs . * smooth euler structures . * let @xmath3 be a compact oriented @xmath4-manifold with @xmath208 , and suppose @xmath3 has euler characteristic @xmath209 ( this is automatic in @xmath45 ) so that it has a global non - vanishing vector field . [ defn : smootheuler ] two non - singular smooth vector fields @xmath210 on @xmath3 are said to be _ homotopic _ ( also called _ _ homologous _ _ through non - singular vector fields can also be analysed @xcite , but we do not consider this in this paper . ] in @xcite ) if for some open ball @xmath211 , the fields @xmath210 are homotopic on @xmath212 in the class of non - singular vector fields on @xmath212 . the set @xmath213 of homotopy classes of non - singular vector fields is called the set of _ smooth euler structures _ on @xmath3 . as explained in section 5 of @xcite , the first obstruction to such a homotopy between @xmath214 is an element ( written as @xmath215 ) of @xmath216 , canonically isomorphic to @xmath75 by poincar duality . it turns out that there is a natural free and transitive action of the group of obstructions @xmath217 on @xmath213 . if we pick a reference nonsingular @xmath218 as the zero , then we can identify @xmath213 with @xmath219 . for @xmath71 , a natural choice for @xmath218 is a constant length vector field pointing along a torus cycle ( this gives a trivial insulating phase ) ; reference to such a @xmath218 will be implicit in this case . we remark that the space of spin@xmath5 structures for @xmath3 is a @xmath154-torsor , and that when @xmath45 , a non - singular @xmath218 determines a spin@xmath5 structure through the unit vector field @xmath220 @xcite , and vice - versa . an equivalent definition of euler structures , which is useful for our generalisation to kervaire structures in section [ sec : kervairestructures ] , was given in @xcite as follows . [ defn : coeuler ] let @xmath3 be a compact oriented @xmath4-manifold with @xmath209 , and let @xmath221 be the sphere bundle for the tangent bundle @xmath222 ( with a riemannian metric ) . define the set of _ euler structures _ to be the subset @xmath223 for which the restriction to each ( oriented ) fibre @xmath224 generates @xmath225 . then there is a free and transitive action of @xmath226 on @xmath227 under pullback to @xmath228 and addition in @xmath229 . to pass from the first picture of smooth euler structures as @xmath213 to this second picture , notice that the unit vector field @xmath36 for a non - singular @xmath29 is a map @xmath230 and so a @xmath4-cycle in @xmath228 . orient @xmath228 by requiring the intersection number of every @xmath231 with this cycle to be @xmath10 . then the @xmath4-cycle @xmath36 poincar dualises to a @xmath86-cocycle which we write as @xmath232 , and the latter represents an euler structure in @xmath233 in the second picture . to emphasize the cohomology definition of euler structure , we will sometimes use the term _ co - euler structure _ , to distinguish it from a homological definition to be given in section [ sec : eulerchain ] . as with @xmath213 , there is an identification of the affine space @xmath234 with @xmath219 upon fixing a reference non - singular @xmath218 . writing @xmath235 for the projection , we note that @xmath236 in the gysin sequence @xmath237 with @xmath238 the pushforward , or integration over the @xmath165 fibers . the subset @xmath233 comprises @xmath239 $ ] and all its translates by @xmath240 , so @xmath241 gives a bijection from @xmath242 taking @xmath239 $ ] to the identity element . there is a dual picture of euler structures involving vector fields with finite singularities , which is closely related to the intuition behind fermi arcs . we sketch the construction here for the case of non - degenerate zeroes , see @xcite for the general case . for a vector field @xmath29 on @xmath3 with finite singularity set @xmath243 , let @xmath244 denote the singular 0-chain @xmath245 which encodes the local charge information . we can also think of @xmath244 as an element of @xmath246 whose weights sum to zero . let @xmath3 be a compact @xmath4-manifold with @xmath208 and @xmath209 . euler chain _ for a 0-chain @xmath247 in @xmath3 whose weights sum to zero , is a 1-chain @xmath248 such that @xmath249 is @xmath247 . an euler chain @xmath250 defines a relative homology class in @xmath251 where @xmath47 is the set of points that @xmath247 is defined on is allowed to be zero on points in @xmath47 . ] . let @xmath252 denote the subset of relative homology classes of 1-chains in @xmath3 whose boundary is @xmath247 . from the exact sequence @xmath253 we see that @xmath254 is a coset of @xmath75 in @xmath251 and so an affine space for @xmath75 . in particular , the difference of @xmath255,[l']\in{\mathfrak{eul}}(t,{\mathcal{w}})$ ] is an element of @xmath75 . note that any 0-chain @xmath247 with total weight zero can be realised as the 0-chain of charges @xmath244 for some singular vector field @xmath29 , by the ( converse of the ) poincar hopf theorem . euler chains can be thought of as extra global data encoding how the local charge configuration for a singular vector field is `` connected '' . as explained in section [ sec : semimetaleulerchain ] , euler chains are in a precise sense poincar dual to cohomological semimetal invariants . * non - degenerate homotopies . * let @xmath210 be two smooth vector fields , then there is a canonical identification of @xmath256 and @xmath257 as affine spaces , using a notion of _ non - degenerate homotopy _ defined as follows . let @xmath258 be the pullback of the tangent bundle @xmath222 under the projection @xmath259\times t\rightarrow t$ ] . non - degenerate homotopy _ between @xmath29 and @xmath260 is a section @xmath261 of @xmath258 transverse to the zero section , which restricts to @xmath29 at @xmath262 and @xmath260 at @xmath263 . such a homotopy exists by perturbing , for instance , a linear homotopy , and allows for the movement @xmath264 of the local charges @xmath265 including the creation and annihilation of pairs of zeros with equal and opposite charges ( fig . [ fig : euler - chain ] ) . for generic @xmath266 , the intermediate vector field @xmath267 intersects the zero vector field transversally ( e.g. transversality of @xmath267 fails when a pair of weyl points are created or annihilated ) . the zero set @xmath268 of @xmath261 is a canonically oriented 1-submanifold with boundary @xmath269 , where the superscript indicates whether the 0-chain lies on @xmath270 or on @xmath271 . let @xmath255\in { \mathfrak{eul}}(t,{\mathcal{w}}_{{\bm{{h}}}})$ ] , then @xmath272 is a 1-cycle on @xmath273\times t$ ] relative to @xmath271 , since @xmath274 . because @xmath275\times t , \{1\}\times t)=0 $ ] , there is a 2-chain @xmath276 on @xmath273\times t$ ] whose boundary is @xmath272 relative to @xmath271 . taking @xmath277 , we see that @xmath278 is a 1-chain on @xmath271 with @xmath279 , which defines a class @xmath280\in { \mathfrak{eul}}(t,{\mathcal{w}}_{{\bm{{h}}}'})$ ] . intuitively , we can think a homotopy @xmath261 as providing the data of how an initial @xmath250 is to be `` carried along @xmath268 '' onto a final @xmath278 . $ ] goes horizontally , while all coordinates for the manifold @xmath3 are suppressed except one cyclic @xmath281-direction . the height function at each @xmath266 ( yellow curve ) indicates the length of a tangent vector on @xmath282 in the aforementioned direction . black arrows represent one component of the initial ( left ) vector field @xmath29 and final ( right ) one @xmath260 , which are taken to be equal in this example . the zero - section of the tangent bundle of @xmath273\times t$ ] is the blue shaded area . a non - degenerate homotopy @xmath261 is shown , with the intermediate vector fields @xmath267 ( indicated by yellow dashed curves ) generically having isolated zeroes with charges @xmath283 . the orange oriented submanifold @xmath268 comprises the singularities of @xmath261 . the euler chains for @xmath267 ( red lines joining @xmath284 to @xmath121 ) are `` carried along '' @xmath268 , disappear when the weyl points coalesce , then reappear when a pair of weyl points are created . the homology class of the euler chain for @xmath29 is unchanged by this homotopy , consistent with what happens if the obvious constant homotopy is chosen instead.,scaledwidth=70.0% ] which translates the initial vector field @xmath29 by half a @xmath281-cycle , showing how the euler chain is carried along the zero set ( orange ) of @xmath261 . rotation by a full cycle will return the euler chain to its original position , with the same homology class.,scaledwidth=70.0% ] which changes the homology class of an euler chain . note that there is no open ball of @xmath3 within which the singular set of @xmath267 stays for all @xmath285 $ ] . rather , such an `` enveloping open set '' contains an @xmath281 cycle.,scaledwidth=70.0% ] one shows @xcite that the above assignment @xmath255\mapsto[l']$ ] is independent of the choices of homotopy and of @xmath102 , yielding an affine map @xmath286 . furthermore , one has the properties @xmath287 , and @xmath288 , so each @xmath256 is canonically isomorphic to a single affine space ( which for oriented @xmath3 can be taken to be the space of smooth euler structures @xmath213 in definition [ defn : smootheuler ] ) over @xmath75 @xcite . in particular , when a non - singular @xmath218 in @xmath213 has been chosen , a possibly singular @xmath29 has an associated euler chain @xmath250 . if there is a @xmath261 such that the local charges @xmath289 are simply moved around smoothly and disjointly without creation / annihilation , i.e. there is a smooth 1-parameter family of diffeomorphisms @xmath290 of @xmath3 taking @xmath291 to @xmath289 , @xmath285 $ ] , then the zero set @xmath292 of @xmath261 is simply a set of disjoint lines representing the trajectories of the local charges . an initial euler chain @xmath250 is `` carried along @xmath292 '' to the final euler chain @xmath278 ( fig . [ fig : euler - chain2 ] ) . suppose @xmath210 have the same 0-chains of local charges , @xmath293 . then the endomorphism @xmath294 can be thought of as the homological change in the euler chains , @xmath255\mapsto [ l']$ ] , under a non - degenerate homotopy @xmath261 taking @xmath29 to @xmath260 . specifically , @xmath295 can be pushed forward under the projection @xmath296 to a 1-chain on @xmath3 , also written @xmath297 , which is actually a cycle . then @xmath280-[l]\in h_1(t,{{\mathbb z}})$ ] is a `` homological difference '' between @xmath29 and @xmath260 ; this difference was called a chern simons class @xmath298 in @xcite . for @xmath299 having different local charges , @xmath298 is defined similarly , as a 1-chain modulo boundaries . suppose there is a non - degenerate homotopy @xmath261 which stays non - singular outside some open ball in @xmath3 ( necessarily containing @xmath300 ) . then an initial euler chain @xmath250 does not pick up any nonzero element of @xmath75 , so @xmath280=[l]$ ] ( fig . [ fig : euler - chain ] , [ fig : euler - chain3 ] ) . the upshot of introducing euler chains is that we can use them to represent globally the topological data of a semimetal band structure , whenever vector fields are used to parameterise semimetal hamiltonians . this is certainly the case for @xmath27 hamiltonians in 3d , and more generally for dirac - type hamiltonians in higher dimensions . the singular points @xmath243 of @xmath29 are the band crossings of @xmath34 , and the local charge information is contained in the 0-chain @xmath244 . each @xmath29 can be associated with an euler chain class in @xmath301 by the prescription of the previous subsection , and vector fields with the same local charges but homologically different euler chains can not he homotoped whilst keeping the singular set within an open ball ( in particular , while keeping the singular set fixed ) . when we generalise to kervaire chains for bilinear hamiltonians later , it will be convenient to use an alternative prescription to associate an euler ( or kervaire ) chain to a semimetal hamiltonian . this alternative prescription uses a definition of euler structures which follows that given in section [ sec : coeuler ] , but which is defined in terms of singular vector fields . let @xmath228 be the sphere bundle of the tangent bundle of @xmath3 , and @xmath302 be the restrictions of @xmath228 to the subspaces of @xmath3 appearing in our mv - sequence , and let @xmath296 denote the various bundle projections . all these sphere bundles have vanishing euler class . there is also an mv - sequence for the cover @xmath303 of @xmath228 , which has intersection @xmath304 , and we write @xmath305 for its connecting maps . let @xmath218 be a non - singular vector field on @xmath3 , then @xmath236 . consider the semimetal mv - sequence and its dual homology sequence , where we have suppressed the integer coefficients , @xmath306&h^{d-1}(t)\ar[d]_{\text{pd}}\ar[rr]^{\iota^ * } & & h^{d-1}(t\setminus w)\ar[d]_{\text{pd}}\ar[rr]^{\beta } & & h^{d-1}(s_w)\ar[d]_{\text{pd}}\ar[rr]^{\sigma } & & h^d(t)\ar[d]_{\text{pd}}\ar[r ] & 0\\ 0\ar[r ] & h_1(t)\ar[rr ] & & h_1(t , w)\ar[rr]^{\partial } & & h_0(w ) \ar[rr]^{\sigma } & & h_0(t)\ar[r ] & 0 } .\ ] ] the local charges can be thought of as a 0-chain @xmath247 on @xmath82 , or dually as an element @xmath307 in @xmath308 ; in either case they are required to have total charge zero . for a 0-chain @xmath247 of local charges on @xmath3 , with total charge 0 and defined on @xmath82 , we write @xmath309 for the image of @xmath254 in @xmath98 under poincar duality ; equivalently this is the inverse image of @xmath307 under @xmath162 . we combine the cohomology mayer vietoris ( horizontal ) and gysin sequences ( vertical ) for @xmath3 and @xmath228 , @xmath310 & 0\ar[d ] & 0\ar[d ] & \\ 0\ar[r ] & h^{d-1}(t)\ar[d]_{p^*}\ar[r]^{\iota^ * } & h^{d-1}(t\setminus w)\ar[d]_{p^*}\ar[r]^{\beta } & h^{d-1}(s_w)\ar[d]_{p^*}\ar[r]^{\quad\sigma } & \cdots\\ 0\ar[r ] & h^{d-1}({\mathcal{s } } ) \ar[r]\ar[d]_{p _ * } & h^{d-1}({\mathcal{s}}|_{t\setminus w})\oplus h^{d-1}({\mathcal{s}}|_{d_w } ) \ar[r]\ar[d]_{p _ * } & h^{d-1}({\mathcal{s}}|_{s_w } ) \ar[r]^{\quad\widetilde{\sigma}}\ar[d]_{p _ * } & \cdots\\ 0\ar[r ] & h^0(t)\ar[r]\ar[d ] & h^0(t\setminus w)\oplus h^0(d_w)\ar[r]\ar[d ] & h^0(s_w ) \ar[r]\ar[d ] & \cdots\\ & 0 & 0 & 0 & } , \label{mvgysin}\ ] ] where we have augmented the middle gysin sequence by a @xmath311 piece needed to make the bottom two horizontal mv sequences exact . the horizontal maps are restriction maps ( or differences of ) and so it is easy to see that the diagram commutes . the middle mv - sequence is , by the knneth theorem , @xmath312 and so the restriction map for the @xmath313 factor has a `` local charge '' component @xmath314 and a second component which is less important for us . furthermore the restriction map for the @xmath315 factor only lands in the @xmath316 factor . the relevant part of is then the commuting diagram of short exact sequences @xmath310 & 0\ar[d ] & 0\ar[d ] & \\ 0\ar[r ] & h^{d-1}(t)\ar[d]_{p^*}\ar[r]^{\iota^*\;\ ; } & h^{d-1}(t\setminus w)\ar[d]_{p^*}\ar[r]^{\beta\;\ ; } & \text{ker}_\sigma(h^{d-1}(s_w))\ar[d]_{p^*}\ar[r ] & 0\\ 0\ar[r ] & h^{d-1}({\mathcal{s } } ) \ar[r]^{\iota^*\;\;}\ar[d]_{p_*}\ar[u]_{\hat{{\bm{{h}}}}_{\text{ref}}^ * } & h^{d-1}({\mathcal{s}}|_{t\setminus w } ) \ar[r]^{\tilde{\beta}\;\;}\ar[d]_{p_*}\ar[u]_{\hat{{\bm{{h}}}}_{\text{ref}}^ * } & \text{ker}_{\widetilde{\sigma}_1}(h^{d-1}(s_w ) ) \ar[r]\ar[d ] & 0\\ 0\ar[r ] & h^0(t)\ar[r]^{\iota^*\;\;}\ar[d ] & h^0(t\setminus w)\ar[r]\ar[d ] & 0 & \\ & 0 & 0 & & } .\label{mvgysin2}\ ] ] let @xmath247 be a 0-chain of local charges defined on the finite subset @xmath82 with total weight zero , and @xmath307 its poincar dual . we define @xmath317 to be the set of @xmath318 such that @xmath319 and @xmath320 . there is an identification of @xmath317 and @xmath309 as affine spaces for @xmath219 . by chasing through the diagram , we see that an element @xmath321 is of the form @xmath322 for some @xmath323 with @xmath324 and some @xmath325 with @xmath326 , i.e. @xmath327 and @xmath328 . clearly , the action of @xmath329 on @xmath317 by pullback @xmath330 and addition is free , and is also transitive since the difference of @xmath331 is the sum of @xmath332 and @xmath333 , which is something in @xmath334 . given a reference non - singular @xmath218 , it is easy to check that the surjective map @xmath335 respects the local charges @xmath336 , and restricts to a @xmath329-equivariant bijection between @xmath317 and @xmath309 . the following diagram summarises the various affine spaces for @xmath337 : @xmath338 \ar[d ] & & { \mathfrak{eul}}(t,{\mathcal{w}})\ar[ll ] \ar[d ] & \subset \quad\quad\;\ ; h_1(t , w ) \\ & h^{d-1}(t)\ar[rr]\ar[d ] \ar[u ] & & { \mathfrak{coeul}}(t,{\mathcal{w}})\ar[ll ] \ar[d]\ar[u ] & \subset \quad h^{d-1}(t\setminus w ) \\ h^{d-1}({\mathcal{s}})\quad\supset & \widetilde{{\mathfrak{coeul}}}(t)\ar[rr ] \ar[u ] & & \widetilde{{\mathfrak{coeul}}}(t,{\mathcal{w}})\ar[ll ] \ar[u ] & \subset \quad h^{d-1}({\mathcal{s}}|_{t\setminus w } ) } .\nonumber\ ] ] if @xmath247 is the zero 0-chain , then @xmath339 coincides under @xmath163 with @xmath234 . let @xmath29 be a vector field with local charges @xmath247 , so @xmath36 defines a @xmath4-cycle on @xmath340 relative to the boundary @xmath304 . its poincar dual is a @xmath86-cocyle on @xmath340 representing an element of @xmath317 . pulling back under @xmath220 gives an element @xmath341 $ ] in @xmath309 , whose poincar dual is an euler chain @xmath342 $ ] in @xmath254 . in this way , a semimetal hamiltonian specified by a vector field with finite singularities can be represented by an euler chain for @xmath247 . [ defn : semimetaltopinv ] suppose @xmath34 is a dirac - type hamiltonian specified by a smooth vector field @xmath29 on @xmath3 with finite singularities @xmath47 . it has a cohomological topological invariant @xmath341\in h^{d-1}(t\setminus w,{{\mathbb z}})$ ] as in the above paragraph , and its _ euler chain representation _ is the poincar dual @xmath342\in h_1(t , w,{{\mathbb z}})$ ] . the euler chain representation of a semimetal is extremely useful because it invariantly characterises and generalises the idea of `` jumps in chern numbers '' as a weyl point is traversed , even for complicated local charge configurations . more precisely , the natural pairing @xmath343 is well - defined for any semimetal with invariant @xmath344 $ ] , and any homology @xmath86-cycle @xmath345 $ ] avoiding @xmath47 . for example , we can take @xmath345 $ ] to be the fundamental classes of @xmath86-spheres surrounding weyl points , or ( when @xmath71 ) of @xmath86 subtori at a fixed coordinate . the pairing poincar dualises to the _ _ intersection pairing _ _ , as in 2.6.17 of @xcite . here @xmath346 is borel moore homology , mentioned in section [ sec : dualmv ] , which is isomorphic to @xmath193 and poincar dual to @xmath98 . ] between @xmath342 $ ] and @xmath345 $ ] , i.e. the ( signed ) intersection between the corresponding euler chain @xmath347 and the @xmath86-cycle @xmath276 in question . as a concrete example in @xmath45 , for those @xmath345 $ ] which are represented by @xmath348-submanifolds @xmath276 , the intersection number computes the integral over @xmath276 of the berry curvature 2-form ( which is defined on @xmath66 ) this yields the familiar first chern numbers ( `` weak '' chern invariants ) and their discontinuities across the weyl points ( fig . [ fig : intersection]-[fig : intersection2 ] ) . for @xmath71 , a projection @xmath349 induces a homology projection @xmath350 , which we can think of as being poincar dual to the bulk - boundary correspondence @xcite . under @xmath351 , an euler chain becomes a surface _ fermi arc _ a 1-chain on @xmath207 whose boundary is the projected 0-chain of local charges . figure [ fig : bulk2boundary ] provides an intuitive picture in @xmath45 explaining how surface fermi arcs are determined to @xmath284 , so the fermi arcs there are _ oppositely oriented _ to those in this paper . ] from bulk euler chains by @xmath351 . the euler chain description is coordinate - free , and determines the fermi arc connectivity for the projection to _ any _ surface . are euler chains encoding , in particular , how the first chern numbers @xmath352 vary with @xmath353 . only the values of @xmath133 are indicated . gapless surface states appear at points in the surface brillouin torus @xmath21 where a first chern number is non - zero , forming a fermi arc connecting the projected weyl points . the fermi arc is determined from the euler chain homologically by projection @xmath351,title="fig : " ] are euler chains encoding , in particular , how the first chern numbers @xmath352 vary with @xmath353 . only the values of @xmath133 are indicated . gapless surface states appear at points in the surface brillouin torus @xmath21 where a first chern number is non - zero , forming a fermi arc connecting the projected weyl points . the fermi arc is determined from the euler chain homologically by projection @xmath351,title="fig : " ] if we are able to access the complete specification of a semimetal hamiltonian as an operator , then its surface fermi arc is uniquely determined in principle . in practice , we have access to the fermi arcs , and different fermi arcs ( as 1-chains ) come from different hamiltonians ( as operators ) . the passage from hamiltonian to fermi arc can be modelled , for example , by transfer matrices @xcite . at the _ topological _ level , a coarser but more appropriate question is how fermi arcs may be used to distinguish _ topologically distinct _ semimetal hamiltonians . for this latter question , we note that _ any _ euler chain representative for a semimetal hamiltonian @xmath34 maps under @xmath351 to a 1-chain which is topologically equivalent to the actual fermi arc for @xmath34 . thus fermi arcs which are topologically distinct must come from topologically distinct hamiltonians ; our slight abuse of language in calling the 1-chain a `` fermi arc '' does not matter at the level of topological invariants . certain scenarios involving tuning semimetal hamiltonians and `` rewiring '' their surface fermi arcs were considered in @xcite . these essentially involve homotopies @xmath261 which fix the local charges @xmath293 ( at least up to a diffeomorphism moving the weyl points ) , and so induce the identity map on the class of euler chains in @xmath354 . under @xmath351 , the surface fermi arcs for @xmath29 and @xmath260 may become `` rewired '' , but in such a way that their class in @xmath205 remains unchanged . rewirings which involve a change in @xmath205 necessarily require the prefiguring euler chain in @xmath254 to change homology class , and this can not be achieved by homotopies @xmath261 which fix the local charges @xmath247 . figures [ fig : intersection]-[fig : intersection2 ] illustrate some examples of rewirings . with one coordinate suppressed ; we can also interpret it as the projected surface brillouin torus @xmath21 . two homologous euler chains ( the left and right pairs of red directed lines ) having the same local charge configuration are shown , and we can also interpret them as the corresponding surface fermi arcs . first chern numbers for 2d subtori in the @xmath355 and @xmath356 directions are shown , and may be computed by counting the signed intersection between the euler chain and the subtorus . the `` straightness '' of the euler chain does not matter , and chern numbers can also be calculated for 2-cycles other than the standard ones with a fixed coordinate.,scaledwidth=90.0% ] representing more semimetal invariants in @xmath357.,scaledwidth=90.0% ] [ remark : higherjumps ] in @xmath46 , the intuition afforded by figures [ fig : chern - jump ] and [ fig : bulk2boundary]-[fig : intersection2 ] carries over in much the same way . for example , in @xmath67 , the blue surfaces represent hyperplane slices ( 3-tori ) on which the semimetal gerbe of section [ sec : semimetalgerbe ] restricts and has a curvature 3-form . there are four independent slice directions . as a slice is moved transversely across a semimetal band crossing with local charge @xmath358 , the dixmier douady invariant of the gerbe on the slice jumps by @xmath358 . in @xmath68 , the slices are 4-tori and and it is the second chern number of the @xmath359-invariant semimetal ( section [ sec:5dsemimetal ] ) which jumps by @xmath358 . [ remark : nonzeroeuler ] it is also possible to define euler structures for @xmath3 even if @xmath360 by introducing a basepoint @xmath361 which `` contains @xmath70 '' , see @xcite . an euler chain for a 0-chain @xmath247 is then a 1-chain @xmath248 such that @xmath362 . every vector field @xmath29 on @xmath3 with isolated zeroes @xmath47 admits an euler chain for its 0-chain @xmath244 of local charges in this sense : take @xmath363 , then @xmath364 by the poincar hopf theorem . euler structures for @xmath3 with basepoint @xmath365 are then defined as classes of pairs @xmath366 which are considered equivalent if @xmath367 up to a boundary . there is an action of @xmath75 on these classes by addition to @xmath250 , which is furthermore free and transitive . so far , we have implicitly assumed that the hamiltonians act on a _ trivial _ bloch bundle @xmath368 and that a trivialisation ( choice of basis ) has been given , so that the dirac - type hamiltonians of the form are well - defined globally . although this default setting suffices to illustrate many interesting features of semimetals , it is worth mentioning that non - trivial @xmath142 can arise physically and can still be handled mathematically . in general one has , after a bloch floquet transform , a _ hilbert bundle _ @xmath369 in which the fibre @xmath370 over @xmath56 comprises the quasi - periodic bloch wavefunctions with quasimomentum @xmath371 @xcite . one typically restricts attention to a low - energy subbundle @xmath142 defined by a finite number of energy bands separated from the rest by spectral gaps . subsequently , a _ bloch bundle _ @xmath142 will refer to such a truncated finite - rank hermitian vector bundle over a manifold @xmath3 of ( quasi)-momenta , on which the _ bloch hamiltonians _ act fiberwise . in the theory of topological band insulators , it is crucial that @xmath142 or its subbundles can be non - trivializable , e.g. 2d chern insulators are essentially specified by the first chern class of a non - trivializable valence line bundle . expressions such as should be understood as _ local _ expressions , which continue to make sense on a bloch bundle @xmath142 as long as @xmath142 has the structure of a clifford module bundle ( or spinor bundle ) so that the @xmath79 operators @xmath372 are well - defined globally . in particular , the analysis of the spectrum of @xmath373 remains the same for _ any _ @xmath142 namely , the @xmath55 eigenspaces degenerate ( i.e. there is a band crossing ) precisely at the zeroes of @xmath29 . on the other hand , the topology of the valence subbundle over @xmath66 , which is defined by the fermi projection @xmath374 , does depend on @xmath142 through the operators @xmath372 . when @xmath375 , the most general @xmath17-band hamiltonian is _ not _ of dirac - type , and well - defined topological invariants should respect the @xmath187 gauge invariance of the rank-@xmath17 bloch bundle @xmath142 . nevertheless , dirac - type hamiltonians are generic when certain time - reversal / particle - hole symmetries are imposed . such additional symmetries restrict the gauge group , and the topological invariants which we will define for insulators / semimetals described by dirac - type hamiltonians , are gauge invariant in this restricted sense . some examples of such gauge restrictions were studied in @xcite , and we also analyse them in section [ sec : higherdimension ] . the properties of dirac - type hamiltonians which are relevant for defining insulator / semimetal invariants can be abstracted as follows . an abstract dirac - type hamiltonian is a section of the vector part of some clifford algebra bundle over a momentum space manifold @xmath3 , i.e. the quantization of a vector field @xmath29 specifying the abstract hamiltonian as in definition [ defn : abstracthamiltonian ] . pointwise in @xmath3 , the spectrum of such a hamiltonian ( as a clifford algebra element ) can be found quite easily ; it depends only on the vector field and has the crucial feature that it degenerates exactly at the zero set @xmath47 of the vector field . when such hamiltonians are represented on some clifford module bundle @xmath142 ( physically the bloch bundle ) , they become concrete families of hermitian operators with fermi projections etc . the local topological invariant at a zero @xmath58 is intrinsic to @xmath29 rather than the particular choice of bloch bundle @xmath142 , and exists even when @xmath142 is not a trivial bundle . furthermore , the symmetries and gauge structure for the hamiltonians become more transparent at this abstract level , and this is especially so in higher dimensions . * hamiltonians as quantized vector fields . * for @xmath77 , let @xmath376 be an oriented rank-@xmath4 real vector bundle over a compact oriented manifold @xmath3 with fibre metric @xmath377 , along with a spin@xmath5 structure ( e.g. take @xmath3 a spin@xmath5 manifold and @xmath376 its tangent bundle , then there is a @xmath154 worth of spin@xmath5 structures to choose from ) . the clifford algebra bundle @xmath378 is a `` quantization '' of the exterior algebra bundle @xmath379 to allow for multiplication of vectors , both bundles having structure group @xmath380 @xcite . with the spin@xmath5 structure , we can construct the ( irreducible ) spinor bundle @xmath142 ( the `` bloch bundle '' ) of complex rank @xmath381 , with @xmath382-invariant hermitian inner product , and clifford multiplication gives a _ real _ bundle homomorphism @xmath383 . thus there is a map @xmath384 in particular , @xmath385 is identified with a subbundle of @xmath378 , and the map @xmath386 takes a section @xmath387 to a concrete dirac - type hamiltonian . more explicitly , an orthonormal frame @xmath388 for @xmath376 becomes @xmath389 in @xmath378 and satisfies the clifford relations @xmath390 . the @xmath391 are represented on @xmath142 as traceless self - adjoint endomorphisms @xmath392 which inherit the familiar relations @xmath393 . thus , a section @xmath394 defines an _ abstract dirac - type hamiltonian _ @xmath395 , which is then represented by a _ concrete dirac - type hamiltonian _ @xmath396 acting on @xmath142 . the concrete _ dirac - type bloch hamiltonian _ above a point @xmath56 is the operator @xmath397 acting on the fiber @xmath398 . in the clifford algebra bundle , it is already ( @xmath380-invariantly ) true that @xmath399 . if follows that the spectrum of a bloch hamiltonian is @xmath400 with each eigenvalue @xmath401-fold degenerate , independently of @xmath142 , and this is @xmath382-invariant . precisely at the zeroes of @xmath29 , the two @xmath401-fold degenerate energy bands cross , and such a crossing is precisely protected by the topological index of @xmath29 there . more generally , we only require @xmath142 to be a clifford module bundle for @xmath378 , for which @xmath395 acts self - adjointly . such clifford module bundles are in fact twisted versions of spinor bundles , obtained by tensoring with some vector bundle @xmath402 , so we shall mainly consider only irreducible spinor bundles . in section [ sec : torsionsemimetal ] we will also consider @xmath403 acting skew - adjointly on @xmath142 , giving rise to `` @xmath51-bilinear hamiltonians '' . physically , one might start off with a given bloch bundle @xmath142 , which is a hermitian @xmath187 vector bundle over @xmath3 obtained by fourier / bloch floquet transform . in order to have a notion of clifford multiplication and dirac - type hamiltonians , we need some assumptions on @xmath142 allowing it to have the structure of a ( irreducible ) clifford module bundle for some @xmath376 . in the first place , @xmath142 should have the correct ( complex ) rank . in most model hamiltonians considered in the physics literature , @xmath71 and the bloch bundle @xmath142 is assumed to be trivial . in this case , we take the trivial bundle @xmath72 , identified with the tangent bundle of @xmath0 , and @xmath387 is a tangent vector field . then the corresponding dirac - type bloch hamiltonian @xmath404 has gamma matrices which can be taken to be constant over @xmath0 . formally , a family of @xmath79 bloch hamiltonians over @xmath3 can also be regarded as a family of hamiltonians on an @xmath17-level quantum mechanical system , with adiabatic phases / holonomies , etc . there is , however , a fundamental difference between @xmath3 as an adiabatic parameter space and as a quasi - momentum space , when antiunitary symmetries are introduced . the brillouin torus @xmath0 is topologically the space of unitary characters for the translation group @xmath406 of a lattice , so the complex conjugation involved in an antiunitary symmetry operator induces an involution @xmath407 ( a @xmath2-action ) on @xmath0 . explicitly , when we parameterise @xmath0 by angles @xmath408 , the character @xmath409 labelled by @xmath371 conjugates to the character labelled by @xmath410 . in the context of bloch hamiltonians , a fermionic time - reversal symmetry @xmath359 is a lift of @xmath407 to a map on the bloch bundle @xmath142 which is antiunitary on fibres , and satisfies @xmath411 . compatibility of @xmath359 with @xmath412 is then the condition @xmath413 . such bundles equipped with a `` quaternionic '' structure @xmath359 were studied in @xcite , and have more recently been investigated in the context of topological insulators @xcite . for bosonic time - reversal , @xmath359 squares to the identity instead , and is sometimes called a `` real structure '' . similarly , an antiunitary particle - hole symmetry @xmath405 _ anticommutes _ with the bloch hamiltonians in a way that respects @xmath407 , namely , @xmath414 . on the other hand , for a family of hamiltonians parametrised by @xmath3 , we simply have @xmath415 and @xmath416 . in either case , the squares @xmath417 may be @xmath418 , and when both @xmath359 and @xmath405 are present , they may be assumed to commute @xcite . the same involution @xmath407 is induced on @xmath0 by a spatial inversion symmetry @xmath419 , since a translation by @xmath420 becomes a translation by @xmath421 . thus we can consider a time - reversal symmetry which concurrently effects spatial inversion and @xmath422 to _ separately _ be symmetries , and indeed , the latter situation requires additional data specifying whether @xmath423 commute or anticommute @xcite , leading in each case to different wigner classes . ] ; we write @xmath424 for such a symmetry operator . then we have @xmath425 where the effect of @xmath407 cancels ; similarly for @xmath426 . thus the operator @xmath424 is an ordinary quaternionic structure , and the bloch bundle @xmath142 has invariants as a _ symplectic bundle_. * convention . * for this paper , we will simply write @xmath427 with the understanding that @xmath428 is meant whenever @xmath3 is a brillouin torus . thus we always have @xmath415 and @xmath429 . let us explain the above abstractions in more detail for two - band hamiltonians in @xmath45 , in which @xmath430 . this is the setting for the basic weyl semimetal . since @xmath431 , any rank-2 hermitian vector bundle @xmath142 can arise as a spinor bundle : take the traceless hermitian endomorphisms @xmath432 of @xmath142 with the hilbert schmidt real inner product @xmath433 . then @xmath432 becomes a real orientable rank-3 bundle with structure group @xmath434 liftable to @xmath435 . an orthonormal frame for @xmath432 gives a set of pauli operators @xmath436 , which is positively oriented if @xmath437 . such a positively - oriented frame has the form of the standard pauli matrices in some local trivialization of @xmath142 . the bloch hamiltonian is given locally by @xmath129 . the @xmath438 has a clifford algebra bundle and a choice of spin@xmath439 structure yields a spinor bundle . @xmath142 is some twisted version of this spinor bundle , and the twisting can be thought of as changing the spin@xmath439 structure under the canonical action of @xmath154 @xcite . thus @xmath436 may be thought of as @xmath440 for an orthonormal frame @xmath441 in @xmath442 . from this point of view , a section @xmath29 is quantized to an _ abstract hamiltonian _ @xmath443 , which acts concretely as @xmath444 on @xmath142 . note that , in particular , the spectrum @xmath445 is invariant under @xmath446 gauge transformations of @xmath142 . [ rem : gauge3d ] a gauge transformation of the bloch bundle @xmath142 implemented by local @xmath27 unitaries @xmath447 turns @xmath448 into the conjugate @xmath449 . the result of this conjugation is known to be equal to @xmath450 where @xmath451 is rotated from @xmath452 by the @xmath453 element @xmath454 in @xmath455 similarly , @xmath456 in case the structure group of @xmath142 can be reduced to @xmath457 . two band - hamiltonians can more generally include a trace term , i.e. @xmath458 , where @xmath459 is not constant and so not eliminable by shifting the overall energy level . indeed , hamiltonians with such @xmath459 terms ( and constant @xmath460 ) can give rise to new phenomena such as `` type - ii weyl semimetals '' @xcite . it is possible to distinguish the two situations , i.e. eliminate the @xmath459 term , by imposing a natural symmetry constraint . * quaternionic structure and @xmath405-symmetry . * let us recall the construction of the spinor representation @xmath461 of @xmath462 . let @xmath463 be orthonormal vectors in @xmath464 , @xmath465 , and @xmath466 . the even part @xmath467 is isomorphic to @xmath468 , and for the complexified algebras , @xmath469 . the spinor representation on @xmath470 is the ( unique complex ) irreducible representation of @xmath471 , and is extended to @xmath472 by setting a chirality condition @xmath473 . @xmath474 lies in @xmath475 as the unit quaternions , and @xmath461 is an irreducible representation of @xmath476 . this is just the fundamental representation of @xmath477 which is well - known to be quaternionic . the quantized bivectors @xmath478 generate @xmath479 . on @xmath461 , they are represented by skew - hermitian operators @xmath480 ( which generate a quaternion algebra ) , and with the chirality element @xmath481 the identity on @xmath461 , we recover the pauli matrices @xmath482 . since @xmath461 is a quaternionic representation for @xmath476 , there is an antiunitary operator @xmath483 on @xmath461 commuting with @xmath484 such that @xmath485 , from which we deduce that @xmath483 _ anticommutes _ with @xmath486 . for instance , we can take @xmath487 where @xmath488 is complex conjugation in the basis where @xmath489 are the standard pauli matrices ( this operator is often taken to be the fermionic time - reversal operator on spin-@xmath490 systems ) . if @xmath72 has a spin ( not just spin@xmath5 ) structure , there is also a notion of spinor bundles @xmath142 associated to the spin structure @xcite , and these have structure group @xmath491 . globalising the above constructions , these spinor bundles come with a quaternionic structure @xmath405 anticommuting with @xmath492 . * reduction to @xmath493 . * suppose we have a reduction of the structure group of the bloch bundle from @xmath494 to @xmath495 . this is possible exactly when @xmath496 is trivial , i.e. @xmath497 ( this is a general result about reducing from @xmath187 to @xmath498 ) . alternatively , since @xmath499 , the reduction from @xmath494 to @xmath457 requires a quaternionic structure on @xmath142 , which provides an isomorphism between the line bundles @xmath500 and @xmath501 for an orthogonal splitting @xmath502 . then @xmath503 . in this case , we can identify @xmath432 with the tangent bundle @xmath504 , as ( trivial ) @xmath453 bundles . therefore , in the two - band case in 3d , the imposition of a @xmath405-symmetry constraint is one possible way to justify ( 1 ) the restriction from arbitrary @xmath27 bloch hamiltonians to the _ traceless _ ones , and ( 2 ) the triviality of the bloch bundle @xmath142 . we now generalize the two - band crossings of the previous section to four - band crossings . these should involve crossings of _ pairs _ of bands , in analogy to the dispersion of massless dirac spinors . again , the adjective `` dirac '' is loaded and possibly misleading , and our constructions below are more analogous to weyl fermions in 1 + 5 d ( which is still misleading ) . for this subsection , we take @xmath3 to be a compact oriented 5-manifold , and @xmath505 a trivial hermitian vector bundle . the bundle @xmath432 of traceless hermitian endomorphisms of @xmath142 is now a trivial rank-15 real vector bundle whose sections are bloch hamiltonians @xmath506 with inner product @xmath507 . for notational ease , we will sometimes suppress the dependence on @xmath371 when dealing with a single operator . physicists often restrict attention to `` dirac - type '' hamiltonians @xmath404 where @xmath508 is an orthonormal set of @xmath509 hermitian operators on @xmath461 satisfying the clifford algebra relation @xmath52 , and @xmath29 is a 5-component vector field . the pragmatic justification is ease of manipulation , since the spectrum is again easily found to be @xmath510 , with doubly - degenerate eigenvalues that become four - fold degenerate precisely at the zeroes of @xmath29 ( fig . [ fig : symmetric spectrum ] ) . however , we should note that arbitrary gauge transformations will not preserve this particular form of @xmath509 hermitian matrices . it is therefore useful to find a symmetry which reduces the allowed gauge transformations so that the dirac - form of @xmath34 is automatically preserved . we do this by invoking a @xmath359-symmetry constraint ( a quaternionic structure ) on the hamiltonians , which will pick out the dirac - type hamiltonians as a rank-5 subbundle of @xmath432 . in fact , the observation that the proper gauge groups and berry connections / phases for time - reversal invariant hamiltonians are _ quaternionic _ , had already been made in @xcite , and in the @xmath509 case such hamiltonians can be constructed by considering @xmath511 as a spin-@xmath512 representation ( so - called `` quadrupole hamiltonians '' ) . we give a more direct account , emphasizing the clifford algebra and spin groups responsible for this structure . consider first a fibre @xmath513 and define the antiunitary operator @xmath514 where @xmath488 is complex conjugation , which satisfies @xmath515 . we verify that ( real ) linear combinations of the following mutually anticommuting @xmath516 , _ commute _ with @xmath517 : @xmath518 a more invariant way to see this is to notice that an anticommuting set of @xmath509 hermitian operators @xmath519 satisfying @xmath520 , generate ( over the reals ) a copy of @xmath521 inside @xmath522 which thereby commutes with a @xmath523 action generated by @xmath524 ( two anticommuting square roots of @xmath8 ) . the product of two or three different @xmath525 is not hermitian , but @xmath526 is hermitian and also commutes with @xmath517 . the remaining ten - dimensional subspace of traceless hermitian operators is spanned by operators @xmath527 and they _ anticommute _ with @xmath517 instead we will interpret this latter subspace of @xmath509 hamiltonians as those compatible with a particle - hole symmetry @xmath405 , and these form an important class to be analysed in section [ sec : z2invariant ] . therefore , traceless hermitian operators of the form @xmath373 are precisely those which are compatible with a quaternionic structure @xmath517 , which we interpret as a time - reversal operator @xmath359 . conversely , a quaternionic structure @xmath517 identifies @xmath511 with @xmath528 , and @xmath529 is the real subalgebra of @xmath530 which commutes with @xmath517 . globalising the above arguments , we deduce : a quaternionic structure @xmath517 on a rank-@xmath531 bloch bundle picks out a real orientable rank-5 subbundle @xmath532 of commuting dirac - type hamiltonians . the chirality condition @xmath526 for an orthonormal frame @xmath533 for @xmath534 defines the positive orientation . a quaternionic structure always exists on a trivial bloch bundle , and in this case , we identify @xmath534 with the tangent bundle @xmath376 as @xmath535-bundles . thus @xmath536 corresponds to a positively - oriented orthonormal frame @xmath537 for @xmath72 . in terms of clifford modules , the clifford algebra @xmath538 has @xmath539 as the spinor representation in which the central chirality element @xmath540 is the identity operator . a vector field @xmath29 is quantized to an abstract dirac - type hamiltonian @xmath541 , which is represented concretely on @xmath142 as the hamiltonian @xmath542 isometrically : @xmath543 . the observation that the space @xmath534 has structure group @xmath535 was made in section 5 of @xcite ( for trivial bundles @xmath142 ) . generalizing remark [ rem : gauge3d ] , @xmath461 is the spinor representation for @xmath544 , and conjugating @xmath404 by a @xmath545 unitary matrix @xmath447 takes it to @xmath546 where @xmath547 is rotated from @xmath452 by the @xmath535 matrix @xmath548 covered by @xmath447 . the reduction of the gauge group is important : to stay within the class of dirac - type hamiltonians , we can not allow all @xmath549 gauge transformations of the bloch bundle @xmath142 , but only those in @xmath550 . for a 5-component vector field with finite isolated zeroes @xmath82 , the local index at @xmath58 is again given by the degree of the unit vector map @xmath551 restricted to a small sphere around @xmath85 . this @xmath104-valued local index measures the obstruction to `` gapping '' out a four - band crossing _ within the family of dirac - type hamiltonians _ ( or @xmath359-invariant hamiltonians ) . away from @xmath47 , @xmath53 can be spectrally - flattened to @xmath552 , and the fermi projection @xmath553 projects onto a complex rank-2 subspace of @xmath539 . this subspace is actually a _ quaternionic line _ due to the existence of @xmath517 commuting with @xmath53 , i.e. an element of @xmath186 . just as a unit 3-vector defines a point in @xmath554 via @xmath555 @xmath556-subspace of @xmath557 , we also have @xmath558 via \{unit 5-vector @xmath29 } @xmath559 \{@xmath556-subspace of @xmath64}. the valence subbundle @xmath20 of a @xmath359-invariant dirac - type hamiltonian in 5d is thus a quaternionic line bundle over @xmath66 . these quaternionic line bundles are pulled back from the tautological one over @xmath560 , and may have non - trivial _ pontryagin class / second chern class in @xmath561 @xcite . the _ symplectic _ pontryagin classes @xmath562 for @xmath563 bundles @xcite , should be distinguished from the real pontryagin classes @xmath564 for @xmath565 bundles . the former comes from @xmath566 and equals @xmath567 of the underlying complex bundle of a quaternionic bundle , while the latter comes from @xmath568 and equals @xmath567 of the complexification of the underlying real bundle of a complex bundle ( e.g. underlying a quaternionic one ) . for the tautological quaternionic line bundle @xmath569 , the symplectic pontryagin class / second chern class is a generator of @xmath570 . the @xmath571 part of the semimetal mayer vietoris sequence @xmath572 is slightly harder to interpret in terms of characteristic classes of @xmath20 : unlike @xmath573 and complex line bundles , it is not generally true that @xmath574 classifies all quaternionic line bundles ( equivalently @xmath575-bundles ) over a given space . for example , @xmath576 is a non - trivial @xmath457 bundle over @xmath577 but @xmath578 so @xmath579 does nt detect this bundle . nevertheless , @xmath580 with the generators coming from pulling back @xmath185 along the projection @xmath581 for each of the five independent choices of 4-subtori . thus all elements of @xmath582 do arise as the second chern class of some valence quaternionic line bundle @xmath583 ( weak insulator invariants ) , and their restrictions to @xmath584 form the kernel of the local charge map @xmath162 in . the other elements @xmath585 have some non - zero local charges under @xmath162 . from a singular vector field @xmath29 with those local charges @xmath586 , the valence bundle for @xmath64 has @xmath587 . thus for @xmath588 , the @xmath574 groups in can be interpreted as the @xmath579 invariants for the valence bundles @xmath20 of the dirac - type hamiltonians as constructed above . quaternionic line bundles are already stable for dim@xmath589 @xcite , so we can classify the @xmath20 by ( reduced ) quaternionic @xmath1-theory @xmath590 . therefore , the semimetal mv - sequence in @xmath590 can be interpreted in terms of obstructions to extending a semimetal valence bundle ( over @xmath591 ) to an insulator one ( over @xmath592 ) . when we introduced @xmath359 to @xmath509 hamiltonians , we found that they had the form @xmath404 for some 5-component vector @xmath452 . now introduce a further ( unitary ) chiral symmetry @xmath593 , with @xmath594 which a compatible @xmath34 must _ anticommute _ with . without loss , we can take @xmath595 . equivalently , this is an additional ( antiunitary ) @xmath405 symmetry , with @xmath596 and @xmath597 : take @xmath598 . the additional anticommutation requirement @xmath599 ( or @xmath600 ) forces @xmath53 to be a linear combination of only the first _ four _ gamma matrices in . furthermore , the traceless condition on @xmath34 is now automatic . thus in 4d , we may identify the @xmath601-compatible ( or @xmath602-compatible ) @xmath509 hamiltonians with the tangent bundle of @xmath603 , via @xmath404 , where @xmath604 now have four components . similarly , @xmath539 is the spinor representation for @xmath605 which has a quaternionic structure @xmath606 compatible with @xmath607 . the spectrum is again @xmath400 , vanishing at the zeroes @xmath608 of @xmath29 . the topological invariant for a four - band crossing at a zero @xmath58 is the degree of the the unit vector map @xmath609 . there is also an analysis based on supercommutants . the requirement of having symmetry operators @xmath610 is equivalent to having a _ @xmath2-graded representation _ of the clifford algebra @xmath611 @xmath612 are the three anticommuting clifford generators , acting irreducibly as odd operators on the superspace @xmath613 . the supercommutant , by the super - schur lemma @xcite , is a real superdivision algebra @xmath462 . more explicitly , we can take @xmath614 the @xmath615-compatible hamiltonians are spanned by the above @xmath462 generators and the grading operator . having identified the @xmath359 and @xmath593-symmetric @xmath509 hamiltonians as those which are linear combinations of @xmath616 , we now explain how a gerbe can be associated to such hamiltonians . first , it has become usual to associate a gapped chiral symmetric hamiltonian @xmath34 with a unitary in half the number of dimensions as follows . in a basis in which @xmath593 is @xmath617 , a chiral symmetric hamiltonian @xmath34 is off - diagonal as is @xmath618 . thus @xmath619 where @xmath620 is unitary since @xmath621 . note that this association @xmath622 is basis - dependant ( under @xmath623 transformations which respect @xmath593 ) , and for families of hamiltonians @xmath506 , the topological invariants of the map @xmath624 , which is defined wherever @xmath625 is defined ( i.e. away from @xmath47 ) , can change under `` large '' gauge transformations . we require a further @xmath359 symmetry such that @xmath359 commutes with @xmath593 , so it is of the form @xmath626 for some @xmath27 unitaries @xmath627 . since @xmath411 , up to a unitary transformation preserving @xmath593 , we can take @xmath628 as in . having done so , a basis for the @xmath359 and @xmath593 compatible hamiltonians is @xmath616 given in . note that the unitary transformations which preserve both @xmath593 and @xmath359 form a @xmath629 subgroup of @xmath549 , and the matrices @xmath616 can be conjugated by such unitaries . in the basis where @xmath630 as in , the spectrally - flattened hamiltonian is @xmath631 with @xmath632 we recognise as a parametrisation of @xmath457 . thus @xmath624 is a map from @xmath633 , from which we can associate a gerbe and analyse the gerbe extension problem to @xmath3 , using the semimetal mv - sequence as in section [ sec : extension ] . gauge transformations live in @xmath634 , and they effect @xmath635 for @xmath636 . the roles of @xmath593 and @xmath618 can be exchanged , in the sense that we can write @xmath637 and @xmath638 instead . from this point of view , @xmath618 is the difference of the spectral projections @xmath639 onto the positive and negative eigenbundles @xmath640 ( over @xmath66 ) . since @xmath641 , @xmath620 gives an isomorphism between @xmath642 and @xmath643 , c.f . section iv of @xcite . there is another interesting class of @xmath509 bloch hamiltonians in 5d : those _ quadratic _ , or _ bilinear _ , in @xmath51 . as explained in section [ sec : dirac5d ] , a quaternionic structure @xmath517 singles out @xmath534 in @xmath432 as the @xmath517-commuting ( traceless ) hamiltonians , and we identify @xmath534 with the tangent bundle @xmath72 of @xmath3 . furthermore , the @xmath517-_anticommuting _ hamiltonians form a complementary rank-10 subbundle @xmath644 to @xmath534 in @xmath432 , also with structure group @xmath535 . in terms of the @xmath51-matrices , an orthonormal basis for @xmath644 is given by @xmath645 where @xmath646 is a multi - index and e.g. @xmath647 $ ] . we should think of @xmath648 as coming from @xmath649 , just as @xmath372 came from an orthonormal frame @xmath650 . in accordance with terminology in the physics literature , elements of @xmath644 are said to be particle - hole symmetric , with @xmath651 . thus @xmath405-symmetric hamiltonians are exactly the `` @xmath51-bilinear '' ones they are ( @xmath652 times of ) quantizations of @xmath653 , and we will refer to such hamiltonians simply as _ bilinear hamiltonians_. * hamiltonians parametrised by a pair of vector fields . * for example , a _ pair _ of vector fields @xmath654 defines an abstract hamiltonian @xmath655 $ ] in @xmath656 . this is represented on @xmath142 as the hamiltonian @xmath657={\text{i}}({\bm{{a}}}\wedge{\bm{{b}}})_i\gamma_i$ ] . note that @xmath658 is geometrically just an oriented area element , and is invariant under rotations in the plane of @xmath659 . furthermore , @xmath660 iff @xmath659 are linearly dependent . following @xcite , a pair of vector fields @xmath654 over @xmath3 which are linearly independent everywhere is called a _ tangent @xmath348-field _ ; if @xmath654 is defined only on the complement of a finite number of points @xmath47 , it is called a _ tangent @xmath348-field over @xmath3 with finite singularities _ , or a _ tangent @xmath348-field over @xmath66_. [ rmk : twoterm ] the general element of @xmath661 is not a simple tensor @xmath658 but a sum of two such tensors @xmath662 , which can be taken to be orthogonal to each other this is a linear algebraic fact about canonical forms of antisymmetric matrices @xmath663 thus we can more generally consider @xmath664 for mutually orthogonal pairs of vector fields @xmath654 and @xmath665 . interestingly , the spectrum of @xmath51-bilinear hamiltonians @xmath664 can be found in much the same way as the linear ones , utilising only the clifford algebra relations . let @xmath664 be a bilinear hamiltonian as in remark [ rmk : twoterm ] , and let @xmath666 . then @xmath667 , and all four signs occur . let @xmath668 . at the clifford algebra level , @xmath669 thus , the square of @xmath670 is @xmath671 . if @xmath672 or @xmath673 is zero , we are done . otherwise , @xmath674 for some positively - oriented orthonormal frame @xmath675 , where we have suppressed the wedge product notation and used the chirality condition @xmath676 . since @xmath677 has spectrum @xmath418 with each doubly - degenerate , it follows that @xmath678 and @xmath679 , where we have suppressed the dependence on @xmath56 . the zeroes of @xmath34 are easily read off , and there are two types : ( 1 ) when @xmath680 , and ( 2 ) when @xmath681 . the former involves a four - fold degenerate zero , so all four bands cross , whereas the latter involves crossing between only a pair of bands ( fig . [ fig : four - band - crossing ] ) . -quadratic hamiltonians . if @xmath682 , the spectrum degenerates to the dotted blue lines , which are each 2-fold degenerate.,scaledwidth=60.0% ] when @xmath673 is zero , the four - band spectrum becomes a pair of doubly - degenerate bands , and these degenerate pairs cross exactly at the points where @xmath683 , much like the spectrum of the dirac - type hamiltonians which are linear in @xmath51 . we are interested in such `` bilinear hamiltonians '' , and they are precisely the pure tensor ones @xmath684={\text{i}}\sum_i ( { \bm{{a}}}(k)\wedge{\bm{{b}}}(k))_i\gamma_i$ ] , specified by a _ single _ pair of vector fields @xmath685 , with @xmath686 . we will _ assume _ that the singularities ( points where @xmath658 vanishes ) form a finite set of isolated points @xmath47 . in other words , @xmath654 defines a tangent 2-field over @xmath66 . at a singularity @xmath58 , there may be topological obstructions to extending @xmath654 across @xmath85 , in which case , we can not open up a gap at the four - band crossing at @xmath85 by simply deforming @xmath672 into a strictly positive function . despite the spectrum resembling that of a ( linear ) dirac - type hamiltonian , this type of four - band crossing can be topologically protected by a very different and much more subtle mechanism , see section [ sec : atinvariant ] . we may also consider bilinear hamiltonians @xmath687 specified by a pair of vector fields @xmath654 over a @xmath4-manifold with @xmath688 . however , it is no longer the case that a @xmath405-symmetry singles out such hamiltonians , and one needs to motivate such model hamiltonians in a different way . they may nevertheless be studied as a toy model . a detailed study of tangent 2-fields with finite singularities was carried out in @xcite , where analogues of the poincar hopf theorem for vector fields were obtained . for this , we need the notion of the local index at a singularity of a tangent 2-field @xmath654 on @xmath66 . let @xmath689 be a small oriented 4-sphere surrounding @xmath58 , then @xmath654 gives a map @xmath690 to the non - compact stiefel manifold @xmath691 of 2-frames in @xmath692 . there is a homotopy equivalence between @xmath693 and the ( compact ) stiefel manifold @xmath694 of orthonormal 2-frames in @xmath692 . the _ local index of @xmath654 at @xmath85 _ is defined to be the homotopy class of @xmath695 in @xmath696 , and is the local obstruction to deforming @xmath654 to remove the singularity at @xmath85 . recall from the previous subsection the construction of a bilinear dirac hamiltonian @xmath687 from a tangent 2-field @xmath654 over @xmath3 with finite singularities @xmath47 , and that four - band crossings occur precisely at @xmath47 . the local index at @xmath85 of @xmath654 has the physical interpretation as a @xmath2 local topological charge protecting the band crossing @xmath58 . to understand @xmath697 better , recall the fibration @xmath698 where the @xmath699 base parametrises the choice of a first normalised vector @xmath700 , and the fiber parametrises ( locally ) the choice of @xmath701 orthonormal to @xmath700 . from the homotopy exact sequence , @xmath702 we see that @xmath703 comes from the inclusion of the fiber @xmath704 which has the famous @xmath705 . thus , we can construct an explicit example of a @xmath2-protected four - band crossing as follows . choose some local euclidean coordinates @xmath371 for a neighbourhood @xmath706 of @xmath85 centred at 0 , and a trivialization @xmath707 of the tangent bundle above @xmath706 . take @xmath708 , and for @xmath709 , we choose @xmath701 orthonormal to @xmath700 by letting @xmath710 be given by a generator of @xmath711 . extend to orthogonal vectors @xmath712 to all of @xmath706 by scaling by @xmath713 . thus @xmath714 on @xmath715 and @xmath716 . then @xmath85 is a four - band crossing with non - trivial @xmath2-charge . by taking @xmath717 we obtain a _ _ 2-plane field__-distributions _ , see @xcite for a guide to @xmath371-distributions on manifolds . ] defining for each @xmath718 a point in the oriented grassmannian @xmath719 of 2-planes in @xmath692 . for a 2-plane field over @xmath66 , there is similarly a map @xmath720 , and a local index can be defined at @xmath58 . note that @xmath721 , and at the level of @xmath722 the exact sequence @xmath723 says that the index is in @xmath724 . although we can consider bilinear hamiltonians specified by a field of oriented area elements , i.e. a 2-plane field along with areas ( magnitudes ) , which will also have @xmath2 local indices , we only consider those specified by a tangent 2-field ( with finite singularities ) in this paper . for tangent 2-fields in @xmath77-dimensions , the local topological index at a point singularity is defined similarly , as an element of @xmath725 . these homotopy groups are known to be @xmath2 when @xmath4 is odd and @xmath726 when @xmath4 is even @xcite . unlike the case of vector fields , the global constraint for these indices depends on @xmath4 ( mod 4 ) @xcite and in the following subsection , we will restrict to @xmath727 ( mod 4 ) . for this section , we assume that @xmath3 is a compact oriented @xmath4-manifold @xmath3 with @xmath727 ( mod 4 ) and @xmath728 . most of the constructions of section [ sec : euler ] can be carried out for @xmath2 coefficients , and there is also a generalisation of the poincar hopf theorem for tangent 2-fields and 2-plane fields with finite singularities @xcite . this suggests that hamiltonians parametrised by a tangent 2-field , such as the 5d @xmath2-dirac semimetals introduced in section [ sec : z2invariant ] , can have topologically protected crossings which are furthermore constrained by the global atiyah dupont condition . if @xmath729 , we expect there to be `` @xmath2-charge cancellation '' and also a notion of `` @xmath2-fermi arcs '' . let us simplify notation and write @xmath730 for the tangent 2-field @xmath654 over @xmath66 , and @xmath731 for its local @xmath2 charge at @xmath58 . the sum @xmath732 is a global obstruction to deforming @xmath733 to a tangent 2-field on all of @xmath3 . in @xcite , atiyah dupont showed that @xmath734 is equal to the _ _ real kervaire semicharacteristic__. ] @xmath735 , defined as @xmath736 thus @xmath737 for any @xmath730 . this is not a vacuous result since conditions on @xmath3 , under which tangent 2-fields with finite singularities exist , are known , see @xcite . in particular , it is sufficient that the stiefel whitney class @xmath738 vanishes ( e.g. @xmath71 ) . although we have assumed the bilinear hamiltonians to be specified by tangent 2-fields , we could also work more generally with 2-fields for other vector bundles using a generalisation of the atiyah dupont theory in @xcite . let @xmath3 be a compact oriented @xmath4-manifold @xmath3 with @xmath727 ( mod 4 ) and @xmath729 . for a 0-chain @xmath739 defined on @xmath47 and with total weight zero , a _ kervaire chain _ for @xmath247 is a 1-chain @xmath740 such that @xmath741 , thus @xmath255\in h_1(t , w,{{\mathbb z}}_2)$ ] . we write @xmath742 for the subset of homology classes of kervaire chains for @xmath247 . the @xmath2-versions of the semimetal mv sequence and the dual homology sequence are @xmath306&h^{d-1}(t,{{\mathbb z}}_2)\ar[d]_{\text{pd}}\ar[r ] & h^{d-1}(t\setminus w,{{\mathbb z}}_2)\ar[d]_{\text{pd}}\ar[rr]^{\beta } & & h^{d-1}(s_w,{{\mathbb z}}_2)\ar[d]_{\text{pd}}\ar[rr]^{\sigma } & & h^d(t,{{\mathbb z}}_2)\ar[d]_{\text{pd}}\ar[r ] & 0\\ 0\ar[r ] & h_1(t,{{\mathbb z}}_2)\ar[r ] & h_1(t , w,{{\mathbb z}}_2)\ar[rr]^{\partial } & & h_0(w,{{\mathbb z}}_2 ) \ar[rr]^{\sigma } & & h_0(t,{{\mathbb z}}_2)\ar[r ] & 0 } .\nonumber\ ] ] as before , we define @xmath743 to be the poincar dual image of @xmath742 . we will then explain how a tangent 2-field with finite singularity set @xmath47 has a topological invariant in @xmath744 and therefore a kervaire chain representation . let @xmath3 be a compact oriented @xmath4-manifold @xmath3 with @xmath727 ( mod 4 ) and @xmath729 , and @xmath228 be the sphere bundle for its tangent bundle . a _ kervaire structure _ for @xmath3 is a class in @xmath745 whose restriction to each fibre @xmath746 generates @xmath747 . define @xmath748 to be the set of kervaire structures for @xmath3 . from the gysin sequence with @xmath2-coefficients , we see that @xmath749 acts on @xmath748 freely and transitively by pullback and addition . let @xmath750 be the bundle of stiefel manifolds @xmath751 associated to the orthonormal frame bundle of @xmath3 ( endowed with some riemannian metric ) . corresponding to the fibration @xmath752 , there is a fibration @xmath753 a ( non - singular ) tangent 2-field @xmath730 on @xmath3 gives rise to a kervaire structure as follows . let @xmath754 denote the section of @xmath755 determined by @xmath730 , which is a @xmath4-cycle on @xmath755 . the poincar dual is a @xmath756-cocycle on @xmath755 , and pushforward under @xmath757 gives a @xmath86-cocycle in @xmath228 . furthermore , the construction of the cocycle is such that its restriction to @xmath231 generates @xmath747 for each @xmath56 . now consider tangent 2-fields on @xmath3 with finite singularities @xmath47 . as in section [ sec : semimetaleulerchain ] , we can define a local charge operator @xmath758 . then for each 0-chain @xmath247 with total charge zero , we define @xmath759 to be the subset of @xmath760 whose restriction to each fibre over @xmath718 generates @xmath747 , and whose local charges is @xmath307 . each @xmath759 is an affine space for @xmath749 . suppose @xmath730 is a tangent 2-field on @xmath66 with local charge 0-chain @xmath247 , then it determines a @xmath4-cycle in @xmath761 relative to the boundary @xmath762 , which poincar dualises to a @xmath756-cocycle on @xmath761 . pushforward under @xmath757 gives a @xmath86-cocycle on @xmath340 representing a class in @xmath759 . fix a reference non - singular vector field @xmath218 , then @xmath763 identifies @xmath759 with @xmath744 . the following diagram summarises the various affines spaces for @xmath764 : @xmath765 \ar[d ] & & { \mathfrak{kerv}}(t,{\mathcal{w}})\ar[ll ] \ar[d ] & \subset \quad\quad\;\ ; h_1(t , w,{{\mathbb z}}_2 ) \\ & h^{d-1}(t,{{\mathbb z}}_2)\ar[rr]\ar[d ] \ar[u ] & & { \mathfrak{cokerv}}(t,{\mathcal{w}})\ar[ll ] \ar[d]\ar[u ] & \subset \quad h^{d-1}(t\setminus w,{{\mathbb z}}_2 ) \\ z}}_2)\quad\supset & \widetilde{{\mathfrak{cokerv}}}(t)\ar[rr ] \ar[u ] & & \widetilde{{\mathfrak{cokerv}}}(t,{\mathcal{w}})\ar[ll ] \ar[u ] & \subset \quad h^{d-1}({\mathcal{s}}|_{t\setminus w},{{\mathbb z}}_2 ) } .\nonumber\ ] ] let @xmath34 be a bilinear hamiltonian specified by a tangent 2-field @xmath730 over a compact oriented 5-manifold @xmath3 with @xmath729 , with finite singularities @xmath47 . it has a cohomological topological invariant @xmath766\in h^4(t\setminus w,{{\mathbb z}}_2)$ ] defined as in the previous paragraph , and its _ kervaire chain representation _ is the poincar dual @xmath767\in h_1(t , w,{{\mathbb z}}_2)$ ] . the bulk - boundary correspondence proceeds exactly as before , taking a kervaire chain for @xmath247 to a `` @xmath2-fermi arc '' with topological invariant in @xmath768 . unlike the usual @xmath104-fermi arcs , the @xmath2-fermi arcs cancel in pairs so that there is at most one arc between two projected band crossings , and the arc never fully winds around a 1-cycle in @xmath769 . the topological semimetal - topological insulator relationship encoded in the semimetal mv sequence can , in principle , be refined using an mv sequence in _ differential cohomology _ @xcite . another refinement comes from additional spatial symmetries . for example , a spatial inversion symmetry @xmath419 induces an action on the brillouin torus @xmath3 under which @xmath47 should be invariant . the same action arises if @xmath359 rather than @xmath770 symmetry is present . the vector field @xmath29 needs to be equivariant , and the local index at @xmath58 is an equivariant degree . finally , the @xmath1-theory mv sequence can be used ( with @xmath66 replaced by @xmath771 ) to analyse stable topological semimetal invariants . this paves the way for the development of a new type of t - duality transformation @xmath772 , which would provide a t - dual picture of semimetals along the lines of @xcite in the case of insulators . we end by posing the question : associated to the mv sequence analysis in the case of kervaire semimetals ( section [ sec : kervairestructures ] ) , is there a geometric extension problem ? _ acknowledgements_. we thank keith hannabuss for some helpful discussions , and siye wu for his useful criticisms . this work was supported by the australian research council via arc discovery project grant dp150100008 and arc decra grant de170100149 . to be self - contained , we recall here some universal constructions of geometric objects with connections that are used in the main text . + * the hopf line bundle over @xmath40 . * ( sections [ sec : physback ] and [ sec : extension ] ) recall that the 2d - sphere @xmath40 is defined as the space of complex lines in @xmath773 . the hopf line bundle @xmath144 over @xmath40 is defined as the sub - bundle of the trivial bundle @xmath774 given by @xmath775 as a sub - bundle of the trivial bundle @xmath774 ( which has the obvious hermitian metric and unitary connection ) @xmath144 comes equipped with a hermitian metric and unitary connection @xmath776 . explicitly , a natural projection defining @xmath144 is given by @xmath777 , where @xmath778 is the unit vector in @xmath464 and @xmath779 are the pauli matrices , @xmath780 for @xmath781 . then @xmath782 and the curvature is @xmath783 * the basic gerbe over @xmath785 . * ( sections [ sec : semimetalgerbe ] and [ sec : extension ] ) we identify @xmath182 and consider @xmath786 defined by @xmath787 then the projection to the 2nd factor is a submersion @xmath788 and the fibred product @xmath789 } = \{(z_1 , z_2 , g ) \in \text{u}(1)^2 \times \text{su}(2 ) \big| z_1 , z_2 \ne 1 , z_1 , z_2 \notin { \rm spec}(g)\}.\ ] ] for @xmath790 , let @xmath791 denote the @xmath672 eigenspace of @xmath377 and @xmath792 where @xmath672 is an eigenvalue lying in the open arc component in @xmath793 joining @xmath794 that does not contain @xmath795 . let @xmath796 denote the determinant line , which is the highest exterior power . then @xmath797 is a bundle gerbe description of the basic gerbe over @xmath182 , see @xcite , @xcite . it comes with gerbe connection whose 3-curvature is the wess witten ( wzw ) integrand @xmath798 and whose dixmier - douady number classifying the gerbe up to equivalence is , @xmath799 + * the quaternionic hopf line bundle over @xmath186 . * ( sections [ sec:5dsemimetal ] and [ sec : extension ] ) recall that the 4d - sphere @xmath186 is defined as the space of quaternionic lines in @xmath528 and that the quaternionic hopf line bundle @xmath185 over @xmath186 is defined as the sub - bundle of the trivial bundle @xmath800 given by @xmath801 as a sub - bundle of the trivial bundle @xmath802 ( which has the obvious hermitian metric and unitary connection ) @xmath185 comes equipped with a hermitian metric and unitary connection @xmath803 . explicitly , a natural projection defining @xmath185 is given by @xmath804 , where @xmath805 is the unit vector in @xmath692 and @xmath806 are gamma matrices , with @xmath807 for @xmath808 generating the clifford algebra and @xmath526 . then @xmath809 and the curvature is @xmath810 the 2nd chern number classifying @xmath185 up to isomorphism is , @xmath811 + * the stiefel manifold @xmath812 . * ( section [ sec : torsionsemimetal ] ) it can be defined as the space of all orthonormal 2-frames in @xmath692 . as a homogeneous space , @xmath813 , where we notice that it is a compact , connected , oriented @xmath814-dimensional manifold , which can be identified with the unit sphere @xmath815 subbundle of the tangent bundle of the sphere @xmath699 . so @xmath812 is a non - principal @xmath816-bundle over @xmath699 with euler class equal to @xmath348 . the pontryagin number of the bundle @xmath817 is just the pontryagin number of @xmath699 , which is zero since the sphere @xmath699 is the oriented boundary of the @xmath43 dimensional ball . by @xcite , the euler and pontryagin numbers together determine the bundle up to isomorphism . via the long exact sequence for the fibration @xmath817 , we see that there is a natural isomorphism , ( this is related to @xcite ) @xmath818 the generator of @xmath819 determines the non - trivial @xmath785-bundle over @xmath577 via the clutching construction , @xmath820 by the pontryagin thom construction , @xmath821 comes from @xmath822 and so relates to the t hooft polyakov @xmath453 monopole @xcite . bradlyn , b. , cano , j. , wang , z. , vergniory , m.g . , felser , c. , cava , r.j . , bernevig , b.a . : beyond dirac and weyl fermions : unconventional quasiparticles in conventional crystals . science * 353*(6299 ) ( 2016 ) korba , j. : distributions , vector distributions , and immersions of manifolds in euclidean spaces . in : handbook of global analysis ( edited by d. krupka and d. saunders ) , elsevier science , amsterdam , pp . 665724 ( 2008 ) turner , a.m. , vishwanath , a. : beyond band insulators : topology of semimetals and interacting phases . in : m. franz and l. molenkamp ( eds . ) , contemp . concepts cond . mat . * 6 * , _ topological insulators _ , pp . 293324 , elsevier , amsterdam ( 2013 ) wigner , e.p . : unitary representations of the inhomogeneous lorentz group including reflections . in group theoretical concepts in elementary particle physics , ( ed . f. grsey ) , vol . 3780 , gordon and breach , new york , ( 1964 )

the subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields . part of this story is the relationship between cohomological semimetal invariants , euler structures , and ambiguities in the torsion of manifolds . dually , semimetal invariants can be represented by euler chains from which the surface fermi arc connectivity can be deduced . these dual pictures , and the link to insulators , are organised using geometric exact sequences . we go beyond dirac - type hamiltonians and introduce new classes of semimetals whose local charges are subtle atiyah dupont thomas invariants globally constrained by the kervaire semicharacteristic , leading to the prediction of torsion fermi arcs .

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the classical problems of statistical physics , and physical chemistry , is to find a macroscopic statistical description for the diffusion of a dissolved molecule or ion in a uniform fluid solvent . the microscopic state of such a system has very many degrees of freedom , possibly even more than an avogadro number of degrees of freedom . this gives rise to an equally large number of coupled equations of motion . it is completely impractical to solve such a large system with rigor . we must abandon the idea of an exact solution . we are forced to use a statistical description , where we can only describe the probability of certain events . we denote the probability of finding a brownian particle at a certain point on space , @xmath0 , and time , @xmath1 , by @xmath2 . the time - evolution of @xmath3 is governed by a partial differential equation called the fokker - planck equation : @xmath4 the functions @xmath5 and @xmath6 are referred to as the infinitesimal first and second moments of diffusion . in practice , the infinitesimal second moment does sometimes depend on concentration of the solute , @xmath3 , but is usually regarded as constant and is called the `` fick s law constant . '' a typical value ( for a hydrated sodium ion in water ) would be of the order @xmath7 . the infinitesimal first moment depends on the magnitude of externally imposed forces and on the mobility of the brownian particle which is given by @xmath8 where @xmath9 is the electrical charge on the particle , @xmath10 is the kinematic viscosity of the solvent and @xmath11 is the effective radius of the particle . a typical value for the mobility ( of a hydrated sodium ion in water ) would be @xmath12 . further descriptions and numerical data may be found in books on physical chemistry and statistical physics @xcite . if we apply an electrical potential , or voltage , of @xmath13 then the infinitesimal first moment is given by @xmath14 the theory behind equations [ eq : mobility ] and [ eq : infinitesimal_first_moment ] is due to stokes and einstein @xcite . more information about the methods of solution and the applications of the fokker - planck equation can be found in risken @xcite . when we take into account the functional forms of @xmath15 and @xmath16 then we can rewrite the fokker - planck equation as : @xmath17 this is the form of the fokker - plank equation which we will sample at regular intervals in time and space , to yield finite difference equations . many partial differential equations , or pdes , including equation [ eq : expanded_fokker_planck ] , can be very difficult to solve analytically . one well established approach to this problem is to sample possible solutions to a pde at regular intervals , called mesh points @xcite . the true solution is approximated locally by a collocating polynomial . the values of the derivatives of the true solution are approximated by the corresponding derivatives of the collocating polynomial . we can define local coordinates , expanded locally about a point @xmath18 we can map points between a real space @xmath19 and an integer or discrete space @xmath20 . time , @xmath1 , and position , @xmath0 , are modelled by real numbers , @xmath21 and the corresponding sampled position , @xmath22 , and sampled time , @xmath23 , are modelled by integers @xmath24 . we sample the space using a simple linear relationship @xmath25 where @xmath26 is the sampling length and @xmath27 is the sampling time . in order to map equation [ eq : expanded_fokker_planck ] into discrete space , we need to make suitable finite difference approximations to the partial derivatives . the notation is greatly simplified if we define a family of difference operators : @xmath28 in principle , this is a doubly infinite family of operators but in practice we only use a small finite subset of these operators . this is determined by our choice of sampling points . this choice is not unique and is not trivial . the set of sampling points is called a `` computational molecule @xcite . '' some choices lead to over - determined sets of equation with no solution . some other choices lead to under - determined sets of equations with infinitely many solutions . we chose a computational molecule called `` explicit '' computation with the following sample points : @xmath29 . we also need to make a choice regarding the form of the local collocating polynomial . this is not unique and inappropriate choices do not lead to unique solutions . a polynomial which is quadratic in @xmath0 and linear in @xmath1 is the simplest feasible choice : @xmath30 where @xmath31 , @xmath32 and @xmath33 are the real coefficients of the polynomial . equations [ eq : sample ] , [ eq : difference ] and [ eq : polynomial ] imply a simple system of linear equations that can be expressed in matrix form : @xmath34 \left [ \begin{array}{c } a_1 \\ a_2 \\ b_1 \end{array } \right ] = \left [ \begin{array}{c } \delta_{-1,-1 } \\ \delta_{0,-1 } \\ \delta_{+1,-1 } \end{array } \right]~. \label{eq : finite_difference_conversion}\ ] ] these can be solved algebraically , using cramer s method to obtain expressions for @xmath31 , @xmath32 and @xmath33 : @xmath35 and @xmath36 and @xmath37 these are all intuitively reasonable approximations but their choice is not arbitrary . equations [ eq : a_1 ] , [ eq : a_2 ] , [ eq : b_1 ] form a complete and consistent set . we could not change one without adjusting the others . we can evaluate the derivatives of equation [ eq : polynomial ] to obtain a complete and consistent set of finite difference approximations for the partial derivatives : @xmath38 and @xmath39 and @xmath40 we can apply the same procedure to @xmath5 to obtain @xmath41 equations [ eq : dpdz ] , [ eq : d2pdz2 ] , [ eq : dpdt ] and [ eq : dalphadz ] can be substituted into equation [ eq : expanded_fokker_planck ] to yield the required finite partial difference equation : @xmath42 where @xmath43 and @xmath44 and @xmath45 we can overload the arguments of @xmath46 and write them in terms of the discrete space @xmath20 using the mapping defined in equation [ eq : sample ] . @xmath47 the meaning of the arguments should be clear from the context and from the use of subscript notation , @xmath48 , rather than function notation , @xmath49 . equation [ eq : integer_partial_difference_equation ] is precisely the form required for parrondo s games . in the original formulation , the conditional probabilities of winning or losing depend on the state , @xmath22 , of capital but not on any other information about the past history of the games : * game a is a toss of a biased coin : @xmath50 where @xmath51 is an adverse external bias that the game has to `` overcome '' . this bias , @xmath51 , is typically a small number such as @xmath52 , for example @xcite . * game b depends on the capital , @xmath22 : + if @xmath53 : : , then the odds are unfavorable . @xmath54 if @xmath55 : : , then the odds are favorable . @xmath56 it is straightforward to simulate a randomized sequence of these games on a computer using a very simple algorithm @xcite . we can write the requirements for game a in the form of equation [ eq : integer_partial_difference_equation ] . @xmath57 this implies a constraint that @xmath58 which implies that @xmath59 which defines the relative scales of @xmath26 and @xmath27 so we can give it a special name : @xmath60 the constraints on @xmath61 and @xmath62 imply a value for parrondo s `` @xmath51 '' parameter : @xmath63 which can be related back to an externally imposed electric field , @xmath64 using equations [ eq : mobility ] and [ eq : infinitesimal_first_moment ] : @xmath65 the small bias , @xmath51 , is proportional to the applied external field which justifies parrondo s original intuition . there is still zero probability of remaining in the same state which implies a constraint that @xmath58 which implies that we still have the same scale , @xmath66 . if we are in state @xmath22 then we can denote the probability of winning by + @xmath67 . we can write the difference equations for game b in the form : @xmath68 which , together with equations [ eq : am1 ] , [ eq : a0 ] and [ eq : ap1 ] , gives @xmath69 which implies that @xmath70 this can be combined with equation [ eq : infinitesimal_first_moment ] and then directly integrated to calculate the required voltage profile . we can approximate the integral with a riemann sum : @xmath71 so we can construct the required voltage profile for the ratchet which means that , given the values of @xmath72 , it is possible to construct a physical brownian ratchet that has a finite difference approximation which is identical with parrondo s games . we can conclude that parrondo s games are literally a finite element model of a flashing brownian ratchet . we note that game b , as defined here , is quite general and actually includes game a as a special case . we would like to think that as long as @xmath73 is preserved then the solution to the finite partial difference equation [ eq : integer_partial_difference_equation ] would converge to the true solution of the partial differential equation [ eq : expanded_fokker_planck ] , as the mesh size , @xmath26 goes to zero . fortunately , there is a theorem due to obrien , hyman and kaplan @xcite which establishes that the numerical integration of a parabolic pde , in explicit form , will converge to the correct solution as @xmath74 and @xmath75 provided @xmath76 . similar results may also be found in standard texts on numerical analysis @xcite . we see that parrondo s choice of diffusion operator , with @xmath66 is at the very edge of the stable region . there is a possible range of values for @xmath77 . as @xmath78 we require the time step @xmath75 which means that the number of time steps required to simulate a given time interval , @xmath79 , increases without bound @xmath80 . it is computationally infeasible to perform simulations with very small values of @xmath77 . on the other hand , the value of @xmath81 implied in parrondo s games is at the very limit of stability . in fact , the presence of small roundoff errors in the arithmetic could cause the the discrete simulation to diverge significantly from the continuous solution . we propose that choosing @xmath82 , in the middle of the feasible range , is most appropriate . if we consider the case of pure diffusion , with @xmath83 , then equation [ eq : integer_partial_difference_equation ] reduces to @xmath84 and if we choose @xmath82 then this reduces to @xmath85 which is the same as pascal s triangle with every second row removed . the solution to the case where the initial condition is a kronecker delta function , @xmath86 is easy to calculate : @xmath87 which is a half period , or double frequency , binomial . we can invoke the laplace and de moivre form of the central limit theorem which establishes a correspondence between binomial ( or bernoulli ) distribution and the gaussian distribution to obtain @xmath88 this expression is only approximate but is true in the limiting case as @xmath89 . in the case where @xmath90 ; the fokker planck equation [ eq : expanded_fokker_planck ] reduces to a diffusion equation : @xmath91 einstein s solution to the diffusion equation is a gaussian probability density function : @xmath92 where the variance , @xmath93 , is a linear function of time : @xmath94 it is possible to verify that this is a solution by direct substitution : @xmath95 if we sample this solution in equation [ eq : gaussian ] using the mapping in equation [ eq : sample ] then we obtain equation [ eq : laplace_demoivre ] again . this is an exact result . we conclude that the choice of @xmath96 is very appropriate for the solution to the diffusion equation . we suggest that this would also be true for the fokker - planck equation , in the case where @xmath16 is `` small . '' the appropriate choice of @xmath77 , given arbitrarily large , or rapidly varying , @xmath16 is still an unsolved problem . in general , we would expect that much smaller values , @xmath97 , would be needed to accommodate more extreme choices of alpha . we simulated a physically reasonable ratchet with a moderately large modulo value , @xmath98 . ( the value for the original parrondo s games was @xmath99 . ) we used the value of @xmath82 . the simulation was based on a direct implementation of equation [ eq : integer_partial_difference_equation ] in matlab . we chose a sampling time of @xmath100 and a sampling distance of @xmath101 . the result is shown in figure [ fig : simulation ] , where we indicate how the expected position of a particle can move within a brownian flashing ratchet during four cycles of the modulating field . we can see a steady drift of the mean position of the particle in response to the ratchet action . this simulation includes a total of 500 time samples . note that the average rate of transport quickly settles down to a steady value , even after only four cycles of the ratchet . we acknowledge the similar , but independent , work of heath @xcite et al . the focus of our paper is different . we seek to establish the physical , and mathematical , basis of parrondo s games and to derive a practical numerical technique for simulation . we conclude that parrondo s games _ are _ a valid finite - element simulation of a flashing brownian ratchet , which justifies parrondo s original intuition . we have established that parrondo s `` @xmath51 '' parameter is a reasonable way to simulate a gradual externally imposed electric field , or voltage gradient . we have established that parrondo s implied choice of the @xmath77 parameter does lead to a stable simulation but we suggest that the choice of @xmath82 is more appropriate from a mathematical point of view . finally , we have generalised parrondo s games , in the form of a set of finite difference equations [ eq : integer_partial_difference_equation ] and we have shown that these can be implemented on a computer . j. s. bader , r. w. hammond , s. a. henk , m. w. deem , g. a. mcdermott , j. m. bustillo , j. w. simpson , g. t. mulhern and j. m. rothberg , _ dna transport by a micromachined brownian ratchet device _ , _ pnas _ * 96 * \ { 23 } ( 1999 ) 1316513169 . d. heath , d. kinderlehrer , kowalczyk _ discrete and continuous ratchets : from coin toss to molecular motor _ , _ website : http//math.smsu.edu / journal _ _ discrete and continuous dynamical systems series b _ * 2 * \ { 2 } ( 2002 ) 153167 .

several authors @xcite have implied that the original inspiration for parrondo s games was a physical system called a `` flashing brownian ratchet @xcite '' the relationship seems to be intuitively clear but , surprisingly , has not yet been established with rigor . the dynamics of a flashing brownian ratchet can be described using a partial differential equation called the fokker - planck equation @xcite , that describes the probability density , of finding a particle at a certain place and time , under the influence of diffusion and externally applied fields . in this paper , we apply standard finite - difference methods of numerical analysis @xcite to the fokker - planck equation . we derive a set of finite difference equations and show that they have the same form as parrondo s games . this justifies the claim that parrondo s games are a discrete - time , discrete - space version of a flashing brownian ratchet . parrondo s games , are in effect , a particular way of sampling a fokker - planck equation . our difference equations are a natural and physically motivated generalisation of parrondo s games . we refer to some well established theorems of numerical analysis to suggest conditions under which the solutions to the difference equations and partial differential equations would converge . the diffusion operator , implicitly assumed in parrondo s original games , reduces to the schmidt formula for the integration of the diffusion equation . there is actually an infinite continuum of possible diffusion operators . the schmidt formula is at one extreme of the feasible range . we suggest that an operator in the middle of the feasible range , with half - period binomial weightings , would be a better representation of the underlying physics . physical brownian ratchets have been constructed and have worked @xcite . it is hoped that the finite element method presented here will be useful in the simulation and design of flashing brownian ratchets .

You are an expert at summarizing long articles. Proceed to summarize the following text: galactic winds are observed to be ubiquitous in galaxies that have recently experienced significant amounts star formation ( see e.g. , * ? ? ? * for a review ) . these outflows represent a fundamental part of galaxy formation models , because the absence of outflows galaxy star formation rates ( sfrs ) are much higher than those observed ( e.g. , * ? ? ? * ) and baryon fractions in the disk are close to the universal value ( e.g. , * ? ? ? * ) , much higher than inferred from observations . in contrast , models that include a variety of feedback effects predict much lower sfrs and baryon fractions . additionally , outflows are required to drive metal - enriched gas out of galaxies , as suggested by both observational ( e.g. * ? ? ? * ) and theoretical ( e.g. * ? ? ? * ) work . however , despite their key role in galaxy formation , the exact processes driving winds remain an open question . plausible driving mechanisms include core collapse supernovae ( sn , * ? ? ? * ) and radiation pressure @xcite . sn - driven winds are now routinely included in semi - analytic and numerical simulations . however , it has long been known that in the disk of the galaxy there is a rough equipartition of the magnetic and cosmic ray ( cr ) energy densities ( e.g * ? ? ? this indicates that crs play a significant role in dynamics of interstellar medium . only relatively recently have the effects of crs have been considered in the context of galaxy formation ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) and galaxy cluster @xcite simulations . a tight link between crs and star formation is evidenced by the correlation between a galaxy s infra - red luminosity , closely related to its sfr , and the luminosity of its radio halo ( e.g. * ? ? ? * ; * ? ? ? the relationship is almost linear , has very little scatter , and does not evolve with redshift @xcite , indicating that the coupling between star formation and crs is robust over a wide variety of conditions . although the energy injection rate of crs is small when compared to the other sources of energy from star formation , the rate at which they inject momentum is not @xcite . this is because the crs that supply most of the pressure in the galaxy generate alfvn waves in the ism @xcite , which then scatter the crs with a mean free path of @xmath3pc . crs are self - confined ( e.g. * ? ? ? * ) , and it takes @xmath4myr for a typical cr to escape its parent galaxy . theoretical models of dynamical haloes in which crs diffuse and are advected out in a galactic wind predict steady , supersonic galaxy - scale outflows driven by a combination of cr and thermal pressure @xcite . in this letter we present high - resolution hydrodynamical simulations of isolated disk galaxies , including a model for the injection , transport and decay of crs , to investigate how outflows are driven by crs and the properties of the outflowing gas . our simulations are performed with the adaptive - mesh - refinement ( amr ) code ramses , described in @xcite . the detailed description of physical processes included in our simulations star formation , radiative cooling , and metal enrichment from type ia sne , type ii sne and intermediate mass stars can be found in @xcite . sn feedback is modelled by injecting a total of @xmath5ergs of thermal energy per sn into the cells neighbouring the star particle . we do not employ any delay of dissipation for the injected energy in these runs ( the runs are equivalent to the energy only run in * ? ? ? a full description of the cr field would require modelling the distribution function of crs as a function of position , momentum and time . however , if the cr mean free path is shorter than the length scale of the problem , the cr field can be described as a fluid @xcite . we thus take the approach of modelling the cr energy density , @xmath6 , as an additional energy field that advects passively with the gas density ( e.g. * ? ? ? * ) and exerts a pressure @xmath7 . thus , the total pressure entering the momentum and energy equations governing gas evolution is @xmath8 . we assume throughout that the cr fluid is an ultra relatavistic ideal gas with @xmath9 . as described above , crs undergo a random walk through the ism after their injection . their evolution is thus a combination of advection with the ambient gas flow and diffusion , which we parametrize by the diffusion coefficient , @xmath10 . the evolution of baryon and cr fluids is thus governed by the standard continuity and momentum equations and the following energy equations : @xmath11 @xmath12 where @xmath13 is gas velocity , @xmath14 , @xmath15 and @xmath16 , @xmath17 are the pressure and internal energy of gas and crs , respectively . the @xmath18 indicates energy injection by sn , and @xmath19 is the fraction of this energy that is injected in the form of crs . @xmath20 indicates radiative cooling of gas , while @xmath21 indicates the heating of gas by both crs and uv radiation . finally , @xmath22 corresponds to energy losses by crs both due to decays and coulomb interactions with gas mediated by magnetic fields ( e.g. * ? ? ? * ; * ? ? ? following @xcite we assume that the cr cooling rate is : @xmath23 where @xmath24 is the local electron number density . the ratio of the catastrophic cooling rate to the couloumb cooling rate for our cr population is 3.55 . some fraction of the energy lost by the cr population heats the thermal gas ( e.g. * ? ? ? * ) at a rate given by @xcite @xmath25 equations [ eq : crcool ] and [ eq : crheat ] are solved on every timestep to calculate the rate of decay of the cr energy density along with the corresponding gain in the gas thermal energy . we have tested our cr implementation using a standard shock - tube test involving gas and cr fluids @xcite and found that results accurately match the analytic solution . results of this and other tests will be presented in a forthcoming paper . strong shock waves associated with sn explosions have long been recognized as a likely source of galactic crs ( e.g. * ? ? ? empirically , in order to match the galactic energy density in crs , sne must be capable of transferring a fraction @xmath26 of the explosion kinetic energy into the form of cr energy @xcite . in our models we make the assumption that a certain fraction , @xmath27 , of the sn energy is injected to the cr fluid energy density . the remaining fraction @xmath28 is injected thermally into the gas field . we note that the assumptions that the diffusion of crs is isotropic and that the diffusion coefficient is a constant are necessary simplification in our models , which track neither the direction nor the strength of the magnetic field . on small scales ( @xmath29100 pc ) , the strength of the random component of the galactic magnetic field is several times higher than the average field strength ( e.g. * ? ? ? * ) because galaxy formation processes ( e.g. supernovae and hydrodynamical turbulence ) in the disk @xcite and the turbulent dynamo effect and cr buoyancy in the halo @xcite tangle the magnetic field to the extent that isotropic diffusion is a good approximation(e.g . codes that assume isotropic diffusion are able to predict cr - emitted spectral data down to the few percent level ( e.g * ? ? ? for the purposes of this exploratory work we employ the isotropic diffusion model , but note that investigation of complex models represents an interesting future direction for this work . we simulate isolated , model galaxies of two different masses representing an smc - sized dwarf galaxy and mw - sized disk galaxy with three different feedback models : no feedback , thermal feedback only , and thermal plus cr feedback . the thermal feedback runs inject @xmath30 of the energy released by each sn blast into the gas thermal energy . the cr feedback runs inject @xmath31 of the sn energy into the gas thermal energy and the remaining @xmath32 into the cr energy density field . every simulation models radiative cooling , star - formation and metal enrichment . all runs are evolved for 0.5gyr and throughout this letter we report results for this time . following @xcite and @xcite the galaxy model consists of a dark matter halo , a stellar bulge and an exponential disc of stars and gas . the dark matter halo is modelled as an nfw halo @xcite . the gas and stars are then initialized into an exponential disk , and the bulge is assumed to have a @xcite profile with a scale length that is 10% of the disk scale length . the relevant parameters for each set of initial conditions are given in table [ tab : ics ] . each simulation is run with a maximum spatial resolution of 75pc ( 37.5pc ) for the mw ( smc ) runs . [ cols= " > , < , > , > , > , > , < , < , < , > , > " , ] + notes : from left to right the columns contain : ( 1 ) simulation set name ; ( 2 ) spherical overdensity dm halo mass defined relative to the 200 times the critical density at @xmath33 ; ( 3 ) circular velocity at the virial radius ; ( 4 ) concentration of nfw halo ; ( 5 ) halo spin parameter ; ( 6 ) disk gas fraction ; ( 7 ) mass of gas in the disk ; ( 8) mass of stars in the disk ; ( 9 ) mass of stars in the bulge ; ( 10 ) scale length of exponential disk ; ( 11 ) scale height of gas disk . + [ tab : ics ] the solid curves show the mass loading factor , @xmath34 , of the galactic wind , defined as the ratio of the sfr to the gas outflow rate , as a function of time ( left - hand axis ) . the dotted curves show the galaxy sfr ( right - hand axis ) . the color of each curve denotes the feedback model and the top ( bottom ) panel shows results for the smc ( mw ) simulation . the no - feedback model ( black curves ) is not shown on the mass - loading plot because there is a net inflow of gas at all times . both feedback models predict mass loadings of @xmath35 for the mw galaxy , but the cr feedback is capable of suppressing the sfr by a larger fraction than the thermal feedback model . in the smc galaxy the cr feedback model is capable of driving galactic winds with large ( @xmath36 ) mass loadings and suppresses the sfr significantly more than thermal feedback alone . + ] we begin by considering the sfrs of the simulated galaxies in fig . [ fig : ml ] . the sfr in simulations without feedback is higher than in simulations with feedback and is higher than typically observed sfrs of galaxies of these sizes . simulations with crs suppress sfr compared to simulations with thermal sn feedback only , especially in the smc - sized galaxy . this is because crs act as a source of pressure in the galaxy disk . this significantly changes the density pdf of the gas in the disk reducing the fraction of mass in star forming regions . outflow efficiency can be parametrized by the mass loading factor , @xmath34 , defined as the ratio of the gas outflow rate to the sfr . the solid curves in fig . [ fig : ml ] show @xmath34 as a function of time for different simulations . outflow rates are measured as the instantaneous mass flux through the plane parallel to the galactic disk at a height of 20kpc . in the mw simulation the mass loading is approximately 0.5 in both simulations , whereas in the smc simulation the mass loading is @xmath37 in the simulation with crs and only @xmath38 in the simulation with thermal feedback only . this indicates that crs greatly enhance efficiency of outflows from dwarf galaxies . velocity of the outflowing gas ( @xmath39 ) as a function of halo circular velocity . the gray points show the observations of @xcite ( downward pointing triangles ) and @xcite ( upward pointing triangles ) . the solid points show simulation predictions . the squares ( circles ) show the mw ( smc ) simulations and the colors denote the feedback model . in both galaxies , the outflows in the cr feedback models ( blue points ) have velocities comparable to the obserations , whereas the thermal feedback models ( red points ) overestimate the wind velocity by a large factor . + ] figure [ fig : vz ] shows velocity of the outflowing gas , @xmath40 , as a function of the circular velocity of the halo , @xmath41 , compared to observations of cool wind gas around dwarf galaxies @xcite and @xmath42 starburst dominated galaxies @xcite . we measure outflow velocities by projecting the gas field perpendicular to the disk and calculating the velocity that contains 90% of the cool ( @xmath43k ) gas . in each galaxy the thermal feedback simulation predicts outflow velocities that are significantly larger than those observed whereas the cr runs are comparable to the observations . finally , fig . [ fig : im ] shows the temperature of the outflowing gas in a thin slice through the centre of the simulated galaxies ( left ) . the notable difference between simulations is that wind in the cr simulation is considerably cooler , especially in the smc simulation . the panels to the right of this figure show the profiles of velocity and outward pressure gradient . the thermal feedback run has winds that accelerate abruptly from the galactic disk up to @xmath44km / s and thereafter have a constant velocity . the cr simulations , however show a wind that accelerates smoothly into the halo . the reason for this is revealed in the right - hand panels , where it is immediately apparent that the pressure gradient in the halo with crs is a factor of 3 - 10 larger in the cr simulation than in the thermal feedback simulation ( the difference is particularly striking in the smc simulation ) . these results illustrate that the wind properties in the simulations with crs are qualitatively different properties to the wind driven by thermal sn feedback . our simulations show that energy injection in the form of crs is a promising feedback process that can substantially aid in driving outflows from star - forming galaxies . first , we find that cr injection can suppress the sfr by providing an extra source of pressure that stabilizes the disk . turbulent and cr pressure are in equipartition in the disk , thus the cr pressure can significantly affect most of the volume of the disk , but will be sub - dominant inside supersonic molecular clouds , where turbulent pressure dominates over both cr and thermal presure , particularly in the dwarf galaxy . the sfrs measured in our galaxies with cr feedback are comparable to observed sfrs for both the mw and the smc . second , we find that addition of the cr feedback increases the mass loading factor , @xmath34 , in the dwarf galaxy by a factor of ten compared to the simulation with sn only feedback . as a result , the smc and mw - sized galaxies ( circular velocities of @xmath45 and @xmath46 km / s , respectively ) have mass loading factors that differ by a factor of @xmath47 , depending on the stage of evolution . this is in rough agreement with expectations from theoretical models based on simulations and semi - analytic models , which show that dependence @xmath48 with @xmath49 is needed to reproduce the observed faint end of the galaxy stellar mass function and other properties of the galaxy population ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . moreover , the wind velocities in the smc and mw - sized simulated galaxies are consistent with the observed trend for galaxies in this mass range @xcite both in normalization and slope . although we have reported only two models , these results are encouraging , especially because simulation parameters have not been tuned to reproduce these observations . perhaps the most intriguing difference of the cr - driven winds compared to the winds driven by thermal sn feedback is that they contain significantly more `` warm '' @xmath2 k gas . this is especially true for the dwarf galaxy , which develops a wind strikingly colder than in the sn - only simulation ( see fig . [ fig : im ] ) . the cr - driven wind has a lower velocity , and is accelerated gradually with vertical distance from the disk . the reason for these differences is that the gas ejected from the disk is accelerated not only near star - forming regions , as is the case in sn - only simulations , but is continuously accelerated by the pressure gradient established by crs diffused outside of the disk ( see fig . [ fig : im ] ) . the diffusion of crs is thus a key factor in ejecting winds and in their resulting colder temperatures . the cooler temperatures of the ejected gas may be one of the most intriguing new features of the cr - driven winds , as this may provide a clue on the origin of ubiquitous warm gas in gaseous halos of galaxies ( e.g. , * ? ? ? * and references therein ) . detailed predictions of cgm properties will require cosmological galaxy formation simulations incorporating cr feedback , which we will pursue in future work . several studies have explored effects of cr injection on galaxies . @xcite found that crs suppress the sfr in dwarf galaxies by an amount comparable to our simulations , but have almost no effect on the sfr of mw - sized systems . we find significant sfr suppression for both masses . additionally , @xcite found that crs did not generate winds with diffusion alone and in a recent study using a similar model @xcite argued that to launch winds cr streaming is crucial . in contrast , we find that cr - driven winds are established with cr diffusion alone . these differences likely arise due to assumption of equilibrium between the sources of crs ( star formation @xmath50 ) and the sinks ( catastrophic losses @xmath51 ) in the subgrid model of @xcite . the subgrid model thus predicts that cr pressure scales as @xmath52 and is subdominant to the thermal ism pressure at densities @xmath53@xmath54 ( see fig . 7 of * ? ? ? this assumption of equilibrium , which is likely true only in the deepest parts of the galaxy potential well ( see e.g. the discussion in * ? ? ? * ) , breaks down in lower density gas . in our simulations we do not assume such equilibrium and we find significant pressure contributions from crs up to much higher densities . our results thus indicate that crs , even in the diffusion limit , not only suppress star formation but also drive outflows efficiently . thus , the effects of cr feedback on the properties of galaxies of different masses should be significantly stronger and span a wider range of masses than simulations that use the @xcite model ( e.g. * ? ? ? while this manuscript was in a late stage of preparation @xcite appeared as a preprint . these authors have presented simulations of a mw - sized galaxy , similar to the model presented here , albeit without accounting for cr cooling losses and with a much larger sfr in their model galaxy ( up to @xmath55 . where our results overlap ( e.g. , mass - loading factor ) with those of @xcite we find remarkably good agreement . these authors also find that outflows are efficiently generated with cr diffusion alone . our study extends the results of @xcite by presenting the differences between wind properties in dwarf and mw - sized systems . the results of @xcite and our study indicate that crs can significantly suppress star formation in galaxies and efficiently drive outflows with significant mass loading factors and velocities comparable to observed outflows . a detailed exploration of the effects of such feedback on the galaxy population in a full cosmological setting is therefore extremely interesting . ng and ak were supported via nsf grant oci-0904482 . ak was supported by nasa atp grant nnh12zda001n and by the kavli institute for cosmological physics at the university of chicago through grants nsf phy-0551142 and phy-1125897 and an endowment from the kavli foundation and its founder fred kavli .

we present results from high - resolution hydrodynamic simulations of isolated smc- and milky way - sized galaxies that include a model for feedback from galactic cosmic rays ( crs ) . we find that crs are naturally able to drive winds with mass loading factors of up to @xmath0 in dwarf systems . the scaling of the mass loading factor with circular velocity between the two simulated systems is consistent with @xmath1 required to reproduce the faint end of the galaxy luminosity function . in addition , simulations with cr feedback reproduce both the normalization and the slope of the observed trend of wind velocity with galaxy circular velocity . we find that winds in simulations with cr feedback exhibit qualitatively different properties compared to sn driven winds , where most of acceleration happens violently in situ near star forming sites . the cr - driven winds are accelerated gently by the large - scale pressure gradient established by crs diffusing from the star - forming galaxy disk out into the halo . the cr - driven winds also exhibit much cooler temperatures and , in the smc - sized system , warm ( @xmath2 k ) gas dominates the outflow . the prevalence of warm gas in such outflows may provide a clue as to the origin of ubiquitous warm gas in the gaseous halos of galaxies detected via absorption lines in quasar spectra .

You are an expert at summarizing long articles. Proceed to summarize the following text: in high mass x - ray binaries ( hmxbs ) a neutron star or black hole orbits a massive early - type star and accretes matter either via roche - lobe overflow or from the stellar wind which powers the x - ray emission ( for recent reviews see nagase 1989 , white et al . 1995 , bildsten et al . one divides the class of hmxbs according to the stellar type of the mass donor star into supergiant x - ray binaries with luminosity class i - ii ob star and be / x - ray binaries with luminosity class iii - iv be star companions . be / x - ray binaries form the larger sub - group of hmxbs . balmer emission lines in the optical spectrum and a characteristic infrared excess are attributed to the presence of circum - stellar material , probably forming a disk in the equatorial plane of the be star . be / x - ray binaries often show transient behaviour with two types of outbursts . x - ray outbursts repeating with the orbital period are most likely associated with the passage of the neutron star through the circum - stellar disk in an eccentric orbit while giant outbursts , often lasting longer than a binary period , probably arise from an expansion of the disk . currently about 100 hmxbs and candidates are known . nearly one third were found in the magellanic clouds ( mcs ) from which the majority is located in the small magellanic cloud ( smc , coe 1999 ) . most of the be / x - ray binaries in the smc were discovered in recent years by x - ray missions like asca , bepposax , rosat and rxte ( nagase 1999 ) . from 20 optically identified hmxbs in the smc only one is securely associated with a supergiant system ( the x - ray pulsar smcx1 ) and from 11 of the 19 be / x - ray binaries x - ray pulsations were detected . five additional x - ray pulsars are yet to be identified , but are most likely also be systems . the location of such a large number of hmxbs at a similar distance makes the smc ideally suited for statistical and in particular spatial distribution studies of the population of hmxbs in a galaxy as a whole . recent surveys to look for h@xmath0 emission - line objects in the smc were performed by meyssonier & azzopardi ( 1993 , hereafter ma93 ) and murphy & bessell ( 1999 , mb99 ) . the survey of ma93 mainly covers the main body and eastern wing of the smc and their catalogue lists 1898 emission - line stars . the catalogue of mb99 covers nearly all the area where rosat pspc observations of the smc are available ( except the southern half of the most south - east observation ) but is less sensitive ( 372 objects , partially in common with ma93 ) . a main goal of ma93 and mb99 was to identify planetary nebulae in the smc , however , the catalogues also contain be stars . ma93 state that all three at the time of publication known b[e ] supergiants which were covered by the survey were detected . very few be / x - ray binaries were known in the smc until 1993 and it was not noticed that the be stars proposed as optical counterparts for smcx-3 , 2e0050.17247 and 2e0051.17304 were listed in ma93 as emission - line stars ( lin 198 , azv 111 and azv 138 , respectively ) . a correlation of the larger sample of be / x - ray binaries known today in the smc shows that most of them are found in the catalogues of ma93 and mb99 . in this paper we use the identification of x - ray sources with emission - line stars to propose new very likely candidates for be / x - ray binaries in the smc ( sect . 2 ) . x - ray source catalogues of the smc which we used for our correlations with the emission - line star catalogues were published by wang & wu ( 1992 ) based on einstein ipc observations ( seward & mitchel 1981 , inoue et al . 1983 , bruhweiler et al . 1987 ) and by haberl et al . ( 2000 , hfpk00 ) produced from rosat pspc data . we also used a preliminary version of the rosat hri catalogue of sasaki et al . ( 2000b ) . a cross - correlation of the 517 pspc x - ray sources in the smc region ( hfpk00 ) with the catalogue of h@xmath0 emission - line stars published by ma93 ( 1898 entries ) yielded 46 possible optical counterparts ( distance @xmath2 @xmath3 to account for systematic uncertainties in x - ray and optical positions , where r@xmath4 denotes the 90% statistical uncertainty of the x - ray position in arc seconds ) . an additional object correlating with a pspc source was found within the catalogue of candidate emission - line objects of mb99 . from this sample of 47 objects one coincides with a known supernova remnant , three with supersoft sources and ten with optically identified be stars proposed as optical counterparts for the x - ray sources . extending the search by using additional smc x - ray sources such as from the rosat hri catalogue of sasaki et al . ( 2000b ) , the einstein ipc catalogue of wang & wu ( 1992 ) and x - ray pulsars discovered by instruments on asca , bepposax and rxte yields another three emission - line stars identified with known be stars . our correlation results are summarized in table [ tab - ma93 ] which is sorted in right ascension . columns 2 - 4 of table [ tab - ma93 ] give the source numbers in the x - ray catalogues for rosat pspc ( hfpk00 ) , rosat hri ( sasaki et al.2000b ) and einstein ipc ( wang & wu 1992 ) . for sources detected by rosat coordinates with statistical 90% error ( from hfpk00 when detected by the pspc or from sasaki et al . 2000b when detected by hri only ) are given in columns 5 - 7 . for the group of ipc sources which were not detected by rosat , the ipc coordinates and a 40 error as published in ww92 is given . the three digits in column 8 denote the number of pspc detections in the energy bands 0.1 - 0.4 kev , 0.5 - 0.9 kev and 0.9 - 2.0 kev . with a few exceptions most of the sources were detected mainly in the higher energy bands which indicates a hard x - ray spectrum . hfpk00 used count ratios in the different energy bands , the hardness ratios , for a spectral classification of the pspc sources . however , very hard sources without detection in the lower energy bands were not classified in hfpk00 because of large errors on the hardness ratios . column 9 lists the maximum observed x - ray luminosity for be / x - ray binaries and candidates derived in this work . the values are selected from literature or computed from rosat count rates using the conversion factor 1.67@xmath5 erg s@xmath6/cts s@xmath6 ( see kahabka & pietsch 1996 , hereafter kp96 ) , typical for x - ray binaries with hard spectrum at the distance of the smc . hri count rates were converted to pspc rates using a multiplication factor of 3.0 ( sasaki et al . luminosities derived from count rates are indicated with colon . for all given x - ray luminosities we assume a distance of 65 kpc to the smc . column 10 of table [ tab - ma93 ] lists the entry number of the nearest object in ma93 ( mb99 in one case ) and column 11 the ma93 classification type ( 2 = snr ; 5 = planetary nebula , pn ; 9 = late type star ) . the distance between x - ray and optical position as listed in ma93 ( mb99 ) is listed in column 12 . b , v and r magnitudes found in the literature for identified sources are given in columns 13 - 15 . when available b and r obtained from the usno a2.0 catalogue are listed for the remaining sources . in the last column identifications are given together with references and new proposals are marked with ? behind the source type . from the total of 18 high mass x - ray binaries in the smc with known be star as proposed counterpart 13 are found in the emission - line catalogues of ma93 and mb99 . for completeness the remaining five be / x - ray binaries are also listed in table [ tab - ma93 ] . the identification of most known be / x - ray binaries with stars in the emission - line catalogues of ma93 and mb99 suggests that un - identified x - ray sources with emission - line star counterparts are most likely also be / x - ray binary systems . other objects like ssss and snrs can be recognized by their very soft x - ray spectrum ( in contrast to the be / x - ray binaries with hard spectrum ) or by their x - ray source extent , respectively . in the following section we summarize the 18 x - ray sources optically identified as be / x - ray binary . we then propose emission - line stars as likely be counterparts for the five un - identified pulsars and in addition for 25 hard x - ray sources . three supersoft sources detected by rosat were identified with emission - line objects in the catalogue of ma93 . two of them are associated with planetary nebulae ( pn ) while the remaining one is identified with a symbiotic star in the smc . more detailed information on the individual sources and finding charts with x - ray error circles can be found in sect . [ sect - notes].1 . for 19 x - ray sources in the smc nearby be stars were optically identified and proposed as counterparts , suggesting a be / x - ray binary nature . eighteen were covered by rosat observations and information on the individual sources is summarized in sect . [ sect - notes ] where also finding charts are found . the heao source 1h0103762 was not observed with rosat ( see kp96 and references therein ) . also we do not include the hmxb candidates rxj0106.27205 ( hughes & smith 1994 ) and exo0114.67361 in our summary . for rxj0106.27205 no optical spectrum from the suggested counterpart is published yet , which would confirm its proposed be star nature . for exo0114.67361 wang & wu ( 1992 ) propose the b0ia star azv488 as counterpart , however , azv477 , also a b0ia star is even closer to the x - ray position . both candidates suggest a supergiant type of hmxb . it is remarkable , that together with the only other known supergiant hmxb smcx1 , exo0114.67361 is located in the eastern wing of the smc . fourteen of the identified be / x - ray binaries were detected by the rosat pspc and their x - ray properties can be found in hpfk99 . axj0051722 , smcx3 and rxj0058.27231 were detected by the rosat hri and are listed in the catalogue of hri sources in the smc ( sasaki et al . only 2e0051.17304 was not detected by rosat . thirteen of the proposed be star counterparts are listed in the catalogues of ma93 and mb99 and only for five x - ray sources the be counterparts have no entry in ma93 and mb99 ( axj0049729 , smcx2 , rxj0032.97348 , rxj0058.27231 and xtej0111.2 - 7317 ) . rxj0032.97348 was not covered by the ma93 survey . five x - ray pulsars in the smc were reported for which no optical identifications are published up to day . four of them were detected by rosat pspc and/or hri , yielding more accurate positions ( hfpk00 , sasaki et al . 2000b ) and for the fifth case , xtej0054720 , several rosat sources are found within the large rxte error circle . in or very close to the rosat error circles emission - line objects from ma93 are found and we propose these as optical counterparts . literature , finding charts and other information on the x - ray binary pulsars is presented in sect . [ sect - notes ] . most of the be stars proposed as optical counterparts for x - ray sources in the smc , as summarized in the previous section , are found as emission - line objects in the catalogue of ma93 . this strongly supports that the unknown emission - line objects within the error circles of the unidentified pulsars are also be star counterparts of the x - ray pulsars forming be / x - ray binaries . from the correlation of x - ray source and emission - line object catalogues 34 hard x - ray sources were found with an h@xmath0 emission - line object as possible optical counterpart in the x - ray error circle ( see table [ tab - ma93 ] ) . the 34 x - ray sources were investigated in detail to obtain more information which can help to identify the nature of the object . finding charts and notes to the individual sources are compiled in sect . [ sect - notes ] . many sources were observed more than once by rosat and we looked for long - term time variability . in the case of the pspc we used the 0.9 2.0 kev band because of higher sensitivity for hard sources like be / x - ray binaries . to combine detections from the different instruments we convert hri to pspc count rates by multiplying with 3.0 and ipc to pspc count rates by multiplying with 1.1 ( appropriate for a 5 kev bremsstrahlung spectrum with 4.3@xmath7 @xmath8 absorption column density ) . given the uncertainties in the count rate conversions , variability is only treated as significant above a factor of 3 . none of the sources was observed with sufficient counting statistics in order to perform a detailed temporal analysis on shorter time scales ( within an observation ) and to detect x - ray pulsations . we discuss all un - identified x - ray sources with emission - line object in or close to the error circle in the following and indicate very promising candidates for be / x - ray binaries with be / x ? " in the remark column of table [ tab - ma93 ] . to estimate the number of false identifications of x - ray sources with emission - line objects in ma93 we shifted the x - ray positions of the sources in an arbitrary direction and cross - correlated again with the ma93 catalogue . to get statistically more reliable results this was repeated with different distances between 1 10 arc minutes . for this purpose we used the pspc catalogue which is most complete . after application of our selection criteria for accepting be / x - ray binary candidates ( likelihood of existence for the x - ray source @xmath9 13 , no other identification , distance @xmath2 @xmath3 ) we find on average about seven expected chance coincidences between pspc sources and emission - line objects in ma93 . we emphasize that these are mainly caused by the pspc sources with the largest position uncertainties . the pspc sources in table [ tab - ma93 ] with the largest errors on the x - ray position and therefore most likely chance coincidences are 99 , 248 , 295 and 404 . indeed three of them were rejected as be / x - ray binary candidates due to the presence of other likely counterparts . similarly pspc sources 77 and 253 were disregarded . other x - ray sources with large position errors in table [ tab - ma93 ] are the ipc sources which were not securely detected with rosat . also here two were not regarded as be / x - ray binary candidates . for the 25 new be / x - ray binary candidates we therefore estimate that about two to three may be misidentifications , most likely among those with position error r@xmath4 @xmath9 15 . the 1st asca catalogue of x - ray sources in the smc was compiled by yokogawa ( 1999 ) . the sources were classified according to their hardness ratios and be / x - ray binaries were detected as sources with hard x - ray spectrum in the asca energy band . from this classification yokogawa ( 1999 ) proposed eight new be / x - ray binary candidates ( binary pulsar candidates , bpc ) . we correlated our list of be / x - ray binary candidates with the asca catalogue and find for five of the eight bpc a likely counterpart within 2 ( the maximum asca position uncertainty ) . in addition one out of the nine ( probably because of its low flux ) unclassified asca sources also correlates with a pspc be / x - ray binary candidate . in table [ tab - asca ] the be / x - ray binary candidates with likely asca counterpart are summarized . the first three columns show asca source number , classification and observed x - ray luminosity ( 0.7 10 kev , but corrected for the distance of 65 kpc used throughout this paper ) from yokogawa ( 1999 ) . the next two columns contain pspc source number and distance between asca and pspc position and the last two columns give the same for hri sources . the x - ray luminosities observed by asca are generally a factor of 1.2 2.4 higher than the rosat values ( table [ tab - ma93 ] ) which may only partly be explained by x - ray variability . finding a systematically higher luminosity with asca is probably caused by the different sensitive energy bands and/or different intrinsic source spectrum . in particular the pspc count rate to luminosity conversion is very sensitive to the assumed absorption . in the case of asca sources 36 ( pspc 279 ) and 7 ( pspc 468 ) the asca / rosat luminosity ratio is much higher than average ( 4.2 and 17 , respectively ) , indicating strong flux variability . rlcrrrr & & + no & class & [ erg s@xmath6 ] & no & d [ ] & no & d [ ] + 2 & bpc & 3.5@xmath11 & 434 & 30.1 & & + 6 & bpc & 2.9@xmath11 & 511 & 39.8 & 28 & 27.8 + 7 & bpc & 4.6@xmath11 & 468 & 52.1 & & + 27 & bpc & 8.6@xmath11 & 159 & 15.6 & 95 & 19.0 + 28 & bpc & 2.6@xmath11 & 220 & 11.3 & 97 & 20.1 + 36 & un & 9.6@xmath12 & 279 & 56.5 & & + [ tab - asca ] an optical identification campaign of a selected sample of hard rosat pspc sources from the catalogue of hfpk00 was started independently to the present work ( keller et al . in preparation ) . from three of the x - ray sources presented here , spectra were taken which in all cases revealed a be star nature of the proposed counterpart from the ma93 catalogue . this confirms rxj0057.87207 ( pspc 136 , sect . [ sect - notes].4.8 ) as be / x - ray binary and also the proposed counterpart for the pulsar axj0105722 ( pspc 163 , sect . [ sect - notes].3.2 , see also filipovi et al . 2000a ) as be star . rxj0051.97311 ( pspc 424 , sect . [ sect - notes].2.8 ) was independently identified by schmidtke et al . these results can be taken as further evidence that the emission - line objects we propose for counterparts of x - ray sources are indeed be stars . the spatial distribution of the smc hmxbs including the new candidates from this work is shown in fig . [ fig - distrib ] . nearly all new candidates are located along the main body of the smc where most of the optically identified be / x - ray binaries are concentrated . only one new candidate is found in the eastern wing near the supergiant hmxb smcx1 , where already two other be / x - ray pulsars are known . the distribution is not biased due to incomplete coverage , neither in the optical nor in x - rays and makes the strong concentration of be / x - ray binaries in certain areas of the smc more pronounced . the x - ray luminosity distribution of be / x - ray binaries and candidates in the smc is compared to that of systems in the galaxy in fig . [ fig - lx ] . to do this we intensively searched the literature on galactic be / x - ray binary systems . we derived 31 galactic sources with @xmath13 erg s@xmath6 which are summarized in table [ tab - begal ] ( which should be mostly complete ) . the luminosity estimates of galactic systems are often hampered by uncertain distances and different energy bands of the observing instrument . this may cause luminosity uncertainties by a factor of @xmath1410 in some cases but should not change the overall distribution drastically . the new candidates in the smc mainly raise the number of be / x - ray binaries with luminosities log ( ) @xmath2 35.5 ( 21 out of 24 are new candidates ) . this can easily be explained by the high sensitivity of the rosat instruments which allowed to detect be / x - ray binaries in their low - state while most of the higher luminosity be / x - ray binaries were discovered during outburst . x - ray luminosities derived from detectors sensitive at higher energies ( typically 0.5 10 kev ) might be up to a factor @xmath142 higher than those derived from rosat count rates ( see sect . 3 ) which would shift the low - luminosity end in fig . [ fig - lx ] by 0.3 dex to the right . however , such a shift would not change the overall distribution significantly . recently , in the galaxy several likely low - luminosity be / x - ray binaries were discovered by bepposax and asca ( 1saxj1324.4 - 6200 , angelini et al . 1998 ; axj1820.5 - 1434 , kinugasa et al . 1998 ; axj1749.2 - 2725 , torii et al . 1998 ; 1saxj1452.8 - 5949 , oosterbroek et al . 1999 ; axj1700062 - 4157 , torii et al . rosat also contributed new low - luminosity systems ( rxj0440.9 + 4431 , rxj0812.4 - 3114 , rxj1037.5 - 5647 , rxj0146.9 + 6121 , motch et al . however , the high absorption in the galactic plane makes the detection of low - luminosity x - ray sources in the rosat x - ray band and their optical identification difficult . this might explain the smaller number of low - luminosity be / x - ray binaries discovered so far in the galaxy compared to the smc and might suggest that the luminosity distribution of be / x - ray binaries is very similar in the smc and our galaxy . in this case many more such systems are expected to be found in our galaxy which would significantly contribute to the hard x - ray galactic ridge emission ( warwick et al . 1985 ) . various authors have suggested the existence of a population of low - luminosity systems which are usually persistent x - ray sources showing moderate outbursts and long pulse periods ( e.g. kinugasa et al . 1998 , mereghetti et al . 2000 ) , somewhat different to the high - luminosity systems with strong outbursts and shorter pulse periods . the smc results suggest that the low - luminosity sources even dominate the be / x - ray binaries in number . from the fact that about one third of the already identified be / x - ray binaries is not listed in current emission - line catalogues even more such systems are expected to be found in the rosat x - ray source catalogues of the smc . on the other hand some of them will be observed with higher maximum luminosity in future outbursts , but if they indeed form a class of low - luminosity be / x - ray binaries like xper , the outbursts are expected to be small changing the luminosity distribution immaterial . there is a large difference in the number of ob supergiant hmxbs between the galaxy and the smc . in the smc at most two such systems are identified ( smcx-1 and maybe exo0114.67361 ) resulting in an overall ratio of be to supergiant x - ray binaries of more than 20 . in the galaxy this proportion is more of order 2 ( 12 supergiant systems in the galaxy are listed in the reviews of white et al . 1995 and bildsten et al . it is remarkable that the smc supergiant hmxbs are all located in the eastern wing giving rise to a local be / supergiant hmxb ratio similar to that in the galaxy . in contrast no supergiant hmxb is known in the smc main body making the difference more extreme . one possible explanation is a different star formation history . be / x - ray binaries evolve from binary star systems with typical total mass of @xmath1420 m@xmath15 within about 15 my ( van den heuvel 1983 ) while the more massive supergiant hmxbs are formed on shorter time scales . the latter therefore would trace more recent epochs of star formation than the be / x - ray binaries . the comparatively large number of be / x - ray binaries in the smc in this view suggests a burst of star formation about 15 my ago while relatively few massive early - type stars were born during the last few million years . it is remarkable that the large magellanic cloud ( lmc ) may also have experienced a burst of star formation about 16 my ago as was derived from optical photometric surveys by harris et al . lmc and smc resemble in the relative composition of their x - ray binary populations , both rich in hmxbs but very few old low - mass x - ray binaries ( cowley et al . 1999 ) , suggesting a common star formation history triggered by tidal interaction during close encounters of lmc , smc and milky way . however , according to present day modeling the last encounter occurred @xmath140.2 gy ago ( gardiner & noguchi 1996 ) , too early for the formation of the be / x - ray binaries we see today in x - rays . therefore , the event which caused the origin of the frequent smc ( and lmc ? ) be / x - ray binaries remains still unclear . also the different numbers of hmxbs detected in lmc and smc relative to their total mass need to be explained . in the following notes x - ray source numbers refer to the catalogues of wang & wu ( 1992 ) for einstein ipc , of hfpk00 for rosat pspc and of sasaki et al.(2000b ) for rosat hri . finding charts with x - ray error circles are shown in figs . [ fig - fcsss ] , [ fig - fcobex ] , [ fig - fcpbex ] , and [ fig - fcnbex ] , for ssss , already identified be / x - ray binaries , unidentified x - ray pulsars and new be / x - ray binary candidates , respectively . the order within each group of sources follows table [ tab - ma93 ] and for faster access to the table entry the running number is given together with source name . \7 ) rxj0048.47332 : the sss rxj0048.47332 was discovered by kahabka et al . ( 1994 ) and identified as the symbiotic m0 star smc3 by morgan ( 1992 ) . this star is listed in ma93 as object 218 and classified as late type star , consistent with the spectral type determined by morgan ( 1992 ) . the accurate hri position ( source 23 ) confirms the identification of the pspc source ( 512 ) . \39 ) 1e0056.87154 : this sss was discovered in einstein data ( inoue et al . 1983 ) and was detected with rosat pspc ( source 47 ) and hri ( 79 ) . it coincides in position with the smc planetary nebula n67 ( aller et al . 1987 ) which is listed as object 1083 in ma93 . \41 ) rxj0059.67138 : this very soft source was discovered by hfpk00 ( pspc 51 ) and proposed as new sss due to its positional coincidence with the planetary nebula lin357 ( 1159 in ma93 ) in the smc . \1 ) rxj0032.97348 : stevens et al . ( 1999 ) identified two be stars within the pspc error circle of rxj0032.97348 , discovered by kp96 as variable source with hard x - ray spectrum . the a factor of @xmath145 smaller error radius obtained from a different pspc observation by hfpk00 ( source 567 ) , however , still contains both be stars which are very close to each other ( fig . [ fig - fcobex ] ) . the two stars were not covered by the survey of ma93 . \9 ) axj0049729 : yokogawa & koyama ( 1998a ) reported x - ray pulsations in asca data of this source . kahabka & pietsch ( 1998 ) suggested the highly variable source rxj0049.17250 ( kp96 ) as counterpart . stevens et al . ( 1999 ) identified two be stars , one only 3 from the x - ray position and one just outside the error circle given by kp96 . the revised position of pspc source 351 in hfpk00 makes the more distant be star further unlikely as counterpart ( see fig . [ fig - fcobex ] ) . none of the two be stars turns up in the list of emission - line objects of ma93 with the nearest entry ( 279 ) 58 away . \13 ) axj0051733 : yokogawa & koyama ( 1998b ) discovered x - ray pulsations from this source in asca data . the x - ray source was detected in einstein ipc , rosat pspc and hri archival data and the 18 year history shows flux variations by at least a factor of 10 ( imanishi et al . cowley et al . identified already 1997 a be star as optical counterpart of the rosat hri source rxj0050.87316 ( hri 34 ) which is located within the asca error circle . cook ( 1998 ) reported a 0.708 d period from this star using data from the macho collaboration . the source was also detected by the pspc ( source 444 ) and coincides with object 387 in ma93 . \16 ) axj0051722 : corbet et al . ( 1998b ) reported 91 s x - ray pulsations from asca observations of this pulsar which was originally confused with the nearby 46 s pulsar xtej0053 - 724 in xte data . axj0051722 was not detected by the pspc . an hri detection reduced the position uncertainty and stevens et al . ( 1999 ) identified a be star as likely optical counterpart . the x - ray source is found as source 37 in the hri catalogue and the star is identical to the only emission - line object from ma93 ( 413 ) in the asca error circle . lrrll name & pulse & d & & reference for + & period [ s ] & [ kpc ] & [ erg s@xmath6 ] & + 4u0115 + 63 & 3.6 & 4 & 3.0@xmath5 & tamura et al . 1992 + v0332 + 53 & 4.4 & 3 & 3.2@xmath5 & takeshima et al . 1994 + 2s1553 - 542 & 9.3 & 10 & 7.0@xmath16 & apparao 1994 + gs0834 - 430 & 12.3 & 5 & 1.1@xmath5 & wilson et al . 1997 + xtej1946 + 274 & 15.8 & 5 & 5.4@xmath16 & campana et al . 1999 + 2s1417 - 624 & 17.6 & 10 & 8.0@xmath16 & apparao 1994 + rxj0812.4 - 3114 & 31.9 & 9 & 1.1@xmath16 & reig & roche 1999a + exo2030 + 375 & 41.8 & 5 & 1.0@xmath17 & parmar et al . 1989 + gs2138 + 56 & 66.3 & 3.8 & 9.1@xmath11 & schulz et al . 1995 + groj1008 - 57 & 93.5 & 2 & 2.9@xmath11 & macomb et al . 1994 + gs1843 - 02 & 94.8 & 10 & 6.0@xmath16 & finger et al . 1999 + 4u0728 - 25 & 103 & 6 & 2.8@xmath11 & corbet et al . 1997 + a0535 + 26 & 105 & 2.4 & 2.0@xmath5 & apparao 1994 + axj1820.5 - 1434 & 152 & 4.7 & 9.0@xmath12 & kinugasa et al . 1998 + 1saxj1324.4 - 6200 & 171 & 10 & 9.8@xmath12 & angelini et al . 1998 + groj2058 + 42 & 198 & 7 & 2.0@xmath16 & wilson et al . 1998 + rxj0440.9 + 4431 & 203 & 3.2 & 3.0@xmath12 & reig & roche 1999b + axj1749.2 - 2725 & 220 & 8.5 & 2.6@xmath11 & torii et al . 1998 + gx304 - 1 & 272 & 2.4 & 1.0@xmath16 & apparao 1994 + 4u1145 - 619 & 292 & 0.5 & 7.4@xmath12 & apparao 1994 + 4u2206 + 54 & 392 & 2.5 & 2.5@xmath11 & saraswat & apparao 1992 + a1118 - 616 & 405 & 5 & 5.0@xmath16 & apparao 1994 + 1saxj1452.8 - 5949 & 437 & 9 & 8.7@xmath18 & oosterbroek et al . 1999 + axj1700062 - 4157 & 715 & 10 & 7.2@xmath12 & torii et al . 1999 + x persei & 836 & 0.83 & 1.9@xmath11 & haberl 1994 + rxj1037.5 - 5647 & 860 & 5.0 & 4.5@xmath11 & reig & roche 1999b + rxj0146.9 + 6121 & 1412 & 2.5 & 3.5@xmath11 & haberl et al . 1998 + a0114 + 65 & & 2.6 & 1.7@xmath12 & motch et al . 1997 + gamma cas & & 0.188 & 3.9@xmath12 & apparao 1994 + 1e0236.6 + 6100 & & 3.1 & 2.0@xmath12 & motch et al . 1997 + 1h0521 + 373 & & & 4.0@xmath18 & apparao 1994 + [ tab - begal ] \19 ) rxj0051.97311 : this x - ray source was detected by cowley et al . ( 1997 ) during rosat hri observations of einstein ipc source 25 and identified with a be star by schmidtke et al . it is identical to pspc source 424 and hri 41 . the be star is found as object 504 in ma93 . \20 ) rxj0051.87231 : this source was reported as strongly x - ray variable by kp96 and is associated with the x - ray pulsar 2e0050.17247 ( israel et al . 1997 ) . observed x - ray luminosities range between 5@xmath12 erg s@xmath6 and 1.4@xmath16 erg s@xmath6 ( israel et al . the star azv111 ( object 511 in ma93 ) was proposed as counterpart for 2e0050.17247 while israel et al . ( 1997 ) identified another h@xmath0 active star within their error circle of rxj0051.87231 which is larger than that of kp96 . also the position error given for the corresponding pspc source 265 by hfpk00 is large . the detection of the source in the pspc observation 600453p ( used by kp96 ) where the source was bright was rejected by the semi - automatic analysis of hfkp99 because the detection was close to the support structure of the detector entrance window . a careful analysis ( and using the latest processed data of 600453p ) of the photons of the source in the detector frame shows , however , that it moved nearly parallel to the window support structure and that it was not affected by it . in table [ tab - ma93 ] therefore the parameters derived from this pspc observation are given . they confirm the results of kp96 with small error circle ( see fig . [ fig - fcobex ] ) . both azv111 and star 1 of israel et al . ( 1997 ) are outside this error circle which , however , contains a be star identified by stevens et al . this star is found as object 506 in ma93 and is the most probable counterpart of rxj0051.87231 . \22 ) smcx3 : this long - known x - ray source was not detected by the rosat pspc but is included in the hri catalogue as source 43 . the be star counterpart ( e.g. crampton et al . 1978 ) corresponds to object 531 in ma93 . \23 ) rxj0052.17319 : lamb et al . ( 1999 ) reported x - ray pulsations from the variable source rxj0052.17319 ( kp96 ) found in rosat hri and cgro batse data . israel et al . ( 1999 ) identified a be star as likely optical counterpart . it is found in ma93 as object 552 and was detected as x - ray source by ipc ( 29 ) , pspc ( 453 ) and hri ( 44 ) . the strong x - ray variability by a factor of @xmath14200 between different hri observations ( kahabka 2000 ) strongly supports the identification as be / x - ray binary . \24 ) 2e0051.17304 : for this source , listed as entry 31 in the einstein ipc source catalogue of wang & wu ( 1992 ) , the be star azv138 ( garmany & humphreys 1985 ) was proposed as optical counterpart . azv138 corresponds to object 618 in ma93 . 2e0051.17304 was not detected in rosat observations . \25 ) rxj0052.97158 : this source was detected as x - ray transient by cowley et al . ( 1997 ) during rosat hri observations of einstein ipc source 32 ( the largest circle in the finding chart of fig . [ fig - fcobex ] . upper limits derived from pspc observations imply flux variations by at least a factor of @xmath14350 ( cowley et al . the strong variability and the hard x - ray spectrum imply a be / x - ray transient consistent with the suggested be star counterpart ( schmidtke et al . the be star is identical to object 623 in ma93 . the x - ray source was detected by rosat ( pspc 94 and hri 46 , the hri position is most accurate as indicated by the smallest error circle in the finding chart of fig . [ fig - fcobex ] ) and is located near the edge of the error circle of xtej0054 - 720 . due to the large position uncertainty of xtej0054 - 720 it is , however , not clear if they are identical . \28 ) smcx2 : the long known be / x - ray binary smcx2 was caught in outburst with 0.4 cts s@xmath6 by the rosat pspc ( source 547 , see kp96 and references therein ) . another pspc observation yielded an upper limit indicating x - ray variability of more than a factor of 670 . optical spectra of the be counterpart were taken by e.g. crampton et al . the be star is located on the rim of the pspc error circle ( fig . [ fig - fcobex ] ) and is not contained in the ma93 catalogue . \31 ) xtej0055724 : x - ray pulsations from this source were discovered by rxte ( marshall & lochner 1998 ) and confirmed in a sax observation ( santangelo et al . santangelo et al . ( 1998 ) also report pulsations from archival rosat data reducing the positional uncertainty . stevens et al . ( 1999 ) identified a be star as optical counterpart which corresponds to object 810 in ma93 and which is inside the error circle of pspc source 241 and hri source 58 . \38 ) rxj0058.27231 : rxj0058.27231 was detected as weak hri source by schmitdke et al . ( 1999 ) and identified with a be star . it is contained in the hri catalogue ( source 76 ) but not found in the pspc catalogue of hfpk00 . the be star is not detected in the emission - line star surveys of ma93 and mb99 . \40 ) rxj0059.27138 : this transient x - ray pulsar with peculiar soft component in the x - ray spectrum was discovered by hughes ( 1994 ) during an outburst with a 0.2 2.0 kev luminosity of 3.5@xmath5 erg s@xmath6 . the x - ray source was identified with a be star by southwell & charles ( 1996 ) as star 1 in their finding chart which is identical to the emission - line object 179 in mb99 . \42 ) rxj0101.07206 : this source was suggested as x - ray transient by kp96 with a flux variability of at least a factor of 30 . stevens et al . ( 1999 ) identified a be star as optical counterpart . object 1 in their fig.1f corresponds to entry 1240 in ma93 . \49 ) saxj0103.27209 : israel et al . ( 1998 ) reported x - ray pulsations from this source consistent in position with the einstein source 1e0101.57225 . they confirm the be star suggested as counterpart for the einstein source by hughes & smith ( 1994 ) as the only object in the sax error circle showing strong h@xmath0 activity . ogle observations presented by coe & orosz ( 2000 ) confirm this . the be star corresponds to object 1367 in ma93 and was also detected by pspc ( source 143 ) and hri ( 101 ) . \58 ) xtej0111.27317 : chakrabarty et al . ( 1998a ) reported x - ray pulsations found in rxte data from this source located about 30 from smcx1 . wilson & finger ( 1998 ) confirmed the pulsations from cgro batse data and chakrabarty et al . ( 1998b ) derived an improved position from asca data . two be stars were identified by israel et al . ( 1999 ) within or near the asca error circle of 30 . the closer of the two was concluded as most likely counterpart of xtej0111.27317 by coe et al . this be star has no counterpart in ma93 . a week source with existence likelihood of 14.5 is found in the pspc catalogue ( 446 ) . the large error circle of 61 overlaps with the asca one and includes the position of the be star . there is an additional ma93 object ( 1731 ) within the rxte error circle and the second be star found by israel et al . ( 1999 ) is identical to object 1747 in ma93 but both are outside the asca and pspc confidence regions ( see fig . [ fig - fcobex ] ) . \59 ) rxj0117.67330 : similar to the previous source x - ray pulsations were discovered from the x - ray transient rxj0117.67330 ( clark et al . 1997 ) in rosat pspc and cgro batse data ( macomb et al . 1999 ) . between two pspc observations , about 8 months apart , the count rate diminished by a factor of 270 . clark et al . ( 1997 ) identified a be star counterpart which is identical to object 1845 in ma93 and also within the error circle of x - ray source 506 in the smc pspc catalogue . \10 ) axj0049732 : axj0049732 was discovered as x - ray pulsar by imanishi et al . filipovi et al . ( 2000b ) reported two hard x - ray point sources from the catalogue of hfpk00 as possible counterparts to the asca pulsar . they suggest one of them ( pspc source 427 ) as the more likely counterpart due to its identification with an emission - line object in ma93 ( number 300 ) . \25 ) xtej0054720 : the position of this x - ray pulsar could only be determined to an accuracy of 10 radius ( lochner et al . there are at least five x - ray sources detected by the hri within that circle ( labeled 1 through 5 in fig . [ fig - fcpbex ] which correspond to the catalogue sources 55 , 50 , 62 , 59 and 46 , respectively ) . object 2 , 4 and 5 were also detected by the pspc ( 104 , 157 and 94 ) . the southern of the three ( also detected by ipc , 36 ) is proposed as active galactic nucleus ( agn ) by hfpk00 and the northern ( pspc 94 , hri 46 , ipc 32 ? ) was identified as be / x - ray transient rxj0052.9 - 7158 ( see sect . [ sect - notes].2.2 ) . the be star counterpart of rxj0052.9 - 7158 coincides with object 623 in ma93 . it is not clear if this be / x - ray binary is identical to the rxte pulsar . a final identification requires the detection of pulsations from rxj0052.9 - 7158 . \27 ) xtej0053724 : corbet et al . ( 1998a ) discovered this pulsar and report a rosat source within the error box . the pulse period , originally confused with axj0051722 , was clarified by corbet et al . ( 1998b ) . hfpk00 give source 242 as likely counterpart of xtej0053724 . a single emission - line object from ma93 ( 717 ) is found inside the intersecting error circles of ipc source 34 and the pspc source . \35 ) axj0058720 : x - ray pulsations from this source were discovered by yokogawa & koyama ( 1998b ) in asca observations . the source was detected in archival einstein ipc , rosat pspc and hri data which span 18 years and showed flux variations by more than a factor of 100 ( tsujimoto et al . this high variability already strongly suggests a be / x - ray binary . a single emission - line object from ma93 ( 1036 ) is found within the pspc error circle ( source 114 ) which is also consistent with the hri position ( 73 ) . it is not clear whether ipc source 41 originates from the same x - ray source . it may also be associated with another emission - line object ( 1039 of ma93 ) closer to the ipc position or completely unrelated . \53 ) axj0105722 : yokogawa & koyama ( 1998c ) reported axj0105722 as x - ray pulsar . from several nearby objects in ma93 number 1517 is closest to the x - ray position of pspc source 163 . this pspc source was identified as likely counterpart of the asca pulsar in an area of complex x - ray emission by filipovi et al . ( 2000a ) combining the rosat x - ray and radio data . the star 1517 in ma93 is the northern and bluer component of a pair of stars close to the error circles of pspc and hri detection ( 110 ) . the nearby ipc source 53 , 77 to the north - east is most likely associated to the snr dems128 ( filipovi et al . 2000a ) . \2 ) rxj0041.27306 : hfpk00 classified pspc source 404 as foreground star based on the hardness ratios . an emission - line object in the error circle is classified as planetary nebula by ma93 indicating a chance positional coincidence . this makes the identification with the bright star just outside the error circle most likely . \3 ) rxj0045.67313 : this source ( pspc 436 ) was detected once in the 0.9 2.0 kev band of the pspc . an emission - line object in the error circle suggests an be / x - ray binary . \5 ) rxj0047.37239 : the pspc error circle of rxj0047.37239 ( source 295 ) overlaps with that of ipc source 19 . an emission - line object ( 168 in ma93 and classified as late type star ) and two radio sources from the catalogue of filipovi et al . ( 1998 ) are located in the x - ray confidence region . a point - like radio source as counterpart would favour an agn identification leaving the nature of rxj0047.37239 ambiguous . \6 ) rxj0047.37312 : rxj0047.37312 ( pspc 434 ) is most likely identified with the emission - line star 172 in ma93 . the fluxes derived from pspc detections show a factor of nine variations , supporting that the x - ray source is a be / x - ray binary . rxj0047.37312 is probably identical to ipc source 18 , which showed an intensity within the range observed by the pspc . it is also the likely counterpart of asca source 2 in yokogawa ( 1999 ; see sect . 3 ) , an x - ray binary candidate detected with similar intensity . \8 ) rxj0048.57302 : the emission - line object 238 in ma93 is the brightest optical object in the error circle of rxj0048.57302 ( pspc 392 ) . a be / x - ray binary is suggested . \11 ) rxj0049.57331 : an hri detection ( source 28 ) with much improved x - ray position compared to the pspc ( source 511 ) confirms the identification with the emission - line object 302 in ma93 . rxj0049.57331 is the probable counterpart of asca source 6 in yokogawa ( 1999 ; see sect . 3 ) further supporting the likely be / x - ray binary nature . \12 ) rxj0049.77323 : this source ( pspc 468 ) was detected once in the 0.9 2.0 kev band of the pspc . an emission - line object in the error circle suggests an be / x - ray binary . rxj0049.77323 is also the likely counterpart of asca source 7 in yokogawa ( 1999 ) , classified as x - ray binary candidate ( see sect . 3 ) . \14 ) rxj0050.77332 : rxj0050.77332 was only once detected by the pspc ( 514 ) and the emission - line object in the error circle suggests a be / x - ray binary identification . \15 ) rxj0050.97310 : hri ( source 36 ) and pspc ( source 421 ) detections are consistent with the identification of rxj0050.97310 with the emission - line object 414 in ma93 , suggesting a be / x - ray binary . \17 ) rxj0051.37250 : two close emission - line objects suggest rxj0051.37250 ( pspc 349 ) as be / x - ray binary , but make the identification ambiguous . \18 ) rxj0051.87159 : the emission - line object 502 ( ma93 ) found in the error circle of rxj0051.87159 ( pspc 99 ) is classified as late type star in ma93 . an active corona of this star may be producing the x - ray emission . the large error circle contains , however , another bright object which could also be responsible for the x - rays . the nature of rxj0051.87159 remains therefore unclear . \21 ) ww 26 : two emission - line objects from ma93 are found near ipc source 26 ( hardness ratio 0.51 , ww92 ) . object 521 is located inside the error circle while 487 can not be completely ruled out as counterpart . no rosat detection could improve on the position . a be / x - ray binary nature is suggested . \26 ) rxj0053.47227 : a precise hri position ( source 48 at the rim of the error circle of pspc 246 ) with the emission - line star 667 ( ma93 ) as brightest object in the error circle makes rxj0053.47227 a likely be / x - ray binary . \29 ) rxj0054.57228 : the uncertainty in the position of rxj0054.57228 ( pspc 248 ) is relatively large and six emission - line objects from ma93 are found as possible counterparts to the x - ray source . it is therefore a likely be / x - ray binary but the optical counterpart remains ambiguous . \30 ) rxj0054.97245 : precise rosat x - ray positions ( pspc 324 = hri 57 ) include an emission - line star ( 809 in ma93 ) with typical be star magnitudes as brightest object in the error circles . a factor of five x - ray flux variability ( the source was bright during a hri observation ) strengthens the identification as be / x - ray binary . \32 ) ww 38 = 2e0054.47237 : an emission - line object ( 904 in ma93 ) is found inside the error circle of ipc source 38 suggesting a be / x - ray binary . the source was not detected by rosat . \33 ) rxj0057.27233 : this weak pspc source ( 270 ) was marginally detected once in the hard 0.5 2.0 kev band with a likelihood of 10.4 . unlike all other hard sources in table [ tab - ma93 ] it was not detected in the 0.9 2.0 kev band and therefore is unlikely a be / x - ray binary . \34 ) ww 40 = 2e0055.87229 : the error circle of ipc source 40 contains two emission - line objects from ma93 . object number 1021 is identified as be star azv111 while 1016 , located further north , is of unknown type . rosat detected an x - ray source inside the ipc error circle ( hri 71 and pspc 117 with consistent positions ) which , however , is located between the two emission - line objects . the relation between the rosat and the einstein source and the emission - line objects is unclear . ipc , hri and pspc count rates are consistent within a factor of two , which may indicate that they come from the same x - ray source . however , the accurate rosat positions make an association with one of the nearby objects from ma93 unlikely . \36 ) rxj0057.87207 : again small error circles from rosat hri ( source 74 ) and pspc ( source 136 ) observations make the identification of rxj0057.87207 with an emission - line star ( 1038 in ma93 ) very likely . pspc detections with factor of eight different intensities and an hri detection during an x - ray bright state which increases the variability to a factor of about 37 , make a be / x - ray binary nature highly probable . \37 ) rxj0057.97156 : be / x - ray binary candidate from positional coincidence of pspc source 87 with emission - line object 1044 in ma93 which shows typical optical brightness . \43 ) rxj0101.37211 : pspc detections of this source ( pspc 159 = hri 95 ) indicate flux variations by at least a factor of 15 and the source was not detected in other observations ( upper limit a factor of 100 below the maximum count rate ) . this high variability and the presence of an emission - line star ( 1257 in ma93 ) in the small x - ray error circles likely exclude any other explanation than a be / x - ray binary . it also is the likely counterpart of asca source 27 in yokogawa ( 1999 ; see sect . 3 ) , an x - ray binary candidate . \44 ) rxj0101.67204 : two accurate positions from hri ( source 96 ) and pspc ( source 121 ) observations suggest the identification of rxj0101.67204 with object 1277 in ma93 . the factor of three variability supports a be / x - ray binary nature of rxj0101.67204 which is probably identical to the ipc source 46 in ww92 . \45 ) rxj0101.87223 : rxj0101.87223 ( pspc 220 = hri 97 ) shows x - ray flux variations of a factor of three . the emission - line star 1288 ( ma93 ) exhibits magnitudes typical for a be star in the smc and is located near the overlapping area of hri and pspc error circles . we suggest rxj0101.87223 as be / x - ray binary as it is also the probable counterpart of asca source 28 in yokogawa ( 1999 ; see sect . 3 ) , an x - ray binary candidate . \46 ) rxj0102.87157 : this weak pspc source ( 92 ) was only once marginally detected in the broad 0.1 2.4 kev band . the low detection likelihood of 10.5 and the non - detection in the hard bands indicates that it may not be real , or is at least not a hard source . a be / x - ray binary nature is therefore unlikely . \47 ) ww 49 : the ipc source 49 ( ww92 give a hardness ratio of 0.21 ) contains a faint emission - line object ( 1357 in ma93 ) classified as planetary nebula . the spectral hardness of the ipc source is inconsistent with an sss interpretation . the positional coincidence is likely by chance . \48 ) rxj0103.17151 : this source was detected only once by the pspc ( source 77 ) and the lowest upper limit indicates variability by at least a factor of five , suggesting the detection of a single outburst . the emission - line object near the rim of the pspc error circle is , however , the optically weakest ( see table [ tab - ma93 ] ) , unusual in comparison with identified be / x - ray binaries and candidates derived from this work . we therefore do not regard rxj0103.17151 as prime candidate for a be / x - ray binary . \50 ) rxj0103.67201 : small error circles from hri ( source 105 ) and pspc ( source 106 ) observations make the identification with object 1393 in ma93 very likely . rxj0103.67201 shows variability by a factor of three between the rosat observations , consistent with a be / x - ray binary . 51 ) rxj0104.17243 : two emission - line objects and a radio source from the catalogue of filipovi et al . ( 1998 ) close to rxj0104.17243 ( pspc 317 ) make the identification somewhat ambiguous . the most likely identification with emission - line star 1440 in ma93 suggests rxj0104.17243 as be / x - ray binary . \52 ) rxj0104.57121 : this source was not detected by the pspc but the accurate hri position ( source 108 ) includes only the emission - line object 1470 from ma93 as bright object in the error circle . rxj0104.57121 is therefore very likely a be / x - ray binary . \54 ) rxj0105.77226 : an emission - line star ( 1544 in ma93 ) in the pspc error circle ( 737 ) suggests rxj0105.77226 as be / x - ray binary . \55 ) rxj0105.97203 : a single bright object ( the emission - line star 1557 in ma93 ) is found in the small pspc error circle ( source 120 ) , which makes the identification of rxj0105.97203 as be / x - ray very likely . \56 ) rxj0107.17235 : the probable pspc detection ( 279 ) of ipc source 56 improves the x - ray position and allows to identify it with the emission - line star 1619 in ma93 . the source was a factor of 10 brighter during the einstein observation and is also the likely counterpart of asca source 36 in yokogawa ( 1999 ; see sect . 3 ) detected with a factor @xmath144 higher intensity . a be / x - ray binary nature is likely . \57 ) rxj0109.07229 : the emission - line object 1682 in ma93 is classified as planetary nebula . x - ray sources associated with planetary nebulae appear as ssss which is not compatible with the hard spectrum of rxj0109.07229 ( pspc 253 ) . the positional coincidence of rxj0109.07229 is therefore by chance and the nature of the x - ray source is unclear . \60 ) rxj0119.67330 : this source ( pspc 501 ) was detected once in the 0.9 2.0 kev band of the pspc . an emission - line object in the error circle suggests a be / x - ray binary . we reviewed the identification of eighteen known be / x - ray binaries in the smc and found that thirteen of them are listed in emission - line object catalogues of meyssonier & azzopardi ( 1993 ) and murphy & bessell ( 1999 ) . from a general correlation of smc x - ray source and h@xmath0 emission - line object catalogues we propose optical counterparts for the five optically unidentified x - ray pulsars and present 25 new be / x - ray binary candidates together with their likely optical counterparts . this more then doubles the number of know high mass x - ray binary systems in the smc . the rosat project is supported by the german bundesministerium fr bildung , wissenschaft , forschung und technologie ( bmbf / dlr ) and the max - planck - gesellschaft . the finding charts are based on photographic data obtained using the uk schmidt telescope . the uk schmidt telescope was operated by the royal observatory edinburgh , with funding from the uk science and engineering research council , until 1988 june , and thereafter by the anglo - australian observatory . original plate material is copyright ( c ) the royal observatory edinburgh and the anglo - australian observatory . the plates were processed into the present compressed digital form with their permission . the digitized sky survey was produced at the space telescope science institute under us government grant nag w-2166 . rrrrccrrcrcrlllp75mml 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 + & no & no & no & ra & dec & r@xmath4 & det & l@xmath19 & no & t & dist & b & v & r & remarks + & rp & rh & ei & & ( j2000.0 ) & [ ] & & erg s@xmath6 & ma & & [ ] & & & & + 1 & 567 & & & 00 32 56.1 & -73 48 19 & 12.9 & 012 & 1.3@xmath5 & & & & & & & be / x rxj0032.9 - 7348 , 2 be stars ( kp96 , scb99 ) + 2 & 404 & & & 00 41 16.4 & -73 06 41 & 35.6 & 100 & & 22 & 5 & 31.5 & 18.4 : & & 17.0 : & [ fg star ] gsc 9141.4223 ? + 3 & 436 & & & 00 45 37.9 & -73 13 54 & 29.4 & 001 & 1.2@xmath11 : & 114 & & 23.4 & 18.3 : & & 16.9 : & be / x ? + 4 & 413 & & 16 & 00 47 12.2 & -73 08 26 & 6.8 & 012 & & 165 & 2 & 4.5 & & & & snr 0045 - 73.4 ( rlg94 ) 13 cm + 5 & 295 & & 19 & 00 47 18.4 & -72 39 42 & 49.8 & 001 & & 168 & 9 & 21.1 & & & & agn ? radio smc b0045 - 7255 ( fhw98 ) 13 cm + 6 & 434 & & 18 & 00 47 23.4 & -73 12 23 & 4.0 & 012 & 2.4@xmath11 : & 172 & & 1.6 & & & & be / x ? [ hard ] + 7 & 512 & 23 & & 00 48 23.1 & -73 31 43 & 24.1 & 620 & & 218 & 9 & 15.1 & & & & sss rxj0048.4 - 7332 , symbiotic star m0 ( kp96 , m92 ) + 8 & 392 & & & 00 48 33.7 & -73 02 24 & 7.6 & 011 & 1.2@xmath11 : & 238 & & 6.5 & 14.6 : & & 16.9 : & be / x ? + 9 & 351 & & & 00 49 02.5 & -72 50 52 & 13.7 & 002 & 6.1@xmath16 & & & & 17.01 & 16.92 & & be / x axj0049 - 729 , 74.67 s pulsar ( yk98a , kp98 , scb99 , co00 ) + 10 & 427 & & & 00 49 29.6 & -73 10 56 & 5.5 & 002 & 4.1@xmath11 & 300 & & 2.6 & 18.1 : & & 15.6 : & be / x ? axj0049 - 732 , 9.132 s pulsar ( iyk98 , fph00b ) + 11 & 511 & @xmath2028 & & 00 49 30.7 & -73 31 09 & 1.8 & 011 & 1.2@xmath11 : & 302 & & 22.2 & & & & be / x ? + 12 & 468 & & & 00 49 43.8 & -73 23 02 & 14.9 & 001 & 2.7@xmath12 : & 315 & & 14.7 & 17.7 : & & 15.4 : & be / x ? + 13 & 444 & 34 & & 00 50 44.3 & -73 15 58 & 4.9 & 023 & 1.8@xmath16 & 387 & & 6.9 & 15.41 & 15.44 & & be / x axj0051 - 733 , 323.2 s pulsar ( csm97 , yk98b , sc98 , co00 ) + 14 & 514 & & & 00 50 46.8 & -73 32 47 & 33.7 & 010 & 2.4@xmath12 : & 393 & & 9.2 & & & & be / x ? + 15 & 421 & 36 & & 00 50 56.5 & -73 10 09 & 4.5 & 013 & 1.1@xmath11 : & 414 & & 3.5 & & & & be / x ? + 16 & & 37 & & 00 50 56.9 & -72 13 31 & 1.4 & & 2.9@xmath5 & 413 & & 2.7 & 15.8 : & & 16.4 : & be / x axj0051 - 722 91.12 s pulsar ( cml98b , l98 , scb99 ) + 17 & 349 & & & 00 51 19.5 & -72 50 43 & 15.6 & 001 & 3.6@xmath12 : & 447 & & 14.6 & & & & be / x ? + 18 & 99 & & & 00 51 51.4 & -71 59 50 & 47.7 & 001 & & 502 & 9 & 22.3 & 14.6 : & & 13.3 : & ac ? + 19 & 424 & 41 & 25 & 00 51 51.5 & -73 10 31 & 2.2 & 055 & 4.7@xmath11 : & 504 & & 4.2 & 13.1 : & 14.4 & 14.4 : & be / x rxj0051.9 - 7311 ( csm97 , scc99 ) + 20 & @xmath21265 & &27 & 00 51 53.1 & -72 31 50 & 2.0 & 112 & 1.4@xmath16 & 506 & & 1.3 & 13.12 & 13.4 & & be / x rxj0051.8 - 7231 , 8.9 s pulsar ( isa97 , scb99 ) + 21 & & & 26 & 00 51 54.2 & -72 55 36 & 40.0 & & 6.0@xmath12 & 521 & & 28.6 & & & & be / x ? + 22 & & 43 & & 00 52 05.4 & -72 25 55 & 15.8 & & 5.6@xmath5 & 531 & & 7.1 & 15.0 & & 15.4 : & be / x smc x-3 + 23 & 453 & 44 & 29 & 00 52 13.9 & -73 19 13 & 1.9 & 011 & 1.3@xmath5 & 552 & & 5.1 & & 14.62 & 14.54 & be / x rxj0052.1 - 7319 , 15.3 s pulsar ( lpm99 , isc99 ) + 24 & & & 31 & 00 52 52.7 & -72 48 22 & 40.0 & & 1.6@xmath11 & 618 & & 7.8 & 14.28 & 14.28 & 15.6 : & be / x 2e0051.1 - 7304 , azv138 ( gh85 ) + 25 & 94 & @xmath2046 & 32 & 00 52 54.7 & -71 58 08 & 2.0 & 004 & 2.0@xmath5 & 623 & & 13.2 & 15.39 & 15.46 & 14.7 : & be / x rxj0052.9 - 7158 ( csm97,scc99 ) = ? xtej0054 - 720 169.3 s pulsar + 26 & 246 & @xmath2048 & & 00 53 24.1 & -72 27 14 & 2.1 & 001 & 6.4@xmath12 : & 667 & & 31.0 & 17.0 : & & 15.8 : & be / x ? + 27 & 242 & & 34 & 00 53 53.3 & -72 27 01 & 25.7 & 012 & 7.4@xmath16 & 717 & & 22.7 & & & & be / x ? xtej0053 - 724 , 46.63 s pulsar ( cml98b ) + 28 & 547 & & & 00 54 30.8 & -73 40 55 & 10.5 & 011 & 8.4@xmath5 & & & & 15.7 & 16.0 & & be / x smcx-2 ( kp96 ) + 29 & 248 & & & 00 54 33.1 & -72 28 08 & 45.7 & 001 & 1.5@xmath16 : & 772 & & 26.6 & & & & be / x ? [ hard ] + 30 & 324 & 57 & & 00 54 55.3 & -72 45 06 & 5.6 & 012 & 3.0@xmath11 : & 809 & & 4.7 & 16.8 : & & 14.8 : & be / x ? + 31 & 241 & 58 & 35 & 00 54 55.4 & -72 26 46 & 3.5 & 023 & 3.0@xmath5 & 810 & & 3.1 & 15.24 & 15.28 & & be / x xtej0055 - 724 , 59.07 s pulsar ( ml98 , sci98 , scb99 , co00 ) + + @xmath22 b denotes 11 detections + @xmath23 entry in catalogue of mb99 + @xmath21 parameters from observation 600453 when source was bright + @xmath20 hri position and error + @xmath24 asca position and error [ tab - ma93 ] rrrrccrrcrcrlllp75mml 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 + & no & no & no & ra & dec & r@xmath4 & det & l@xmath19 & no & t & dist & b & v & r & remarks + & rp & rh & ei & & ( j2000.0 ) & [ ] & & erg s@xmath6 & ma & & [ ] & & & & + 32 & & & 38 & 00 56 03.3 & -72 21 32 & 40.0 & & 4.0@xmath12 & 904 & & 29.6 & 15.7 : & & 14.3 : & be / x ? + 33 & 270 & & & 00 57 15.4 & -72 33 38 & 25.9 & 000 & & 993 & & 27.1 & 17.2 : & & 16.7 : & ? + 34 & 117 ? & 71 ? & 40 & 00 57 32.4 & -72 13 17 & 40.0 & & & 1021 & & 23.6 & 15.4 : & & 13.7 : & ? + 35 & 114 & 73 & 41 & 00 57 48.4 & -72 02 42 & 7.9 & 001 & 1.6@xmath16 & 1036 & & 9.5 & 16.8 : & & 17.7 : & be / x ? axj0058 - 720 , 280.4 s pulsar ( yk98b ) + 36 & 136 & 74 & & 00 57 50.1 & -72 07 56 & 5.1 & 012 & 4.3@xmath11 : & 1038 & & 1.3 & 16.7 : & & 15.8 : & be / x ? [ hard ] + 37 & 87 & & & 00 57 59.5 & -71 56 37 & 19.2 & 001 & 5.7@xmath12 : & 1044 & & 21.1 & 17.1 : & & 15.5 : & be / x ? + 38 & & 76 & & 00 58 12.9 & -72 30 45 & 3.1 & & 2.1@xmath11 : & & & & 16.7 : & 14.9 & 15.5 : & be / x rxj0058.2 - 7231 ( scc99 ) + 39 & 47 & 79 & 43 & 00 58 37.2 & -71 35 50 & 1.3 & @xmath22b30 & & 1083 & 5 & 1.8 & 16.8 : & & 15.3 : & sss 1e0056.8 - 7154 , pn lin333 ( kp96 ) + 40 & 53 & & & 00 59 11.3 & -71 38 45 & 2.8 & 111 & 5.0@xmath5 & @xmath23179 & & 10.1 & 14.21 & 14.08 & 14.03 & be / x rxj0059.2 - 7138 , 2.763 s soft pulsar ( h94 , sc96 ) + 41 & 51 & & & 00 59 41.7 & -71 38 15 & 14.5 & 300 & & 1159 & 5 & 5.7 & 16.8 : & & 16.5 : & sss ? rxj0059.6 - 7138 , pn lin357 ( kpfh99 ) + 42 & 132 & 93 & & 01 01 01.1 & -72 06 57 & 2.8 & 011 & 1.3@xmath16 & 1240 & & 8.5 & & & & be / x rxj0101.0 - 7206 ( kp96 , scb99 ) + 43 & 159 & @xmath2095 & & 01 01 20.5 & -72 11 18 & 1.6 & 013 & 6.6@xmath11 : & 1257 & & 4.7 & & & & be / x ? [ nonstar ] + 44 & 121 & 96 & 46 & 01 01 37.0 & -72 04 19 & 7.2 & @xmath2206b & 3.8@xmath11 : & 1277 & & 5.5 & 17.7 : & & 16.4 : & be / x ? + 45 & 220 & 97 & & 01 01 51.0 & -72 23 26 & 8.1 & 025 & 2.2@xmath11 : & 1288 & & 9.8 & 14.7 : & & 14.3 : & be / x ? + 46 & 92 & & & 01 02 51.3 & -71 57 43 & 18.2 & 000 & & 1338 & & 14.1 & 15.1 : & & 13.6 : & ? + 47 & & & 49 & 01 03 06.9 & -72 32 59 & 40.0 & & & 1357 & 5 & 4.8 & & & & ? + 48 & 77 & & & 01 03 07.1 & -71 51 47 & 11.8 & 001 & & 1365 & & 14.6 & 18.8 : & & 17.8 : & ? + 49 & 143 & 101 & 50 & 01 03 14.0 & -72 09 16 & 3.4 & 067 & 1.5@xmath16 & 1367 & & 1.1 & 14.73 & 14.80 & 14.74 & be / x saxj0103.2 - 7209 , 345.2 s pulsar ( isc98 , hs94 , co00 ) + 50 & 106 & 105 & & 01 03 37.0 & -72 01 39 & 5.0 & 013 & 3.0@xmath11 : & 1393 & & 7.2 & & & & be / x ? + 51 & 317 & & & 01 04 07.4 & -72 43 59 & 17.7 & 002 & 3.8@xmath12 : & 1440 & & 9.0 & 14.1 : & & 14.4 : & be / x ? or agn ? 13 cm + 52 & & 108 & & 01 04 35.6 & -72 21 43 & 2.3 & & 4.8@xmath12 : & 1470 & & 4.0 & 14.8 : & & 15.1 : & be / x ? + 53 & 163 & 110 & & 01 05 08.9 & -72 11 44 & 6.6 & 012 & 1.5@xmath11 & 1517 & & 7.7 & & & & be / x ? axj0105 - 722 , 3.343 s pulsar ( yk98c , fhp00a ) + 54 & 737 & & & 01 05 42.3 & -72 26 15 & 15.7 & 001 & 1.8@xmath12 : & 1544 & & 12.8 & 14.2 : & & 14.0 : & be / x ? + 55 & 120 & & & 01 05 54.8 & -72 03 54 & 5.5 & 011 & 6.5@xmath12 : & 1557 & & 3.7 & & & & be / x ? + 56 & 279 & & 56 & 01 07 10.9 & -72 35 36 & 11.1 & 001 & 2.3@xmath12 : & 1619 & & 10.1 & 16.6 : & & 15.4 : & be / x ? + 57 & 253 & & & 01 09 01.2 & -72 29 07 & 17.6 & 012 & & 1682 & 5 & 12.3 & & & & ? [ hard ] + 58 & @xmath24446 & & & 01 11 14.5 & -73 16 50 & 30.0 & & 2.0@xmath17 & & & & 15.24 & 15.32 & 15.37 & be / x xtej0111.2 - 7317 , 31.03 s pulsar ( clc98a , wf98 , isc99 , chr99 ) + 59 & 506 & & & 01 17 41.4 & -73 30 49 & 0.6 & 122 & 1.2@xmath17 & 1845 & & 4.3 & 14.1 : & 14.2 & 14.7 : & be / x rxj0117.6 - 7330 , 22.07 s pulsar ( mfh99 , crw97 ) + 60 & 501 & & & 01 19 37.6 & -73 30 06 & 12.9 & 001 & 1.5@xmath12 : & 1867 & & 8.3 & 15.1 : & & 15.8 : & be / x ? + references : + ( chr99 ) @xcite 1999 , ( clc98a ) @xcite 1998a , ( cml98b ) @xcite 1998b , ( co00 ) @xcite 2000 , ( crw97 ) @xcite 1997 , ( csm97 ) @xcite 1997 , ( fhw98 ) @xcite 1998 , ( fhp00a ) @xcite 2000a , ( fph00b ) @xcite 2000b , ( gh85 ) @xcite 1985 , ( h94 ) @xcite 1994 , ( hs94 ) @xcite 1994 , ( isa97 ) @xcite 1997 , ( isc98 ) @xcite 1998 , ( isc99 ) @xcite 1999 , ( iyk98 ) @xcite 1998 , ( kp96 ) @xcite 1996 , ( kp98 ) @xcite 1998 , ( kpfh99 ) @xcite 1999 , ( l98 ) @xcite 1998 , ( lpm99 ) @xcite 1999 , ( m92 ) @xcite 1992 , ( mfh99 ) @xcite 1999 , ( ml98 ) @xcite 1998 , ( rlg94 ) @xcite 1994 , ( sc96 ) @xcite 1996 , ( sc98 ) @xcite 1998 , ( scb99 ) @xcite 1999 , ( scc99 ) @xcite 1999 , ( sci98 ) @xcite 1998 , ( wf98 ) @xcite 1998 , ( yk98a ) @xcite 1998a , ( yk98b ) @xcite 1998b , ( yk98c ) @xcite 1998c

a correlation of x - ray source and h@xmath0 emission - line object catalogues in the small magellanic cloud ( smc ) shows that more than two thirds of the optically identified be stars in be / x - ray binaries are found as emission - line objects in the catalogues . on the basis of this result we propose up to 25 x - ray sources mainly from recent rosat catalogues as new be / x - ray binaries and give their likely optical counterparts . also for the five yet unidentified x - ray pulsars in the smc we propose emission - line stars as counterparts . this more than doubles the number of known high mass x - ray binary systems in this nearby galaxy . the spatial distribution of the new candidates is similar to that of the already identified be / x - ray binaries with a strong concentration along the smc main body and some systems in the eastern wing . the new candidates contribute mainly to the low - luminosity end of the x - ray luminosity distribution of be / x - ray binaries . a comparison with the luminosity distribution in the milky way reveals no significant differences at the high - luminosity end and the large number of low - luminosity systems in the smc suggests that many such systems may still be undetected in the galaxy . the overall ratio of known be to ob supergiant x - ray binaries in the smc is an order of magnitude larger than in the galaxy , however , might show spatial variations . while in the eastern wing the ratio is comparable to that in the galaxy no supergiant x - ray binary is currently known in the main body of the smc . possible explanations include a different star formation history over the last @xmath1 my .

You are an expert at summarizing long articles. Proceed to summarize the following text: with the introduction of microarrays biologist have been witnessing entire labs shrinking to matchbox size . this paper invites quality researchers to join scientists on their _ fantastic journey _ into the world of microscopic high - throughput measurement technologies . building a biological organism as laid out by the genetic code is a multi - step process with room for variation at each step . the first steps , as described by the _ dogma of molecular biology , _ are genes ( and dna sequence in general ) , their transcripts and proteins . substantial factors contributing to their variation in both structure and abundance include cell type , developmental stage , genetic background and environmental conditions . connecting molecular observations to the state of an organism is a central interest in molecular biology . this includes the study of the gene and protein functions and interactions , and their alteration in response to changes in environmental and developmental conditions . traditional methods in molecular biology generally work on a `` one gene ( or protein ) in one experiment '' basis . with the invention of _ microarrays _ huge numbers of such macromolecules can now be monitored in one experiment . the most common kinds are _ gene expression microarrays , _ which measure the mrna transcript abundance for tens of thousands of genes simultaneously . for biologists , this high - throughput approach has opened up entirely new avenues of research . rather than experimentally confirming the hypothesized role of a certain candidate gene in a certain cellular process , they can use genome - wide comparisons to screen for all genes which might be involved in that process . one of the first examples of such an exploratory approach is the expression profiling study of mitotic yeast cells by @xcite which determined a set of a few hundred genes involved in the cell cycle and triggered a cascade of articles re - analyzing the data or replicating the experiment . microarrays have become a central tool in cancer research initiated by the discovery and re - definition of tumor subtypes based on molecular signatures ( see e.g. @xcite , @xcite , @xcite , @xcite ) . in section [ b ] we will explain different kinds of microarray technologies in more detail and describe their current applications in life sciences research . a dna microarray consists of a glass surface with a large number of distinct fragments of dna called probes attached to it at fixed positions . a fluorescently labelled sample containing a mixture of unknown quantities of dna molecules called the target is applied to the microarray . under the right chemical conditions , single - stranded fragments of target dna will base pair with the probes which are their complements , with great specificity . this reaction is called hybridization , and is the reason dna microarrays work . the fixed probes are either fragments of dna called complementary dna ( cdna ) obtained from messenger rna ( mrna ) , or short fragments known to be complementary to part of a gene , spotted onto the glass surface , or synthesized in situ . the point of the experiment is to quantify the abundance in the target of dna complementary to each particular probe , and the hybridization reaction followed by scanning allows this to be done on a very large scale . the raw data produced in a microarray experiment consists of scanned images , where the image intensity in the region of a probe is proportional to the amount of labelled target dna that base pairs with that probe . in this way we can measure the abundance of thousands of dna fragments in a target sample . microarrays based on cdna or long oligonucleotide probes typically use just one or a few probes per gene . the same probe sequence spotted in different locations , or probe sequences complementary to different parts of the same gene can be used to give within array replication . short oligonucleotide microarrays typically use a larger number per gene , e.g. 11 for the hu133 affymetrix array per gene . such a set of 11 is called probeset for that gene , and the probes in a probe set are arranged randomly over the array . in the biological literature , microarrays are also referred to as ( gene ) chips or slides . when the first microarray platforms were introduced in the early 90s , the most intriguing fact about them was the sheer number of genes that could be assayed simultaneously . assays that used to be done one gene at a time , could suddenly be produced for thousands of genes at once . a decade later , high - density microarrays would even fit entire genomes of higher organisms . after the initial euphoria , the research community became aware that findings based solely on microarray measurements were not always as reproducible as they would have liked and that studies with inconclusive results were quite common . with this high - throughput measurement technology becoming established in many branches of life sciences research , scientists in both academic and corporate environments raised their expectations concerning the validity of the measurements . data quality issues are now frequently addressed at meetings of the microarray gene expression database group ( mged ) . the _ microarray quality control project , _ a community - wide effort , under the auspices of the u.s . food and drug administration ( fda ) , is aiming at establishing _ operational metrics _ to objectively assess the performance of seven microarray platform and develop minimal quality standards . their assessment is based on the performance of a set of standardized external rna controls . the first formal results of this project have been published in a series of articles in the september 2006 issue of _ nature biotechnology . _ assessing the _ quality of microarray data _ has emerged as a new research topic for statisticians . in this paper , we conceptualize microarray data quality issues from a perspective which includes the technology itself as well as their practical use by the research community . we characterize the nature of microarray data from a quality assessment perspective , and we explain the different levels of microarray data quality assessment . then we focus on short oligonucleotide microarrays to develop a set of specific statistical data quality assessment methods including both numerical measures and spatial diagnostics . assumptions and hopes about the quality of the measurements have become a major issue in microarray purchasing . despite their substantially higher costs , affymetrix short oligonucleotide microarrays have become a widespread alternative to cdna chips . informally , they are considered the industrial standard among all microarray platforms . more recently , agilent s non - contact printed high - density cdna microarrays and illumina s bead arrays have fueled the competition for high quality chips . scientist feel the need for systematic quality assessment methods allowing them to compare different laboratories , different chip generation , or different platforms . they even lack good methods for selecting chips of good enough quality to be included in statistical data analysis beyond preprocessing . we have observed several questionable practices in the recent past : * skipping hybridization qa / qc all together * discarding entire batches of chips following the detection of a few poor quality chips * basing hybridization qa / qc on raw data rather than data that has already been had large - scale technical biases removed * delaying any qa / qc until all hybridizations are completed , thereby losing the opportunity to remove specific causes of poor quality at an early stage * focussing on validation by another measurement technolgy ( e.g , quantitative pcr ) in publication requirements rather than addressing the quality of the microarray data in the first place * merging of data of variable quality into one database with the inherent risk of swamping it with poor quality data ( as this produced at a faster rate due to few replicates , less quality checks , less re - doing of failed hybridizations etc . ) the community of microarray users has not yet agreed on a framework to measure accurary or precision in microarray experiments . without universally accepted methods for quality assessment , and guidelines for acceptance , statisticians judgements about data quality may be perceived as arbitrary by experimentalists . users expectations as to the level of gene expression data quality vary substantially . they can depend on time frame and financial constraints , as well as on the purpose of their data collection . @xcite , p.120/21 , explained the standpoint of the applied scientist : + _ he knows that if he were to act upon the meagre evidence sometimes available to the pure scientist , he would make the same mistakes as the pure scientist makes in estimates of accuracy and precisions . he also knows that through his mistakes someone may lose a lot of money or suffer physical injury or both . [ ... ] he does not consider his job simply that of doing the best he can with the available data ; it is his job to get enough data before making this estimate . _ + following this philosophy , microarray data used for medical diagnostics should meet high quality standards . in contrast , microarray data collected for a study of the etiology of a complex genetic disease in a heterogeneous population , one may decide to tolerate lower standards at the level of individual microarrays and invest the resources in a larger sample size . scientists need informative quality assessment tools to allow them to choose the most appropriate technology and optimal experimental design for their precision needs , within their time and budget constraints . the explicit goals of quality assessment for microarrays are manifold . which goals can be envisioned depends on the resources and time horizon and on the kind of user single small user , big user , core faculity , multi - center study , or `` researcher into quality '' . the findings can be used to simply exclude chips from further study or recommend to have samples reprocessed . they can be imbedded in a larger data quality management and improvement plan . typical quality phenomena to look for include : * outlier chips * trends or patterns over time * effects of particular hybridization conditions and sample characteristics * changes in quality between batches of chips , cohorts of samples , lab sites etc . * systematic quality differences between subgroups of a study some aspects of quality assessment and control for cdna have been discussed in the literature . among these , @xcite and @xcite emphasize the need for quality control and replication . @xcite define a quality score for each spot based on intensity characteristics and spatial information , while @xcite approach this with baysian networks . @xcite and @xcite suggest explicit statistical quality measures based on individual spot observations using the image analysis software spot from @xcite . @xcite apply multivariate statistical process control to detect single outlier chips . the preprocessing and data management software package arraymagic of @xcite includes quality diagnostics . the book by @xcite is a comprehensive collection of quality assessment and control issues concerning the various stages of cdna microarray experiments including sample preparation , all from an experimentalist s perspective . @xcite suggest spot quality scores based on the variance of the ratio estimates of replicates ( on the same chip or on different chips ) . spatial biases have also been addressed . in examining the relationship between signal intensity and print - order , @xcite reveals a plate - effect . the normalization methodology by @xcite incorporates spatial information such as print - tip group or plate , to remove spatial biases created by the technological processes . @xcite and @xcite found pairwise correlations between genes due to their relative positioning of the spots on the slide and suggest a localized mean normalization method to adjust for this . @xcite proposed a method of identifying poor quality spots , and of addressing this by assigning quality weights . @xcite developed an approach for the visualization and quantitation of regional bias applicable to both cdna and affymetrix microarrays . for affymetrix arrays , the commercial software @xcite includes a _ quality report _ with a dozen scores for each microarray ( see subsection [ d_a ] ) . none of them makes use of the gene expression summaries directly , and there are no universally recognized guidelines as to which range should be considered good quality for each of the gcos quality scores . users of short oligonucleotide chips have found that the quality picture delivered by the gcos quality report is incomplete or not sensitive enough , and that it is rarely helpful in assigning causes to poor quality . the literature on quality assessment and control for short oligonucleotide arrays is still sparse , though the importance of the topic has been stressed in numerous places , and some authors have addressed have looked at specific issues . an algorithm for probeset quality assessment has been suggested by @xcite . @xcite transfer the weight of a measurement to a subset of probes with optimal linear response at a given concentration . @xcite investigate the effect of updating the mapping of probes to genes on the estimated expression values . @xcite define four types of degenerate probe behaviour based on free energy computations and pattern recognition . @xcite evaluated the affymetrix quality reports of over 5,000 chips collected by st . jude children s research hospital over a period of three years , and linked some quality trends to experimental conditions . @xcite extend traditional effect size models to combine data from different microarray experiments , incorporating a quality measure for each gene in each study . the detection of specific quality issues such as the extraction , handling and amount of rna , has been studied by several authors ( e.g.@xcite , @xcite , @xcite , @xcite ) . before deriving new methods for assessing microarray data quality , we will relate the issue to established research into data quality from other academic disciplines , emphasizing the particular characteristics of microarray data ( section [ d_c ] ) . a conceptual approach to the statistical assessment of microarray data quality is suggested in subsection [ d_a ] , and is followed by a summary of the existing quality measures for affymetrix chips . the theoretical basis of this paper is section [ m ] , where we introduce new numerical and spatial quality assessment methods for short oligonucleotide arrays . two important aspects of our approach are : * the quality measures are based on _ all the data _ from the array . * the quality measures are computed after hybridization and data preprocessing . more specifically , we make use of probe level and probeset level quantities obtained as by - products of the robust multichip analysis ( rma / fitplm ) preprocessing algorithm presented in @xcite , @xcite and @xcite . our _ quality landscapes _ serve as tools for visual quality inspection of the arrays after hybridization . these are two dimensional pseudo - images of the chips based on probe level quantities , namely the weights and residuals computed by rma / fitplm . these quality landscapes allow us to immediately relate quality to an actual location on the chip , a crucial step in detecting special causes for poor chip quality . our numerical quality assessment is based on two distributions computed at the probeset level , the _ normalized unscaled standard error ( nuse ) _ and the _ relative log expression ( rle ) . _ given a fairly general biological assumption is fulfilled , these distributions can be interpreted for chip quality assessment . we further suggest ways of conveniently visualizing and summarizing these distributions for larger chip sets and of relating this quality assessment with other factors in the experiment , to permit the detection of special causes for poor quality to reveal biases . quality of gene expression data can be assessed on a number of levels , including that of probeset , chip and batch of chips . another aspect of quality assessment concerns batches of chips . we introduce the _ residual scale factor ( rsf ) , _ a measure of chip batch quality . this allows us to compare quality across batches of chips within an experiment , or across experiments . all our measures can be computed for all available types of short oligonuceotide chips given the raw data ( cel file ) for each chip and the matching cdf file . software packages are described in @xcite and available at www.bioconductor.org . in section [ r ] we extensively illustrate and evaluate our quality assessment methods on the experimental microarray datasets described in section [ s ] . to reflect the fact that quality assessment is a necessary and fruitful step in studies of any kind , we use a variety of datasets , involving tissues ranging from fruit fly embryos to human brains , and from academic , clinical , and corporate labs . we show how quality trends and patterns can be associated with sample characteristics and/or experimental conditions , and we compare our measures with the affymetrix gcos quality report . after the hunt for new genes has dominated genetics in the 80s and 90s of the last century , there has be a remarkable shift in molecular biology research goals towards a comprehensive understanding of the function of macromolecules on different levels in a biological organism . how and to what extend do genes control the construction and maintenance of the organism ? what is the role of intermediate gene products such as _ rna transcripts _ ? how do the macromolecules interact with others ? the latter may refer to _ horizontal _ interaction , such as genes with genes , or proteins with proteins . it may also refer to _ vertical _ interaction , such as between genes and proteins . _ genomics _ and _ proteomics _ in professional slang summarized as _ omics sciences _ have started to put an emphasis on _ functions . _ as the same time , these research areas have become more quantitative , and they have broadened the perspective in the sense of observing huge numbers of macromolecules simultaneously . these trends have been driven by recent biotechnological inventions , the most prominent ones being _ microarrays . _ with these _ high - throughput _ molecular measurement instruments , the relative concentration of huge numbers of macromolecules can be obtained simultaneously in one experiment . this section will give an overview of the biological background and the applications of microarrays in biomedical research . for an extended introduction to _ omics sciences _ and to microarray - based research we refer to the excellent collections of articles in the three nature genetics supplements _ the chipping forecast i , ii , iii _ ( 1999 , 2002 , 2005 ) and to the recent review paper by @xcite . though the popular belief about genes is still very deterministic once they are put into place , they function in a preprogrammed straight forward way for biologists the effect of a gene is variable . most cells in an organism contain essentially the same set of genes . however , cells will look and act differently depending on which organ they belong to , the state of the organ ( e.g.healthy vs.diseased ) , the developmental stage of the cell , or the phase of the cell cycle . this is predominantly the result of differences in the abundance , distribution , and state of the cells proteins . according to the _ central dogma of molecular biology _ the production of proteins is controlled by dna ( for simplicity , the exceptions to this rule are omitted here ) . proteins are polymers built up from 20 different kinds of amino acids . genes are _ transcribed _ into dna - like macromolecules called _ messenger rna ( mrna ) _ , which goes from the chromosomes to the _ ribosomes . _ there , _ translation _ takes place , converting mrna into the amino acid chains which fold into proteins . the term _ gene expression _ is defined as the relative concentration of mrna and protein produced by that gene . depending on the context , however , it is often used to refer to only one of the two . the _ gene expression profile _ of a type of cell usually refers to the relative abundance of each of the mrna species in the total cellular mrna population . from a practical point of view , in particular by many areas of medical research , protein abundance is seen as generally more interesting than mrna abundance . the measurement of protein abundances , however , is still much more difficult to measure on a large scale than mrna abundance . there is one property which is peculiar to nucleic acids : their complementary structure . dna is reliably replicated by separating the two strands , and complementing each of the single strands to give a copy of the original dna . the same mechanism can be used to detect a particular dna or rna sequence in a mixed sample . the first tool to measure gene expression in a sample of cells of a was introduced in 1975 . southern blot _ named for its inventor is a multi - stage laboratory procedure which produces a pattern of bands representing the activity of a small set of pre - selected genes . during the 1980s spotted arrays on nylon holding bacterial colonies carrying different genomic inserts were introduced . in the early 1990s , the latter would be exchanged for preidentified cdnas . the introduction of _ gene expression microarrays _ on glass slides in the mid 1990s brought a substantial increase in feature density . with the new technology , gene expression measurements could be taken in parallel for thousands of genes . modern microarray platforms even assess the expression levels of tens of thousands of genes simultaneously . a gene expression microarray is a small piece of glass onto which _ a priori _ known dna fragments called _ probes _ are attached at fixed positions . in a chemical process called _ hybridization , _ the microarray is brought into contact with material from a sample of cells . each probe binds to its complementary counterpart , an mrna molecule ( or a complementary dna copy ) from the sample , which we refer to as the _ target_. the hybridization reaction product is made visible using fluorescent dyes or other ( e.g.radioactive ) markers , which are applied to the sample prior to hybridization . the readout of the microarray experiment is a scanned image of the labelled dna . microarrays are specially designed to interrogate the genomes of particular organisms , and so there are yeast , fruit fly , worm and human arrays , to name just a few . there are three major platforms for microarray - based gene expression measurement : _ spotted two - color cdna arrays , _ _ long oligonucleotide arrays _ and _ short oligonucleotide arrays . _ in the platform specific parts of this paper we will focus on the latter . on a short oligonucleotide microarray , each gene is represented on the array by a _ probe set _ that uniquely identifies the gene . the individual probes in the set are chosen to have relatively uniform hybridization characteristics . in the affymetrix hu133 arrays , for example , each probe set consists of 11 to 20 probe sequence pairs . each pairs consists of a _ perfect match ( pm ) _ probe , a 25 bases long oligonucleotide that matches a part of the gene s sequence , and a corresponding _ mismatch ( mm ) _ probe , that has the same sequence as the pm except for the center base being flipped to its complementary letter . the mm probes are intended to give an estimate of the random hybridization and cross hybridization signals , see @xcite and @xcite for more details . other affymetrix gene expression arrays may differ from the hu133 in the number of probes per probe set . exon arrays do not have mm probes . most of the arrays produced by nimblegen are composed from 60mer probes , but some are using 25mer probes . the number of probes per probeset is adapted to the total number of probesets on the array to make optimal use of the space . besides being more efficient than the classical gene - by - gene approach , microarrays open up entirely new avenues for research . they offer a comprehensive and cohesive approach to measuring the activity of the genome . in particular , this fosters the study of interactions . a typical goal of a microarray based research project is the search for genes that behave differently between different cell populations . some of the most common examples for comparisons are diseased vs. healthy cells , injured vs. healthy tissue , young vs. old organism , treated vs. untreated cells . more explicitly , life sciences researchers try to find answers to questions such as the following . which genes are affected by environmental changes or in response to a drug ? how do the gene expression levels differ across various mutants ? what is the gene expression signature of a particular disease ? which genes are involved in each stage of a cellular process ? which genes play a role in the development of an organism ? or , more generally , which genes vary their activity with time ? the principle of microarray measurement technology has been used to assess molecules other than mrna . a number of platforms are currently at various stages of development ( see review by @xcite ) . snp chips detect single nucleotide polymorphisms . they are an example for a well developed microarray - based genotyping platform . cgh arrays are based on comparative genome hybridization . this method permits the analysis of changes in gene copy number for huge numbers of probes simultaneously . a recent modification , representational oligonucleotide microarray analysis ( roma ) , offers substantially better resolution . both snp chips and cgh arrays are genome - based methods , which , in contrast to the gene expression - based methods , can exploit the stability of dna . the most common application of these technologies is the localization of disease genes based on association with phenotypic traits . antibody protein chips are used to determine the level of proteins in a sample by binding them to antibody probes immobilized on the microarray . this technology is still considered semi - quantitative , as the different specificities and sensitivities of the antibodies can lead to an inhomogeneity between measurements that , so far , can not be corrected for . the applications of protein chips are similar to the ones of gene expression microarrays , except that the measurements are taken one step further downstream . more recent platforms address multiple levels at the same time . chip - on - chip , also known as genome - wide location analysis , is a technique for isolation and identification of the dna sequences occupied by specific dna binding proteins in cells . the still growing list of statistical challenges stimulated by microarray data is a _ tour dhorizon _ in applied statistics ; see e.g. @xcite , @xcite and @xcite for broad introductions . from a statistical point of view a microarray experiment has three main challenges : ( i ) measurement process as multi - step biochemical and technological procedure ( array manufacturing , tissue acquisition , sample preparation , hybridization , scanning ) with each step contributing to the variation in the data ; ( ii ) huge numbers measurements of different ( correlated ) molecular species being take in parallel ; ( iii ) unavailability of gold - standards covering a representative part of these species . statistical methodology has primarily been developed for gene expression microarrays , but most of the conceptual work applies directly to many kinds of microarrays and many of the actual methods can be transferred to other microarray platforms fitting the characteristics listed above . the first steps of the data analysis , often referred to as _ preprocessing _ or _ low level analysis _ , are the most platform - dependent tasks . for two - color cdna arrays this includes image analysis ( see e.g. @xcite ) and normalization ( see e.g. @xcite ) . for short oligonucleotide chip data this includes normalization ( see e.g. @xcite ) and the estimation of gene expression values ( see e.g. @xcite and @xcite as well as subsequent papers by these groups ) . questions around the design of micorarray experiments are mostly relevant for two - color platforms ( see e.g. ch . 2 in @xcite , @xcite and further references there ) . analysis beyond the preprocessing steps is often referred to as _ downstream analysis . _ the main goal is to identify genes which act differently in different types of samples . exploratory methods such as classification and cluster analysis have quickly gained popularity for microarray data analysis . for reviews on such methods from a statistical point of view see e.g. ch . 2 and ch . 3 in @xcite and ch . 3 - 7 in @xcite . on the other side of the spectrum , hypothesis - driven inferential statistical methods are now well established and used . this approach typically takes a single - gene perspective in the sense that it searches for _ individual _ genes that are expressed differentially across changing conditions ; see e.g. @xcite . the main challenge is the imprecision of the gene - specific variance estimate , a problem that has been tackled by strategies incorporating a gene - unspecific component into the estimate ; see e.g. @xcite , @xcite , @xcite and references therein , and @xcite for the case of microarray time course data . testing thousands of potentially highly correlated genes at the same time with only a few replicates raises a substantial multiple testing problem that has been systematically addressed by various authors incorporating benyamini s and hochberg s _ false discovery rate ( fdr ) _ ; see e.g. @xcite and the review @xcite . the joint analysis of pre - defined groups of genes based on _ a priori _ knowledge has become an established alternative to the genome - wide exploratory approaches and the gene - by - gene analysis ; see e.g. @xcite and @xcite . while methodology for microarray data analysis has become a fast growing research area , the epistemological foundation of this research area shows gaps . among other issues , @xcite addresses the problem of simultaneous validation of research results and research methods . @xcite offer a review of the main approaches to microarray data analysis developed so far and attempt to unify them . many software packages for microarray data analysis have been made publicly available by academic researchers . in particular , there is the bioconductor project , a community - wide effort to maintain a collection of r - packages for genomics applications at www.bioconductor.org . many of the main packages are described in @xcite . data quality is a well established aspect of many quantitative research fields . the most striking difference between assessing the quality of a measurement as opposed to assessing the quality of a manufactured item is the additional layer of uncertainty . concerns around the accuracy of measurements have a long tradition in physics and astronomy ; the entire third chapter of the classical book @xcite is devoted to this field . biometrics , psychometrics , and econometrics developed around similar needs , and many academic fields have grown a strong quantitative branch . all of them facing data quality questions . clinical trials is a field that is increasingly aware of the quality of large data collections ( see @xcite and other papers in this special issue ) . with its recent massive move into the quantitative field , functional genomics gave birth to what some statisticians call _ genometrics . _ we now touch on the major points that characterize gene expression microarray data from the point of view of qa / qc . these points apply to other high - dimensional molecular measurements as well . * unknown kind of data : * being a new technology in the still unknown terrain of functional genomics , microarrays produce datasets with few known statistical properties , including shape of the distribution , magnitude and variance of the gene expression values , and the kind of correlation between the expression levels of different genes . this limits access to existing statistical methods . * simultaneous measurements : * each microarray produces measurements for thousands of genes simultaneously . if we measured just one gene at a time , some version of shewhart control charts could no doubt monitor quality . if we measured a small number of genes , multivariate extensions of control charts might be adequate . in a way , the use of control genes is one attempt by biologists to scale down the task to a size that can be managed by these classical approaches . control genes , however , can not be regarded as typical representatives of the set of all the genes on the arrays . gene expression measures are correlated because of both the biological interaction of genes , and dependencies caused by the common measurement process . biologically meaningful correlations between genes can potentially `` contaminate '' hybridization quality assessment . * multidisciplinary teams : * microarray experiments are typically planned , conducted and evaluated by a team which may include scientists , statisticians , technicians and physicians . in the interdisciplinarity of the data production and handling , they are similar to large datasets in other research areas . for survey data , @xcite names the risk associated with such a _ `` mlange of workers''_. among other things , he mentions : radically different purposes , lack of communication , disagreements on the priorities among the components of quality and concentration on the `` error of choice '' in their respective discipline . the encouragement of a close cooperation between scientists and statisticians in the care for measurement quality goes all the way back to @xcite , p.70/71 : _ `` where does the statistician s work begin ? [ ... ] before one turns over any sample of data to the statistician for the purpose of setting tolerances he should first ask the scientist ( or engineer ) to cooperate with the statistician in examining the available evidence of statistical sontrol . the statistician s work solely as a statistician begins after the scientist has satisfied himself through the application of control criteria that the sample has arisen under statistically controlled conditions . '' _ * systematic errors : * as pointed out by @xcite , and , in the context of clinical trials , by @xcite , systematic errors in large datasets are much more relevant than random errors . microarrays are typically used in studies involving different experimental or observational groups . quality differences between the groups are a potential source of confounding . * heterogenous quality in data collections : * often microarray data from different sources are merged into one data collection . this includes different batches of chips within the same experiment , data from different laboratories participating in a single collaborative study , or data from different research teams sharing their measurements with the wider community . depending on the circumstances , the combination of data typically takes place on one of the following levels : raw data , preprocessed data , gene expression summaries , lists of selected genes . typically , no quality measures are attached to the data . even if data are exchanged at the level of cel files , heterogeneity can cause problems . some laboratories filter out chips or reprocess the samples that were hybridized to chips that did not pass screening tests , others do not . these are decision processes that ideally should take place according to the same criteria . the nature of this problem is well known in data bank quality or _ data warehousing _ ( see e.g. @xcite , @xcite , @xcite ) . * re - using of shared data : * gene expression data are usually generated and used to answer a particular set of biological questions . data are now often being placed on the web to enable the general community to verify the analysis and try alternative approaches to the original biological question . data may also find a secondary use in answering modified questions . the shifted focus potentially requires a new round of qa / qc , as precision needs might have changed and artifacts and biases that did not interfere with the original goals of the experiment may do so now . * across - platform comparison : * @xcite , p. 112 , already values the consistency between different measurement methods higher than consistency in repetition . for microarrays , consistency between the measurements of two or more platforms ( two - color cdna , long oligonucleotide , short oligonucleotide ( affymetrix ) , commercial cdna ( agilent ) , and real - time pcr ) on rna from the same sample has been addressed in a number of publications . some of the earlier studies show little or no agreement ( e.g. @xcite , @xcite , @xcite , @xcite ) , while others report mixed results ( e.g.@xcite , @xcite , @xcite ) . more recent studies improved the agreement between platforms by controlling for other factors . @xcite and @xcite restrict comparisons to subsets of genes above the noise level . @xcite use sequence - based matching of probes instead of gene identifier - based matching . @xcite , @xcite and @xcite use superior preprocessing methods and systematically distinguish the lab effect from the platform effect ; see @xcite and @xcite for detailed reviews and further references . for affymetrix arrays , @xcite , @xcite and @xcite found inter - laboratory differences to be managable . however , merging data from different generations of affymetrix arrays is not as straightforward as one might expect ( e.g. @xcite , @xcite , @xcite , @xcite , @xcite ) . quality assessment for microarray data can be studied on at least seven levels : * the raw chip ( pre - hybridization ) * the sample * the experimental design * the multi - step measurement process * the raw data ( post - hybridization ) * the statistically preprocessed microarray data * the microarray data as entries in a databank the last two items are the main focus of this paper . the quality of the data after statistical processing ( which includes background adjustment , normalization and probeset summarization ) is greatly affected , but not entirely determined by the quality of the preceeding five aspects . the raw microarray data ( 5 ) are the result of a multi - step procedure . in the case of the expression microarrays this includes converting mrna in the sample to cdna , labelling the target mrna via an in vitro transcription step , fragmenting and then hybridizing the resulting crna to the chip , washing and staining , and finally scanning the resulting array . temperature during storage and hybridization , the amount of sample and mixing during hubridization all have a substantial impact on the quality of the outcome . seen as a multi - step process ( 4 ) the quality management for microarray experiments has a lot in common with chemical engineering , where numerous interwoven quality indicators have to be integrated ( see e.g. @xcite ) . the designer of the experiment ( 3 ) aims to minimize the impact of additional experimental conditions ( e.g. hybridization date ) and to maximize accuracy and precision for the quantities having the hightest priority , given the primary objectives of the study . sample quality ( 2 ) is a topic in its own right , strongly tied to the organism and the institutional setting of the study . the question how sample quality is related to the microarray data has been investigated in @xcite based on a variety of rna quality measures and chip quality measures including both affymetrix scores and and ours . the chip before hybridization ( 1 ) is a manufactured item . the classical theory of quality control for industrial mass production founded by @xcite provides the appropriate framework for the assessment of the chip quality before hybridization . the affymetrix software gcos presents some chip - wide quality measures in the expression report ( rtp file ) . they can also be computed by the bioconductor r package simpleaffy described in @xcite . the document `` qc and affymetrix data '' contained in this package discusses how these metrics can be applied . the quantities listed below are the most commonly used ones from the affymetrix report ( descriptions and guidelines from @xcite and @xcite ) . while some ranges for the values are suggested , the manuals mainly emphasize the importance of _ consistency _ of the measures within a set of jointly analyzed chips using similar samples and experimental conditions . the users are also encouraged to look at the scores in conjuction with others scores . * * average background : * average of the lowest 2@xmath1 cell intensities on the chip . affymetrix does not issue official guidelines , but mentions that values typically range from 20 to 100 for arrays scanned with the genechip scanner 3000 . a high background indicates the presence of nonspecific binding of salts and cell debris to the array . * * raw q ( noise ) : * measure of the pixel - to - pixel variation of probe cells on the chip . the main factors contributing to noise values are electrical noise of the scanner and sample quality . older recommendations give a range of 1.5 to 3 . newer sources , however , do not issue official guidelines because of the strong scanner dependence . they recommend that data acquired from the same scanner be checked for comparability of noise values . * * percent present : * the percentage of probesets called _ present _ by the affymetrix detection algorithm . this value depends on multiple factors including cell / tissue type , biological or environmental stimuli , probe array type , and overall quality of rna . replicate samples should have similar percent present values . extremely low percent present values indicate poor sample quality . a general rule of thumb is human and mouse chips typically have 30 - 40 percent present , and yeast and e. coli have 70 - 90 percent present . * * scale factor : * multiplicative factor applied to the signal values to make the 2@xmath1 trimmed mean of signal values for selected probe sets equal to a constant . for the hu133 chips , the default constant is 500 . no general recommendation for an acceptable range is given , as the scale factors depend on the constant chosen for the scaling normalization ( depending on user and chip type ) . * * gapdh 3 to 5 ratio ( gapdh 3/5 ) : * ratio of the intensity of the 3 probe set to the 5 probe set for the gene gapdh . it is expected to be an indicator of rna quality . the value should not exceed 3 ( for the 1-cycle assay ) . * perfect match ( pm ) : * the distribution of the ( raw ) pm values . while we do not think of this as a full quality assessment measure , it can indicate particular phenomena such as brightness or dimness of the image , or saturation . using this tool in combination with other quality measures , can help in detecting and excluding technological reasons for poor quality . a convenient way to look at the pm distributions for a number of chips is to use boxplots . alternatively , the data can be summarized on the chip level by two single values : the median of the pm of all probes on the chip , abbreviated _ med(pm ) , _ and the interquartile range of the pm of all probes on the chip , denoted by _ _ our other assessment tools use probe level and probeset level quantities obtained as a by - product of the robust multichip analysis ( rma ) algorithm developed in @xcite , @xcite and @xcite . we now recall the basics about rma and refer the reader to above papers for details . consider a fixed probeset . let @xmath2 denote the intensity of probe @xmath3 from this probeset on chip @xmath4 usually already background corrected and normalized . rma is based on the model @xmath5 with @xmath6 a _ probe affinity effect _ , @xmath7 representing the log scale expression level for chip @xmath4 and @xmath8 an i.i.d . centered error with standard deviation @xmath9 . for identifiability of the model , we impose a zero - sum constraint on the @xmath10s . the number of probes in the probeset depends on the kind of chip ( e.g.11 for the hu133 chip ) . for a fixed probeset , rma robustly fits the model using iteratively weighted least squares and delivers a probeset expression index @xmath11 for each chip . the analysis produces residuals @xmath12 and weights @xmath13 attached to probe @xmath3 on chip @xmath14 the weights are used in the irls algorithm to achieve robustness . probe intensities which are discordant with the rest of the probes in the set are deemed less reliable and downweighted . the collective behaviour of all the weights ( or all the residuals ) on a chip is our starting point in developing post - hybridization chip quality measures . we begin with a `` geographic '' approach images of the chips that highlight potential poorly performing probes and then continue with the discussion of numerical quality assessment methods . * quality landscapes : * an image of a hybridized chip can be constructed by shading the positions in a rectangular grid according to the magnitude of the perfect match in the corresponding position on the actual chip . in the same way , the positions can be colored according to probe - level quantities other than the simple intensities . a typical color code is to use shades of red for positive residuals and shades of blue for negative ones , with darker shades corresponding to higher absolute values . shades of green are used for the weights , with darker shades indicating lower weights . as the weights are in a sense the reciprocals of the absolute residuals , the overall information gained from these two types of quality landscapes is the same . in some particular cases , the sign of the residuals can help to detect patterns that otherwise would have been overlooked ( see both fruit fly datasets in sections [ r ] for examples ) . if no colors are available , gray level images are used . this has no further implications for the weight landscapes . for the residual landscapes , note that red and blue shades are translated into similar gray levels , so the sign of the residuals is lost . positive and negative residuals can plotted on two separate images to avoid this problem . * normalized unscaled standard error ( nuse ) : * fix a probeset . let @xmath15 be the estimated residual standard deviation in model ( [ rmamodel ] ) and @xmath16 the _ total probe weight _ ( of the fixed probeset ) in chip @xmath14 the expression value estimate for the fixed probeset on chip @xmath4 and its standard error are given by @xmath17 the residual standard deviations vary across the probesets within a chip . they provide an assessment of overall goodness of fit of the model to probeset data for all chips used to fit the model , but provide no information on the relative precision of estimated expressions across chips . the latter , however , is our main interest when we look into the quality of a chips compared to other chips in the same experiment . replacing the @xmath18 by @xmath19 gives what we call the _ unscaled standard error ( use ) _ of the expression estimate . another source of heterogeneity is the number of `` effective '' probes in the sense of being given substantial weight by the rma fitting procedure . that this number varies across probeset is obvious when different numbers of probes per probeset are used on the same chip another reason is dysfunctional probes , that is , probes with high variabiliy , low affinity , or a tendency to crosshybridize . to compensate for this kind of heterogeneity , we divide the use by its median over all chips and call this _ normalized unscaled standard error ( nuse ) . _ @xmath20 an alternative interpretation for the nuse of a fixed probeset becomes apparent after some arithmetic manipulations . for any odd number of positive observations @xmath21 we have @xmath22 since the function @xmath23 is monotone . for an even number @xmath24 this identity is it still approximatively true . ( the reason for the slight inaccuracy is that , for an even number , the median is the average between the two data points in the center positions . ) now we can rewrite @xmath25 the total probe weight can also be thought of as an _ effective number of observations _ contributing to the probeset summary for this chip . its square root serves as the divisor in the standard error of the expression summaries , similarly to the role of @xmath26 in the classical case of the average of @xmath27 independent observations . this analogy supposes , for heuristic purposes , that the probes are independent ; in fact this is not true due to normalization , probe overlap and other reasons . the median of the total probe weight over all chips serves as normalization constant . in the form , we can think of the nuse as the reciprocal of the normalized square root of total probe weight . the nuse values fluctuate around 1 . chip quality statements can be made based on the distribution of all the nuse values of one chip . as with the pm distributions , we can conveniently look at nuse distributions as boxplots , or we can summarize the information on the chip level by two single values : the median of the nuse over all probesets in a particular chip , _ med(nuse ) , _ and the interquartile range of the nuse over all probesets in the chip , _ iqr(nuse ) . _ * relative log expression ( rle ) : * we first need a reference chip . this is typically the _ median chip _ which is constructed probeset by probeset as the median expression value over all chips in the experiment . ( a computationally constructed reference chips such as this one is sometimes called `` virtual chip '' . ) to compute the rle for a fixed probeset , take the difference of its log expression on the chip to its log expression on the reference chip . note that the rle is not tied to rma , but can be computed from any expression value summary . the rle measures how much the measurement of the expression of a particular probeset in a chip deviates from measurements of the same probeset in other chips of the experiment . again , we can conveniently look at the distributions as boxplots , or we can summarize the information on the chip level by two single values : the median of the rle over all probesets in a particular chip , _ med(rle ) , _ and the interquartile range of the rle over all probesets in the chip , _ iqr(rle ) . _ the latter is a measure of deviation of the chip from the median chip . a priori this includes both biological and technical variability . in experiments where it can be assumed that @xmath28 iqr(rle ) is a measure of technical variability in that chip even if biological variability is present for most genes , iqr(rle ) is still a sensitive detector of sources of technical variability that are larger than biological variability . med(rle ) is a measure of bias . in many experiments there are reasons to believe that @xmath29 in that case , any deviation of med(rle ) from @xmath30 is an indicator of a bias caused by the technology . the interpretation of the rle depends on the assumptions ( and ) on the biological variability in the dataset , but it provides a measure that is constructed independently of the quality landscapes and the nuse . for quality assessment , we summarize and visualize the nuse , rle , and pm distributions . we found series of boxplots to be very a convenient way to glance over sets up to 100 chips . outlier chips as well as trends over time or pattern related to time can easily be spotted . for the detection of systematic quality differences related to circumstances of the experiment , or to properties of the sample it is helpful to color the boxes accordingly . typical coloring would be according to groups of the experiment , sample cohort , lab site , hybridization date , time of the day , a property of the sample ( e.g.time in freezer ) . to quickly review the quality of larger sets of chips , shorter summaries such as the above mentioned median or the interquartile range of pm , nuse and rle . these single - value summaries at the chip level are also useful for comparing our quality measures to other chip quality scores in scatter plots , or for plotting our quality measures against continuous parameters related to the experiment or the sample . again , additional use of colors can draw attention to any systematic quality changes due to technical conditions . while the rle is a form of absolute measure of quality , the nuse is not . the nuse has no units . it is designed to detect differences _ between chips within a batch . _ however , the magnitudes of these differences have no interpretation beyond the batch of chips analyzed together . we now describe a way to attach a quality assessment to a set of chips as a whole . it is based on a common residual factor for a batch of jointly analyzed chips , rma estimates a common residual scale factor . it enables us to compare quality between different experiments , or between subgroups of chips in one experiment . it has no meaning for _ single _ chips . * residual scale factor ( rsf ) : * this is a quality measure for batches of chips . it does not apply to individual chips , but assesses the quality of batches of chips . the batches can be a series of experiments or subgroups of one experiment ( defined , e.g.by cohort , experimental conditions , sample properties , or diagnostic groups ) . to compute the rsf , assume the data are background corrected . as the background correction works on a chip by chip basis it does not matter if the computations were done simultaneously for all batches of chips or individually . for the normalization , however , we need to find one target distribution to which we normalize all the chips in all the batches . this is important , since the target distribution determines the scale of intensity measures being analyzed . we then fit the rma model to each batch separately . the algorithm delivers , for each batch , a vector of the estimated _ residual scales _ for all the probesets . we can now boxplot them to compare quality between batches of chips . the median of each is called _ residual scale factor ( rsf ) . _ a vector of residual scales is a heterogeneous set . to remove the heterogeneity , we can divide it , probeset by probeset , by the median over the estimated scales from all the batches . this leads to alternative definitions of the quantities above , which we call _ normalized residual scales _ and _ normalized residual scale factor ( nrsf ) . _ the normalization leads to more discrimination between the batches , but has the drawback of having no units . software for the computation and visualization of the quality measures and the interpretation of the statistical plots is discussed in @xcite . the code is publicly available from www.bioconductor.org in the r package affyplm . note that the implementation of the nuse in affyplm differs slightly from the above formula . it is based on the `` true '' standard error as it is comes from m - estimation theory instead of the total weights expression in [ m : nuse ] . however , the difference is small enough not to matter for any of the applications the nuse has in chip quality assessment . * affymetrix hu95 spike - in experiments : * here 14 human crna fragments corresponding to transcripts known to be absent from rna extracted from pancreas tissue were spiked into aliquots of the hybridization mix at different concentrations , which we call chip - patterns . the patterns of concentrations from the spike - in crna fragments across the chips form a latin square . the chip - patterns are denoted by a , b, ... ,s and t , with a, ... ,l occurring just once , and m and q being repeated 4 times each . chip patterns n , o and p are the same as that of m , while patterns r , s , and t are the same as q. each chip - pattern was hybridized to 3 chips selected from 3 different lots referred to as the l1521 , the l1532 , and the l2353 series . see www.affymetrix.com/support/technical/sample.data/datasets.affx for further details and data download . for this paper , we are using the data from the 24 chips generated by chip patterns m , n , o , p , q , r , s , t with 3 replicates each . * st . jude children s research hospital leukemia data collection : * the study by @xcite was conducted to determine whether gene expression profiling could enhance risk assignment for pediatric acute lymphoblastic leukemia ( all ) . the risk of relapse plays a central role in tailoring therapy intensity . a total of 389 samples were analyzed for the study , from which high quality gene expression data were obtained on 360 samples . distinct expression profiles identified each of the prognostically important leukemia subtypes , including t - all , e2a - pbx1 , bcr - abl , tel - aml1 , mll rearrangement , and hyperdiploid@xmath3150 chromosomes . in addition , another all subgroup was identified based on its unique expression profile . @xcite re - analized 132 cases of pediatric all from the original 327 diagnostic bone marrow aspirates using the higher density u133a and b arrays . the selection of cases was based on having sufficient numbers of each subtype to build accurate class predictions , rather than reflecting the actual frequency of these groups in the pediatric population . the follow - up study identified additional marker genes for subtype discrimination , and improved the diagnostic accuracy . the data of these studies are publicly available as supplementary data . * fruit fly mutant pilot study : * gene expression of nine fruit fly mutants were screened using affymetrix drosgenome1 arrays . the mutants are characterized by various forms of dysfunctionality in their synapses . rna was extracted from fly embryos , pooled and labelled . three to four replicates per mutant were done . hybridization took place on six different days . in most cases , technical replicates were hybridized on the same day . the data were collected by tiago magalhes in the goodman lab at the university of california , berkeley , to gain experience with the new microarray technology . * fruit fly time series : * a large population of wild type ( canton - s ) fruit flies was split into twelve cages and allowed to lay eggs which were transferred into an incubator and aged for 30 minutes . from that time onwards , at the end of each hour for the next 12 hours , embryos from one plate were washed on the plate , dechorionated and frozen in liquid nitrogen . three independent replicates were done for each time point . as each embryo sample contained a distribution of different ages , we examined the distribution of morphological stage - specific markers in each sample to correlate the time - course windows with the nonlinear scale of embryonic stages . rna was extracted , pooled , labeled and hybridized to affymetrix drosgenome1 arrays . hybridization took place on two different days . this dataset was collected by pavel tomanck in the rubin lab at the university of california , berkeley , as a part of their comprehensive study on spatial and temporal patterns of gene expression in fruit fly development @xcite . the raw microarray data ( .cel files ) are publically available at the project s website www.fruitfly.org/cgi-bin/ex/insitu.pl . * pritzker data collection : * the pritzker neuropsychiatric research consortium uses brains obtained at autopsy from the orange country coroner s office through the brain donor program at the university of california , irvine , department of psychiatry . rna samples are taken from the left sides of the brains . labeling of total rna , chip hybridization , and scanning of oligonucleotide microarrays are carried out at independent sites ( university of california , irvine ; university of california , davis ; university of michigan , ann arbor ) . hybridizations are done on hu95 and later generations of affymetrix chips . in this paper , we are looking at the quality of data used in two studies by the pritzker consortium . the _ gender study _ by @xcite is motivated by gender difference in prevalence for some neuropsychiatric disorders . the raw dataset has hu95 chip data on 13 subjects in three regions ( anterior cingulate cortex , dorsolateral prefrontal cortex , and cortex of the cerebellar hemisphere ) . the _ mood disorder study _ described in @xcite is based on a growing collection of gene expression measurements in , ultimately , 25 regions . each sample was prepared and then split so that it could be hybridized to the chips in both michigan and either irvine or davis . we start by illustrating our quality assessment methods on the well known affymetrix spike - in experiments . the quality of these chips is well above what can be expected from an average lab experiment . we then proceed with data collected in scientific studies from variety of tissue types and experimental designs . different aspects of quality analysis methods will be highlighted throughout this section . our quality analysis results will be compared with the affymetrix quality report for several sections of the large publicly available st . jude children s research hospital gene expression data collection . \(a ) * outlier in the affymetrix spike - in experiments : * 24 hu 95a chips from the affymetrix spike - in dataset . all but the spike - in probesets are expected to be non - differentially expressed across the arrays . as there are only 14 spike - ins out of about twenty thousand probesets , they are , from the quality assessment point of view , essentially 24 identical hybridizations . a glance at the weight ( or residual ) landscapes gives a picture of homogenous hybridizations with almost no local defects on any chip but # 20 ( fig . the nuse indicates that chip # 20 is an outlier . its median is well above 1.10 , while all others are smaller than 1.05 , and its iqr is three and more times bigger than it is for any other chip ( fig . the series of boxplots of the rle distributions confirms these findings . the median is well below @xmath32 and the iqr is two and more times bigger than it is for any other chip . chip # 20 has both a technologically caused bias and a higher noise level . the affymetrix quality report ( fig . a3 ) , however , does not clearly classify # 20 as an outlier . its gapdh 3/5 of about 2.8 is the largest within this chip set , but the value 2.8 is considered to be acceptable . according to all other affymetrix quality measures percent present , noise , background average , scale factor chip # 20 is within a group of lower quality chips , but does not stand out . \(b ) * outlier in st . jude s data not detected by the affymetrix quality report : * the collection of mll hu133b chips consists of 20 chips one of which turns out to be an outlier . the nuse boxplots ( fig . b1 , bottom line ) show a median over 1.2 for chip # 15 while all others are below 1.025 . the iqr is much larger for chip # 15 than it is for any other chip . the rle boxplots ( fig . b1 , top line ) as well distinguish chip # 15 as an obvious outlier . the median is about @xmath33 for the outlier chip , while it is very close to @xmath30 for all other chips . the iqr is about twice as big as the largest of the iqr of the other chips . b1 displays the weight landscapes of chip#15 along with those of two of the typical chips . a region on the left side of chip # 15 , covering almost a third of the total area , is strongly down weighted , and the chip has elevated weights overall . affymetrix quality report ( fig . b3 ) paints a very different picture chip @xmath3415 is an outlier on the med(nuse ) scale , but does not stand out on any of common affymetrix quality assessment measures : percent present , noise , scale factor , and gapdh 3/5. \(c ) * overall comparison of our measures and the affymetrix quality report for a large number of st . jude s chips : * fig . d1 pairs the med(nuse ) with the four most common gcos scores on a set of 129 hu133a chips from the st . jude dataset . there is noticable linear association between med(nuse ) and percent present , as well as between med(nuse ) and scale factor . gapdh 3/5 does not show a linear association with any the other scores . \(d ) * disagreement between our quality measures and the affymetrix quality report for hyperdip@xmath3150 subgroup in st . jude s data : * the affymetrix quality report detects problems with many chips in this dataset . for chip a , raw q ( noise ) is out of the recommended range for the majority of the chips : @xmath3412 , @xmath3414 , c1 , c13 , c15 , c16 , c18 , c21 , c22 , c23 , c8 and r4 . background detects chip @xmath3412 as an outlier . scale factor does not show any clear outliers . percent present is within the typical range for all chips . gapdh 3/5 is below 3 for all chips . for chip b , raw q ( noise ) is out of the recommended range for the @xmath3412 , @xmath348 , @xmath3418 and r4 . background detects chip @xmath3412 and @xmath348 as outliers . scale factor does not show any clear outliers . percent present never exceeds 23@xmath1 in this chip set , and it is below the typical minimum of 20@xmath1 for chips @xmath348 , c15 , c16 , c18 , c21 and c4 . gapdh 3/5 is satisfactory for all chips . our measures suggest that , with one exception , the chips are of good quality ( fig . the heterogeneity of the perfect match distributions does not persist after the preprocessing . for chip a , @xmath3412 has the largest iqr(rle ) and is a clear outlier among the nuse distributions . two other chips have elevated iqr(rle ) , but do not stand out according to nuse . for chip b , the rle distributions are very similar with @xmath3412 again having the largest iqr(rle ) . the nuse distributions are consistently showing good quality with the exception of chip @xmath3412 . \(e ) * varying quality between diagnostic subgroups in the st . jude s data : * each boxplot in fig . e1 sketches the residual scale factors ( rsf ) of the chips of all diagnostic subgroups . they show substantial quality differences . the e2a@xmath35pbx1 subgroup has a much higher med(rsf ) than the other subgroups . the t@xmath35all subgroup has a slightly elevated med(rsf ) and a higher iqr(rsf ) than the other subgroups . \(f ) * hybridization date effects on quality of fruit fly chips : * the fruit fly mutant with dysfunctional synapses is an experiment of the earlier stages of working with affymetrix chips in this lab . it shows a wide range of quality . in the boxplot series of rle and nuse ( fig . f1 ) a dependency of the hybridization date is striking . the chips of the two mutants hybridized on the day colored yellow show substantially lower quality than any of the other chips . f2 shows a weight landscape revealing smooth mountains and valleys . while the pattern is particularly strong in the chip chosen for this picture , it is quite typical for the chips in this dataset . we are not sure about the specific technical reason for this , but assume it is related to insufficient mixing during the hybridization . \(g ) * temporal trends or biological variation in fruit fly time series : * the series consists of 12 developmental stages of fruit fly embryos hybridized in 3 technical replicates each . while the @xmath36(pm ) distributions are very similar in all chips , we can spot two kinds of systematic patterns in the rle and nuse boxplots ( fig . one pattern is connected to the developmental stage . within each single one of the three repeat time series , the hybridizations in the middle stages look `` better '' than the ones in the early stages and the chips in the late stages . this may , at least to some extent , be due to biological rather than technological variation . in embryo development , especially in the beginning and at the end , huge numbers of genes are expected to be affected , which is a potential violation of assumption . insufficient staging in the first very short developmental stages may further increase the variability . also , in the early and late stages of development , there is substantial doubt about the symmetry assumption . another systematic trend in this dataset is connected to the repeat series . the second dozen chips are of poorer quality than the others . in fact , we learned that they were hybridized on a different day from the rest . the pairplot in fig . g2 looks at the relationship between our chip quality measures . there is no linear association between the raw intensities summarized as med(pm ) and any of the quality measures . a weak linear association can be noted between med(rle ) and iqr(rle ) . it is worth to note that is becomes much stronger when focusing on just the chips hybridized on the day colored in black . iqr(rle ) and med(nuse ) again have a weak linear association which becomes stronger when looking only at one of the subgroups , except this time it is the chips colored in gray . for the pairing med(rle ) and med(nuse ) , however , there is no linear relationship . finally ( not shown ) , as in the dysfunctional synapses mutant fruit fly dataset , a double - wave gradient , as seen in fig . f2 for the other fruit fly dataset , can be observed in the quality landscapes of many of the chips . although these experiments were conducted by a different team of researchers , they used the same equipment as that used in generating the other fruit fly dataset . \(h ) * lab differences in pritzker s gender study : * we looked at hu95 chip data from 13 individuals in two brain regions , the cortex of the cerebellar hemisphere ( short : cerebellum ) and the dorsolateral prefrontal cortex . with some exceptions , each sample is hybridized in both lab m and lab i. the nuse and rle boxplots ( fig . h1 ) for the cerebellum dataset display an eye - catching pattern : they show systematically much better quality in lab m then in lab i. this might be caused by overexposure or saturation effects in lab i. the medians of the raw intensities ( pm ) values in lab i are , on a @xmath36-scale between about 9 and 10.5 , while they are very consistently about 2 two 3 points lower in lab m. the dorsolateral prefrontal cortex hybridizations show , for the most part , a lab effect similar to the one we saw in the cerebellum chips ( plots not shown here ) . \(i ) * lab differences in pritzker s mood disorder study : * after the experiences with lab differences in the gender study , the consortium went through extended efforts to minimize these problems . in particular , the machines were calibrated by affymetrix specialists . i1 summarizes the quality assessments of three of the pritzker mood disorder datasets . we are looking at hu95 chips from two sample cohorts ( a total of about 40 subjects ) in each of the brain regions anterior cingulate cortex , cerebellum , and dorsolateral prefrontal cortex . in terms of med(pm ) , for each of the three brain regions , the two replicates came closer to each other : the difference between the two labs in the mood disorder study is a third or less of the difference between the two labs in the gender study ( see first two boxes in each of the three parts of fig . i1 , and compare with fig . this is due to lab i dropping in intensity ( toward lab m ) and the new lab d also operating at that level . the consequence of the intensity adjustments for chip quality do not form a coherent story . while for cerebellum the quality in lab m is still better than in the replicate in one of the other labs , for the other two brain regions the ranking is reversed . effects of a slight underexposure in lab m may now have become more visible . generally , in all brain regions , the quality differences between the two labs are still there , but they are much smaller than they in the gender study data . \(j ) * assigning special causes of poor quality for st . jude s data : * eight quality landscapes from the early st . jude s data , a collection of 335 hu133av2 chips . the examples were picked for being particularly strong cases of certain kinds of shortcomings that repeatedly occur in this chip collection . they do not represent the general quality level in the early st . jude s chips , and even less so the quality of later st . jude s chips . the figures in this paper are in gray levels . if the positive residual landscape is shown , the negative residual landscape is typically some sort of complementary image , and vice versa . colored quality landscapes for all st . jude s chips can be downloaded from bolstad s _ chip gallery _ at www.plmimagegallery.bmbolstad.com . fig . j1 `` bubbles '' is the positive residual landscape of chip hyperdip-50 - 02 . there are small dots in the left upper part of the slide , and two bigger ones in the middle of the slide . we attribute the dots to dust attached to the slide or air bubbles stuck in this place during the hybridization . further , there is an accumulation of positive residuals in the bottom right corner . areas of elevated residuals near the corners and edges of the slide are very common , often much larger than in this chip . mostly they are positive . the most likely explanation are air bubbles that , due to insufficient mixing during the hybridization , got stuck close to the edges where they had gotten when this edges was in a higher position to start with or brought up there by the rotation . note that there typically is some air in the solution injected into the chip ( through a little hole near one of the edges ) , but that the air is moved around by the rotation during the hybridization to minimize the effects on the probe measurements . j2 `` circle and stick '' is the residual landscape of hyperdip47 - 50-c17 . this demonstrates two kinds of spatial patterns that are probably caused by independent technical shortcomings . first , a circle with equally spaced darker spots ( approximately ) . the symmetry of the shape suggests it was caused by a foreign object scratching trajectories of the rotation during the hybridization into the slide . second , there are little dots that almost seem to be aligned along a straight line connecting the circle to the upper right corner . the dots might be air bubbles stuck to some invisible thin straight object or scratch . fig . j3 `` sunset '' is the negative residual landscape of hyperdip-50-c6 . this chip illustrates two independent technical deficiencies . first , there is a dark disk in the center of the slide . it might be caused by insufficient mixing , but the sharpness with which the disk is separated from the rest of the image asks for additional explanations . second , the image obviously splits into an upper and a lower rectangular part with different residual , separated by a straight border . as a most likely explanation , we attribute this to scanner problems . j4 `` pond '' is the negative residual landscape of tel - aml1 - 2m03 . the nearly centered disc covers almost the entire slide . it might be caused by the same mechanisms that were responsible for the smaller disc in the previous figure . however , in this data collection , we have only seen two sizes of discs the small disk as in the previous figure and the large one as in this figure . this raises the question why the mechanism that causes them does not produce medium size discs . j5 `` letter s '' is the positive residual landscape of hypodip-2m03 . the striking pattern the letter s with the `` cloud '' on top is a particularly curious example of a common technical shortcoming . we attribute the spatially heterogeneous distribution of the residuals to insufficient mixing of the solution during the hybridization . j6 `` compartments '' is the positive residual landscape of hyperdip-50 - 2m02 . this is a unique chip . one explanation would be that the vertical curves separating the three compartments of this image are long thin foreign objects ( e.g. hair ) that got onto the chip and blocked or inhabited the liquid from being spread equally over the entire chip . j7 `` triangle '' is the positive residual landscape of tel - aml1 - 06 . the triangle might be caused by a long foreign object stuck to the center of the slide on one end and free , and hence manipulated by the rotation , on the other end . j8 `` fingerprint '' is the positive residual landscape of hyperdip-50-c10 . what looks like a fingerprint on the picture might actually be one . with the slide measuring 1 cm by 1 cm , the pattern has about the size of a human fingerprint or the middle part of it . * quality landscapes : * the pair of the positive and negative residual landscapes contains the maximum information . often , one of the two residual landscapes can already characterize most of the spatial quality issues . in the weight pictures , the magnitude of the derivation is preserved , but the sign is lost . therefore , unrelated local defects can appear indistinguishable in weight landscapes . the landscapes allow a first glance at the overall quality of the array : a square filled with low - level noise typically comes from a good quality chip , one filled with high - level noise comes from a chip with uniformly bad probes . if the landscape is reveals any spatial patterns , the quality may or may not be compromised depending on the size of the problematic area . even a couple of strong local defects may not lower the chip quality , as indicated by our measures . the reason lies in both the chip design and the rma model . the probes belonging to one probeset are scattered around the chip assuring that a bubble or little scratch would only affect a small number of the probes in a probeset ; even a larger under- or overexposed area of the chip may affect only a minority of probes in each probeset . as the rma model is fitted robustly , its expression summaries are shielded against this kind of disturbance . we found the quality landscape most useful in assigning special causes of poor chip quality . a quality landscape composed of smooth mountains and valleys is most likely caused by insufficient mixing during the hybridization . smaller and sharper cut - out areas of elevated residuals are typically related to foreign objects ( dust , hair , etc . ) or air bubbles . symmetries can indicate that scratching was caused by particles being rotated during hybridization . patterns involving horizontal lines may be caused by scanner miscalibration . it has to be noted , that the above assignment of causes are educated guesses rather than facts . they are the result of extensive discussions with experimentalist , but there remains a speculative component to them . even more hypothetical are some ideas we have regarding how the sign of the residual could reveal more about the special cause . all we can say at this point is that the background corrected and normalized probe intensity deviate from what the fitted rma model would expect them to be . the focus in this paper is a global one : chip quality . several authors have worked on spatial chip images from a different perspective , that of automatically detecting and describing local defects ( see @xcite , or the r - package harshlight by @xcite ) . it remains an open question how to use this kind of assessment beyond the detection and classification of quality problems . in our approach , if we do not see any indication of quality landscpape features in another quality indicator such as nuse or rle , we suppose that it has been rendered harmless by our robust analysis . this may not be true . * rle : * despite its simplicity the rle distribution turns out to be a powerful quality tool . for a small number of chips , boxplots of the rle distributions of each chip allow the detection of outliers and temporal trends . the use of colors or gray levels for different experimental conditions or sample properties facilitates the detection of more complex patterns . for a large number of chips , the iqr(rle ) is a convenient and informative univariate summary . med(rle ) should be monitored as well to detect bias . as seen in the drosophila embryo data , these assumptions are crucial to ensuring that what the rle suggests really are technical artifacts rather than biological differences . note that the rle is not tied to the rma model , but could as well be computed based on expression values derived from other algorithms . the results may differ , but our experience is that the quality message turns out to be similar . * nuse : * as in the case of the rle , boxplots of nuse distributions can be used for small chip sets , and plots of their median and interquartile ranges serve as a less space consuming alternative for larger chip sets . for the nuse , however , we often observe a very high correlation between median and interquartile range , so keeping track of just the median will typically suffice . again , colors or gray levels can be used to indicate experimental conditions facilitating the detection their potential input on quality . the nuse is the most sensitive of our quality tools , and it does not have a scale . observed quality differences , even systematic ones , have therefore to carefully assessed . even large relative differences do not necessarily compromise an experiment , or render useless batches of chips within an experiment . they should always alert the user to substantial heterogeneity , whose cause needs to be investigated . on this matter we repeat an obvious but important principle we apply . when there is uncertainty about whether or not to include a chip or set of chips in an analysis , we can do both analyses and compare the results . if no great differences result , then sticking with the larger set seems justifiable . * raw intensities and quality measures : * raw intensities are not useful for quality prediction by itself , but they can provide some explanation for the poor performance according to the other quality measures . all the pritzker datasets , for example , suffer from systematic differences in the location of the pm intensity distribution ( indicated by med(pm ) ) . sometimes the lower location was worse too close to underexposure and sometimes the higher was worse too close to overexposure or saturation . we have seen examples of chips for which the raw data give misleading quality assessment . some kinds of technological shortcomings can be removed without trace by the statististical processing , while others remain . * comparison with affymetrix quality scores : * we found good overall agreement between our quality assessment and two of the affymetrix scores : percent present and scale factor . provided the quality in a chips set covers a wide enough range , we typically see at least a weak linear association between our quality measures and these two , and sometimes other affymetrix quality scores . however , our quality assessment does not always agree with the affymetrix quality report . in the st . jude data collection we saw that the sensitivity of the affymetrix quality report could be insufficient . while our quality assessment based on rle and nuse clearly detected the outlier chip in the mll chip b dataset , none of the measures in the affymetrix quality did . the reverse situation occured in the hyperdip chip a dataset . while most of the chips passed according to our quality measures , most of the chips got a poor affymetrix quality scores . * rsf : * the residual scale factor can detect quality differences between parts of a data collection , such as the diagnostic subgroups in the st . jude s data . in the same way , it can be employed to investigate quality differences between other dataset divisions defined by sample properties , lab site , scanner , hybridization day , or any other experimental condition . more experience as to what magnitudes of differences are acceptable is still needed . in this paper , we have laid out a conceptual framework for a statistical approach for the assessment and control of microarray data quality . in particular , we have introduced a quality assessment toolkit for short oligonucleotide arrays . the tools highlight different aspects in the wide spectrum of potential quality problems . our numerical quality measures , the nuse and the rle , are an efficient way to detect chips of unusually poor quality . furthermore , they permit the detection of temporal trends and patterns , batch effects , and quality biases related to sample properties or to experimental conditions . our spatial quality methods , the weight and residual landscapes , add to the understanding of specific causes of poor quality by marking those regions on the chip where defects occur . furthermore , they illustrate the robustness of the rma algorithm to small local defects . the rsf quantifies quality differences between batches of chips . it provides a broader framework for the quality scores of the individual chips in an experiment . all the quality measures proposed in this paper can be computed based on the raw data using publicly available software . deriving the quality assessment directly from the statistical model used to compute the expression values is more powerful than basing it on the performance of a particular set of of control probes , because the control probes may not behave in a way that is representative for the whole set of probes on the array . the model - based approach is also preferable to metrics less directly related to the bulk of the expression values . some of the affymetrix metrics , for example , are derived from the raw probe values and interpret any artifacts as quality problems , even if they are removed by routine preprocessing steps . a lesson from the practical examples in this paper is the importance of a well designed experiment . one of the most typical sources for bias , for example , is an unfortunate systematic connection between hybridization date and groups of the study a link that could have been avoided by better planning . more research needs to be done on the attribution of specific causes to poor quality measurements . while our quality measures , and , most of all , our quality landscapes , are a rich source for finding specific causes of poor quality , a speculative component remains . to increase the credibility of the diagnoses , systematic quality experiments need to be conducted . a big step forward is bolstad s _ chip gallery _ at www.plmimagegallery.bmbolstad.com , which collects quality landscapes from affymetrix chip collections , provides details about the experiment and sometimes offers explanations for the technical causes of poor quality . started as a collection of chip curiosities this website is now growing into a visual encyclopedia for quality assessment . contributions to the collection , in particular those with known causes of defects , are invited ( anonymous if preferred ) . further methodological research is needed to explore the use of the spatial quality for statistically `` repairing '' local defects , or making partial use of locally damaged chips . we are well aware that the range of acceptable values for each quality scores is the burning question for experimentalists . our quality analysis results with microarray datasets from a variety of scientific studies in section [ r ] show that the question about the right threshold for good chip quality does not have a simple answer yet , at least not as the present level of generality . thresholds computed for gene expression measurements in fruit fly mutant screenings can not necessarily be transferred to brain disease research or to leukemia diagnosis . thresholds need to be calibrated to the tissue type , the design , and the precision needs of the field of application . we offer two strategies to deal with this on different levels : 1 . * individual researchers : * we encouraged researchers to look for outliers and artificial patterns in the series of quality measures of the batch of jointly analyzed chips . furthermore , any other form of unusual observations e.g.a systematic disagreement between nuse and rle , or inconsistencies in the association between raw intensities and quality measures potentially hints at a quality problem in the experiment . * community of microarray users : * we recommend the development of quality guidelines . they should be rooted in extended collections of datasets from scientific experiments . complete raw datasets are ideal , where no prior quality screening has been employed . careful documentation of the experimental conditions and properties help to link unusual patterns in the quality measures to specific causes . the sharing of unfiltered raw chip data from scientific experiments on the web and the inclusion of chip quality scores in gene expression databank entries can lead the way towards community - wide quality standards . besides , it contributes to a better understanding how quality measures relate to special causes of poor quality . in addition , we encourage the conduction of _ designed microarray quality experiments . _ such experiments aim at an understanding of the effects of rna amount , experimental conditions , sample properties and sample handling on the quality measures as well as on the downstream analysis . they give an idea of the range of chip quality to be expected under given certain experimental , and , again , they help to characterize specific causes of poor quality . benchmarking of 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quality of microarray gene expression data has emerged as a new research topic . as in other areas , microarray quality is assessed by comparing suitable numerical summaries across microarrays , so that outliers and trends can be visualized , and poor quality arrays or variable quality sets of arrays can be identified . since each single array comprises tens or hundreds of thousands of measurements , the challenge is to find numerical summaries which can be used to make accurate quality calls . to this end , several new quality measures are introduced based on probe level and probeset level information , all obtained as a by - product of the low - level analysis algorithms rma / fitplm for affymetrix genechips . quality landscapes spatially localize chip or hybridization problems . numerical chip quality measures are derived from the distributions of _ normalized unscaled standard errors _ and of _ relative log expressions . _ quality of chip batches is assessed by _ residual scale factors . _ these quality assessment measures are demonstrated on a variety of datasets ( spike - in experiments , small lab experiments , multi - site studies ) . they are compared with affymetrix s individual chip quality report . * _ to be published in technometrics ( with discussion ) _ * * julia brettschneider @xmath0 , franois collin , benjamin m.bolstad , terence p.speed * * quality assessment for * + + keywords : quality control , microarrays , affymetrix chips , relative log expression , normalized unscaled standard errors , residual scale factors .

You are an expert at summarizing long articles. Proceed to summarize the following text: complex hierarchical systems admit several levels of description . at a microscopic level , the interest lies on the dynamical behavior of the constituent elements . thus , at this level of description , one needs to follow the time evolution of very many microscopic variables . on the other hand , at the macroscopic level , one describes the system in terms of just a few collective variables characterizing it as a whole . the laws describing the behavior of collective variables could , in principle , be deduced from the dynamical laws governing the elementary constituents . in practice this is not an easy task as , in general , the dynamical equations for the collective variables can not be obtained from the microscopic dynamics without further approximations . on the other hand , in general , when studying a complex systems , there are just the collective variables which are accessible to experimental measurements . the detailed dynamics of the constituent elements has to be inferred from the information about the collective variables . a fundamental property common to all complex systems made of a collection of interacting elements ( subsystems ) is the existence of feedback mechanisms of the whole system behavior into the dynamics of the subsystems @xcite . the evolution of the system is determined by those of its elements in a statistical way and , at the same time , the feedback loops put the subsystems under the control of the global system . such feedback loops , for example , give rise to the biological laws that characterize the phenomenon of life . in fact , this mutual interlevel control introduces a self - regulation mechanism to the dynamics of the hierarchical system against external forces and boundary conditions . in this work , we have studied two very simple models describing a finite system composed of several subsystems . each of them is characterized by a single stochastic variable @xmath0 . the subsystems have an intrinsic nonlinear bistable dynamics and they are coupled among them either in a global or local form . the whole system is characterized by a collective variable . we will assume that the information about the system comes from the observation of the collective variable . the randomness of the subsystem variables is due to a noise term in their dynamics . we will take this noise to be a white gaussian noise characterized by a noise parameter @xmath1 . this noise can originate for instance , from the weak interactions between the subsystems and a large thermal bath with a temperature @xmath2 . a goal of this work is to study the form of the distribution function @xmath3 for the collective variable and to compare it with the distribution function of a single element @xmath4 for small finite systems . as we will see , in the absence of feedback , both distributions are unique , regardless of the initial preparation of the system . this is a consequence of the fact that the joint probability distribution satisfies a linear fokker - planck equation . if the observations of the collective variable are compatible with such a behavior , then we can accept the microscopic description . but , if one observes that , depending upon the noise value and the strength of the subsystems coupling , the equilibrium distribution might not be unique and it might depend on the initial preparation of the system , then the linear fokker - planck equation ( or equivalently , the set of coupled langevin - like equations ) is not adequate . to deal with such a situation , we propose stochastic dynamics leading to nonlinear fokker - plank equations . the idea is simply an extension of the old weiss mean - field approach to the equilibrium properties of magnetic systems . then , as the system parameters are varied , the dynamics leads us from just a single stationary solution to situations where two distributions are stable and the system goes to one or the other depending upon the initial conditions . we should point out that such a non - uniqueness of a single variable distribution function was studied long ago by several authors @xcite . they were able to derive a nonlinear fokker - planck equation for a single variable in the limit of an infinite system . we emphasize that , in this work , we are interested in small , finite systems . the intrinsic bistable nonlinearity of the subsystems dynamics forbids explicit analytical solutions , but it is essential for the behavior indicated above . a closed , explicit , stochastic evolution equation for the collective variable can not be found from the subsystems dynamics . we will then rely on numerical simulations of the stochastic evolution equations for the subsystems variables . we consider a system formed by a set of @xmath5 subsystems each one described by a single stochastic variable @xmath6 , whose dynamics is given by the coupled langevin equations ( in dimensionless form ) @xmath7 where @xmath8 is the strength of the coupling and @xmath9 are gaussian white noises with @xmath10 this model was introduced some years ago by kometani and shimizu @xcite in a biophysical context and was later analyzed by desai and zwanzig @xcite and dawson @xcite from a more general statistical mechanical perspective . an alternative formulation can be casted in terms of the linear fokker - planck equation for the joint probability distribution @xmath11 , @xmath12 where @xmath13 is the potential energy relief , @xmath14 with the single particle potential @xmath15 in the limit @xmath16 , one can write a nonlinear fokker - planck equation ( nlfpe ) for the one particle probability distribution , @xmath17 @xcite @xmath18p_{1}(x , t)\big\}+d\frac{\partial^{2}p_{1}(x , t)}{\partial x^{2 } } , \label{langinf}\ ] ] where @xmath19 in this limit , the system undergoes an order - disorder phase transition signaled by a change in the form of the equilibrium one - particle probability distribution @xmath4 and by the fact that for some values of the parameters @xmath8 and @xmath1 , the ergodicity is broken and there are more than one stable equilibrium distribution . the system ends up in one of them depending on its initial preparation . in fig . [ fig1 ] we sketch different regions in the parameter space @xmath20 and @xmath21 , separated by a transition line . below the transition line , there is just one single distribution , regardless of the initial preparation of the system , leading to a single equilibrium @xmath4 with two maxima . this is indicated in the figure by drawing a double well potential . on the other hand , above the transition line , two stable monomodal equilibrium distributions are possible . depending upon the initial condition the system relaxes to one or the other . this feature is illustrated in the figure by drawing two asymmetrical single minima potentials . let us define a collective variable @xmath22 then , the langevin equations ( [ eq001 ] ) can be casted in the equivalent form @xmath23 the nonlinearity of equations ( [ langfin ] ) prevent us from writing a closed langevin equation for @xmath24 . note that there is a feedback of the collective dynamics in the langevin dynamics of each degree of freedom . but , as we analyzed in a previous work @xcite , with this type of feedback for the finite model the single variable distribution is unique for any set of parameter values . this is consistent with the fact that the joint probability distribution @xmath25 satisfies a linear fokker - plank equation with a unique distribution function for any values of the parameters . we now propose to replace the model with @xmath26 here , @xmath27 represents a noise average of the collective variable defined in eq . ( [ eq007 ] ) . clearly , the langevin equation , eq . ( [ eq100 ] ) has to be solved concurrently with the evaluation of this average . the proposal amounts to replace in the dynamical evolution of each individual degree of freedom the whole stochastic process @xmath24 by its noise average , in the spirit of weiss mean - field approximation to magnetism . note that within this model , the collective variable is itself a fluctuating quantity , even though just its average value influences the subsystems dynamics . this model is described by the ( dimensionless ) set of langevin equations @xmath28 with the periodic conditions @xmath29 and @xmath30 . the noises @xmath31 are white gaussian with the properties given in eq . ( [ eq002 ] ) . as in the global interaction model , the joint probability distribution function satisfies a linear fokker - planck equation similar to the one in eq . ( [ fpe ] ) but with the potential energy relief , @xmath32 then , also for this case , the joint probability distribution @xmath25 satisfies a linear fokker - plank equation with a unique distribution function for any values of the parameters . as in the global interaction case , we will now replace eq . ( [ nn1 ] ) with @xmath33 as said above , in both situations , even after using the weiss approach , the collective variable @xmath34 for @xmath5 finite is still a random process with a probability distribution defined by @xmath35 notice that within the mean field dynamics , the joint probability distribution @xmath36 factorizes as a product of single particle distributions , each of them satisfying a nlfpe similar to the one in ( [ langinf ] ) for the single particle distribution in the infinite size limit . + defining the characteristic function @xmath37 of a distribution @xmath38 as @xmath39 one can easily see that the generating function of the collective variable and that of a single particle variable are related by @xmath40 thus , the corresponding cumulant generating functions are related by @xmath41 . then , the cumulants @xmath42 associated to the stochastic process @xmath24 , and those associated to the @xmath43 process , @xmath44 are related by @xmath45 in the limit of very large @xmath5 , @xmath46 becomes a very narrow peak around its first moment as indicated in @xcite . we have solved numerically the corresponding langevin equations , eqs . ( [ langfin],[nn1 ] ) and eqs.~ ( [ eq100],[eq200 ] ) for systems with a small number of subsystems using very many noise realizations . the numerical method used has been detailed elsewhere @xcite . after a convenient relaxation time so that the equilibrium probability distributions have been reached , we construct histograms that approximate the equilibrium probability distributions @xmath47 and @xmath4 . notice that the generation of langevin trajectories for all the degrees of freedom for the mean field dynamics requires the knowledge of @xmath48 . thus , at each time step during the evolution , this noise average is evaluated from the instantaneous values of each subsystem variable for each realization of the noise . namely , @xmath49 where @xmath50 is the number of noise realizations considered . let us first consider the case of global coupling with mean - field dynamics . in fig . [ ps ] we depict the equilibrium distribution of the collective variable @xmath47 for several values of the noise strength and two system sizes , @xmath51 ( upper panel ) and @xmath52 ( lower panel ) . in all cases , we have used @xmath53 and initial conditions such that @xmath54 . as seen in the figure , the qualitative behavior is quite independent of @xmath5 , except for the narrowing of the distribution as @xmath5 increases . for the parameter values considered , @xmath47 is always monomodal , with a peak that is displaced from @xmath55 to higher values of @xmath56 as the noise is decreased . note that the peak is located at positive values of @xmath56 because , at the initial time , we located the value of @xmath57 to be positive . had we started from the same initial values , but with a negative sign , the plots for @xmath47 would be as in fig . [ ps ] , but with the peaks located at negative @xmath56 values . in fig . [ px ] , we depict the equilibrium distribution for a single variable @xmath4 for several values of the noise strength and for two systems with @xmath51 ( upper panel ) and @xmath52 ( lower panel ) . in all cases , we have used @xmath53 . again , the qualitative behavior is quite independent of @xmath5 . by contrast with the behavior of @xmath47 , the single variable distribution @xmath4 changes its shape from a bimodal distribution centered at @xmath58 to a monomodal one , with a peak that is displaced to nonzero values of @xmath59 as the noise is decreased . in figs . [ sm ] and [ xm ] we depict the behavior of the equilibrium first and second cumulant moments of the collective variable and that of an individual variable for a system with @xmath51 degrees of freedom and mean - field global dynamics . the parameter values are indicated in the captions . the average values have been obtained from the corresponding numerical integration involving the equilibrium histograms constructed from the langevin simulations . as expected from eq . ( [ cum ] ) , the first moment of the individual and the collective variables are the same , while the second cumulant for the collective variable is smaller than that of the individual one . for the parameter values used in these figures , the probability distribution centered at @xmath55 ( or @xmath58 ) is unstable for @xmath1 smaller than a bifurcation value . in the figures we have just depicted that branch of the stable first moments which are reached from the imposed initial condition @xmath54 . the case of nearest neighbors interactions is qualitatively similar to the case of global interactions . we will just depict as an example the contrast in the behavior of @xmath47 for the finite model in eq . ( [ nn1 ] ) and the corresponding one with the mean field dynamics eq . ( [ eq200 ] ) . in fig . [ nnfin ] we plot the behavior of @xmath47 and @xmath4 for a system with @xmath51 and nearest neighbors coupling . in fig . [ nnmf ] the same distributions are sketched but now with the systems described by the mean - field dynamics . in both dynamics there is a change in the shape of the equilibrium distribution as the noise value is varied . but for the finite case described by eq . ( [ nn1 ] ) the equilibrium distribution is always unique regardless of the initial preparation . on the other hand , the mean field dynamics leads to a bifurcation of the distribution as @xmath1 is varied . for large values of @xmath1 there is a single stable @xmath47 . as @xmath1 is decreased while keeping @xmath8 constant , the zero centered distribution becomes unstable and two stable distributions appear with peaks located at @xmath60 or @xmath61 depending on the initial preparation . we have carried out numerical simulations to study some aspects of the behavior of complex stochastic systems formed by a finite number of subsystems with bistable intrinsic dynamics and mean field couplings . we have contrasted the behavior of each individual degree of freedom with that of a collective variable characterizing the entire system . the systems considered have finite sizes . then , the collective variable is also a stochastic variable with a probability density with a finite width , in contrast with what occurs in the infinite size limit . furthermore , the mean field dynamics is compatible with the possible existence of several probability distributions in some regions of the parameter space . there is a transition from a single stable distribution to the several stable ones as the noise strength is varied for a fixed value of the coupling parameter . in the case of multiple distributions , the one that is observed depends upon the initial preparation . we argue that if the stochastic collective variable is the accessible one , and it is such that its average value depends upon the initial conditions , the modelling of the subsystems dynamics requires the introduction of some sort of feedback of the average collective behavior on the individual dynamics . an example of this feedback is the mean field dynamics considered in this paper . it should be pointed out that in order to have the above mentioned transition between one or several distributions , the intrinsic bistable dynamics of each individual degree of freedom is essential . 99 k. kometani and h. shimizu , _ j. stat . * 13 * , 473 ( 1975 ) . r. c. desai and r. zwanzig , _ j. stat . * 19 * , 1 ( 1978 ) . d. a. dawson , _ j. stat . phys . _ * 31 * , 29 ( 1983 ) . m. shiino , _ phys . a_*36*,2393 ( 1986 ) . t. d. frank , nonlinear fokker - planck equations . fundamentals and applications , springer , ( 2005 ) . r. krsten , u. behn , _ phys . rev . e_*94 * , 062135 ( 2016 ) . j. gmez - ordez , j. m. casado , m. morillo , c. honisch , and r. friedrich , _ europhys . * 88 * 400006 ( 2009 ) . m. morillo , j . gmez - ordez , and j. m. casado,_phys . e _ * 52*,316 ( 1995 ) .

we consider the behavior of a collective variable in a complex system formed by a finite number of interacting subunits . each of them is characterized by a degree of freedom with an intrinsic nonlinear bistable stochastic dynamics . the lack of ergodicity of the collective variable requires the consideration of a feedback mechanism of the collective behavior on the individual dynamics . we explore numerically this issue within the context of two simple finite models with a feedback mechanism of the weiss mean - field type : a global coupling model and another one with nearest neighbors coupling .

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Proceed to summarize the following text: relativistic heavy ion reactions exhibit dominant collective flow behaviour , especially at higher energies where the number of involved particles , including quarks and gluons , increases dramatically . at intermediate stages approximate local equilibrium is reached , while the initial and final stages may be far out of local equilibrium . also , different stages may have different forms or phases of matter , especially when quark gluon plasma ( qgp ) is formed . the need to describe and match different stages of a reaction was realized by the development of the final freeze - out ( fo ) description in landau s fluid dynamical ( fd ) model @xcite . then it was improved by milekhin @xcite , and a covariant simple model was given by cooper and frye @xcite . in all these models the fo happened when the fluid crossed a hypersurface in the spacetime . at early relativistic heavy ion collisions , the initial compression and thermal excitation was described by a compression shock in nuclear matter . this was already pointed out by the first publications of w. greiner and e. teller and their colleagues @xcite , and the shock took place crossing a spacetime hypersurface ( e.g. a relatively thin layer resulting in a mach cone ) . when sudden large changes happen across a spacetime front the conservation laws and the requirement of increasing entropy should be satisfied : @xmath0~=~0\label{nconserve}~;\\ & [ t^{\mu\nu}d\sigma_{\mu}]~=~0\label{tconserve}~;\\ & [ s^{\mu}d\sigma_{\mu } ] ~\geq~0 \label{entropy}\end{aligned}\ ] ] where @xmath1 is the baryon current , @xmath2 is the entropy current , @xmath3 is the energy momentum tensor , which , for a perfect fluid , is given by @xmath4 where @xmath5 is the energy density , @xmath6 is the pressure , @xmath7 is the entropy density , and @xmath8 is the baryon density of matter . these are invariant scalars . the @xmath9 is the normal vector of the transition hypersurface , @xmath10 is the particle four velocity @xmath11 , normalized to @xmath12 . the square bracket means @xmath13=a_1 - a_0 $ ] , the difference of quantity @xmath14 over the two sides of the hypersurface . the metric tensor is defined as @xmath15 . we will also use the following notations : @xmath16 , @xmath17 is the invariant scalar baryon current across the front , @xmath18 is the generalized specific volume , @xmath19 , and @xmath20 , @xmath21 . for a perfect fluid local equilibrium is assumed , thus the fluid can be characterized by an equation of state ( eos ) , @xmath22 . ( [ nconserve],[tconserve ] ) and the eos are 6 equations , and can determine the 6 parameters of the final state , @xmath5 , @xmath8 , @xmath6 , and @xmath23 . later csernai @xcite pointed out the importance of satisfying energy , momentum and particle charge conservation laws across such hypersurfaces and generalized the earlier description of taub @xcite to spacelike and timelike hypersurfaces ( with spacelike and timelike normals respectively ) . in this situation the matter both before and after the shock was near to thermal equilibrium , and thus the conservation laws led to scalar equations connecting thermodynamical parameters of the two stages of the matter : the generalized _ rayleigh line _ and _ taub adiabat _ @xcite : @xmath24(d\sigma^{\mu } d\sigma_{\mu } ) / [ x ] \ , \ [ p ] = [ ( e+p)x ] / ( x_1 + x_0 ) \ . \label{rayligh - taub}\ ] ] at much higher energies , at the first stages of the collision , the matter becomes transparent and the initial state is very far from thermal equilibrium . for this stage other models were needed to handle the initial development , e.g. refs . @xcite . the initial non - equilibrium state in this situation can not be characterized by thermodynamical parameters or an eos , so the previous approach , with the generalized _ rayleigh line _ and _ taub adiabat _ is not applicable . nevertheless , the intermediate ( fluid dynamical ) stage is in equilibrium and has an eos , while the initial state has a well defined energy momentum tensor . in this work we will demonstrate that the final invariant scalar , thermodynamical parameters can be determined in this situation also from the conservation laws . then , bugaev @xcite observed that fo across hypersurfaces with spacelike normals , has problems with negative contributions in the cooper - frye evaluation @xcite of particle spectra , thus the fo must yield an anisotropic distribution , which he could approximate with a cut - jttner distribution @xcite . this is not surprising as in the rest frame of the front ( rff ) all post fo particles must move `` outwards '' , i.e. @xmath25 is required . this condition is not satisfied by any non - interacting thermal equilibrium distribution , which extend to infinity in all directions even if they are boosted in the rff . subsequently , another analytic form was proposed by csernai and tamosiunas , the cancelling - jttner distribution @xcite , which replaced the sharp cutoff by a continuous cutoff , based on kinetic model results . parallel to this development , the fo process was analysed in kinetic , transport approaches @xcite , where the fo happened in an outer layer of the spacetime , or in principle it could be extended to the whole fluid ( although , at early moments of a collision / explosion , from the center of the reaction few particles can escape ) . these transport studies also indicated that the post fo distributions may become anisotropic @xcite even for fo hypersurfaces with timelike normal [ in short : _ timelike surface _ ] , if the normal , @xmath26 , and the velocity four - vector , @xmath27 , are ( very ) different . these studies led to another fo description , where the initial stages of the collision with strongly interacting matter were described by fluid dynamics , while the final , outer spacetime domain ( or later times ) was described by weakly interacting particle ( and string ) transport models , where the final fo was inherently included , as each particle was tracked , until its last interaction . it is important to mention , that in these approaches , the transition from the fd stage to the molecular dynamics ( md ) or cascade stage happens when the matter crosses a spacetime hypersurface , thus the conservations laws @xcite have to be satisfied and the post fo particle phase space distributions @xcite have to be used when the post fo distributions become anisotropic . in this work for the first time we present a simple covariant solution for the transition problem and conservation laws for the situations when the matter after the front is in thermal equilibrium ( i.e. it has isotropic phase space distribution ) and has an eos , but the matter before the front must not be in an equilibrium state . then we discuss the situation where microscopic models are appended to the fluid dynamical model , which are in , or close to thermal equilibrium , but the eos , is not necessarily known . subsequently , we present the way to generalize the problem to anisotropic matter in final state , which is necessary for fo across spacelike surfaces and also for timelike surfaces if the flow velocity is large in the rest frame of the front ( rff ) . this problem was solved in kinetic approach for the bugaev cut - jttner approach @xcite and the csernai - tamosiunas cancelling - jttner approach , @xcite by calculating the energy momentum tensors explicitly from the anisotropic phase space distributions , but no general solution is given for post fo matter with anisotropic pressure tensor . the transition hypersurface between two stages of a dynamical development are most frequently postulated , governed by the requirement of simplicity . thus , such a hypersurface is frequently chosen as a fixed coordinate time in a descartian frame @xmath28 , or at a fixed proper time @xmath29 from a spacetime point , although in a general 3 + 1 dimensional system the choice of such a point is not uniquely defined . it is important that the _ transition hypersurface should be continuous _ , ( without holes where conserved particles or energy or momentum could escape through , without being accounted for ) . to secure that one quantity ( e.g. baryon charge ) does not escape through the holes of a hypersurface is not sufficient , as other quantities may ( e.g. momentum in case if @xmath30 is different on the two sides of a hole ) . again , to construct such a continuous hypersurface in a general 3 + 1 dimensional system is a rather complex task , although , in 1 + 1 or 2 + 1 dimensions it seems to be easy . both the initial state models and the intermediate stage , fluid dynamical models may be such that the calculation could be continued beyond the point where a transition takes place . then spacetime location of the transition to the next stage can or should be decided , based on a physical condition or requirement , which may be external to the development itself . as a consequence , in some cases the determination of transition surface may be an iterative process . numerically , the extraction of a freeze out ( fo ) hypersurface is by no means trivial . one of us , brs , has recently provided a proper numerical treatment regarding the extraction of fo hypersurfaces in two ( 2d ) , three ( 3d ) and four ( 4d ) dimensions @xcite . for instance , in 2d the history , i.e. , the temporal evolution , of a temperature field of a one - dimensional ( 1d ) relativistic fluid can be represented by a gray - level image ( _ cf . _ , fig . 1 ) . in the figure , we use the time @xmath28 and the radius @xmath31 for the temporal and the spatial dimensions , respectively . let bright pixels ( i.e. , picture elements ) refer to high temperatures and dark ones to low temperatures of the fluid . in this example , a 2d freeze - out hypersurface is an iso - therme . in fig . 1.a , we also depict the corresponding co - variant normal vectors @xmath32 . in 2d , the length of each normal vector is equal to the length of each supporting iso - contour vector . each normal vector has its origin at the contra - variant center , @xmath33 , of a given contra - variant iso - contour vector and points to the exterior of the enclosed spacetime region . the latter is also indicated in fig . 1.b , where we show that a contra - variant normal vector @xmath34 can be obtained by reflection of the co - variant normal vector @xmath32 at the time axis ( dashed line ) . finally , in fig . 1.c we show the contra - variant fo contour vectors with their corresponding contra - variant normal vectors @xmath34 . not all of these contra - variant normal vectors point to the exterior of the enclosed spacetime region . note that the sign conventions of the normals of the transition hypersurface are important , and must be discussed , especially if both timelike and spacelike surfaces are studied . in fact , only the timelike contra - variant normal vectors point outwards , whereas the spacelike contra - variant normal vectors point inwards @xcite . if we know the fo hypersurface and the local momentum distribution after the transition the total , measurable momentum distribution can be evaluated by the cooper - frye formula @xcite . let us define the contra - variant and co - variant surface normal four - vectors as @xmath35 where in general @xmath36 , as the surface element can be either spacelike ( - ) or timelike ( + ) . we can also introduce a unit normal to the surface as : @xmath37 so that @xmath38 furthermore @xmath39 where for timelike surfaces @xmath40 and for spacelike surfaces @xmath41 . for the frequently used timelike , one - dimensional case @xmath42 . in the general case the conserved energy - momentum current crossing the surface element is @xmath43 @xmath44 must be continuous across the freeze - out surface , as must the baryon current @xmath45 , @xmath46 where @xmath47 is the invariant scalar baryon charge current . we assume that the initial state , `` @xmath48 '' , and its energy momentum tensor and baryon current before the front is known . we aim for the characteristics of the final state . in total there are six unknowns in the equilibrated final state , these are @xmath23 , @xmath5 , @xmath6 and @xmath8 ( here we drop the index `` @xmath49 '' for the final state for shorter notation ) , however the pressure @xmath6 , a function of @xmath5 and @xmath8 , is given by the eos , @xmath22 . knowing @xmath8 and @xmath5 , the eos , and the particular form of the corresponding equilibrated distribution function , the parameters @xmath50 , and @xmath51 , can also be obtained . thus , we have to solve 5 equations : @xmath52 the l.h.s . represents quantities of the initial state of matter and the corresponding conserved quantities are known . equations ( [ eq0],[eq1 ] ) can be solved for @xmath53 in the calculational frame : @xmath54 using now eq . ( [ eq1],[eq2],[eq3 ] ) one obtains @xmath55 , and in a similar fashion @xmath56 and @xmath57 @xmath58 this results for @xmath59 , in @xmath60 where , @xmath61 is an invariant scalar , and @xmath62 transforms as the 0-th component of the 4-vector @xmath63 . notice that eq . ( [ eq9 ] ) was not used up to this point , thus we can use there results both for the baryon - free and baryon - rich case . we can have an elegant direct solution for the proper energy density , @xmath5 , and pressure , @xmath6 , as both of these quantities are invariant scalars , and we can express these by the covariant , 4-vector equation ( [ em - curr ] ) . from this 4-vector equation we can get two invariant scalar equations by ( i ) taking its norm , @xmath64 , and ( ii ) taking its projection to the normal direction , @xmath65 : @xmath66 now expressing @xmath67 from eq . ( [ ads ] ) and inserting it to eq . ( [ aa ] ) , we obtain our final equation @xmath68 which can be solved straightforwardly if the eos , @xmath69 , is known . the other three elements of the equation , @xmath64 , @xmath65 , and @xmath70 , are known from the normal to the surface and from energy - momentum current from the pre - transition side . then , eqs . ( [ eqg2a]-[eqg2b ] ) can be used to determine the final flow velocity . at the end , after all conservation law equations are solved , we have to check the non - decreasing entropy condition ( [ entropy ] ) to see whether the solution is physically possible . if the overall entropy is decreasing after transition that would mean that the hypersurface is chosen incorrectly . one will need to choose more realistic condition for the transition and repeat the calculations . this result can be used both if the initial state is in equilibrium and if it is not . in case of an ideal gas of massless particles after the front , with an eos of @xmath71 , eq . ( [ aa2 ] ) leads to a quadratic equation , @xmath72 where @xmath73 , is the energy momentum transfer 4-vector across a unit hypersurface element . if the flow velocity is normal to the fo hypersurface , @xmath74 , then for an initial perfect fluid in the local rest ( lr ) frame the above covariant equation takes a simple form , @xmath75 this has two real roots , @xmath76 ( energy density is conserved ) and @xmath77 which does not correspond to a physical solution , as the energy density should not be negative . if the eos depends on the conserved baryon charge density also , then we must exploit in addition eq . ( [ eqn ] ) : @xmath78 and inserting @xmath79 from here to eq . ( [ ads ] ) yields @xmath80 where @xmath81 is the generalized specific volume , well known from relativistic shock and detonation theory @xcite . this equation provides another equation for @xmath82 as @xmath83 \ , \label{ep2nd}\ ] ] which , together with eq . ( [ aa2 ] ) and the eos , @xmath22 , provide three equations to be solved for @xmath84 and @xmath8 . this evaluation of the post fo configuration is in agreement with the theory of relativistic shocks and detonations @xcite allowing for both spacelike and timelike fo hypersurfaces . see also @xcite . this method of evaluation observables is frequently used at the end of fluid dynamical model calculations ( see e.g.@xcite ) . recently a frequently practiced method to describe the final stages of a reaction is to switch the fd model over to a molecular dynamics ( md ) description at a transition hypersurface . this is frequently a fixed time , @xmath28 , or fixed proper time , @xmath29 hypersurface . the generation of the initial state of such an md model is a task , which depends on the constituents of the matter described by the md model . nevertheless , same principles must be satisfied , like the conservation laws , eqs.([nconserve]-[tconserve ] ) . let us assume , although not required by physical laws , that we have thermal equilibrium on both sides of the transition and we know explicitly the corresponding final momentum distribution of particles . then , the fundamental equation to construct the post transition microscopic state , in addition to the conservation laws is the cooper - frye formula , @xmath85 assuming that the local phase space distribution , @xmath86 , is known for all initial components of the md model . if @xmath86 are local equilibrium distributions then ( in principle ) we know the intensive and extensive thermodynamical parameters and the eos of the matter when the md model simulation starts . these must not be the same as the ones before the transition hypersurface . in the usual transition from fd to md models , where the initial state of md is in equilibrium , the eos - s are known on both sides of the transition surface , and thus , both the equations of rayleigh - line and taub - adiabat , eqs . ( [ rayligh - taub ] ) , as well as the invariant scalar equations derived here , eqs.([aa],[ads],[aa2],[ep2nd ] ) can be used to determine all parameters of the matter starting the md simulation . these then determine the phase space distributions , @xmath86 of all components of the md simulation . subsequently eq.([cf - f ] ) can be used to generate randomly the initial constituents of the md simulation . as eq.([cf - f ] ) is a covariant equation applicable in any frame of reference , the most straightforward is to perform the generation of particles in the calculational frame of the md model . this transition is by now performed in many hybrid models combining fluid dynamics with microscopic transport models @xcite . these models at present are the most effective to describe experimental data and make the need for a modified boltzmann transport equation @xcite less problematic . in some cases the first step of the transition , the determination of the parameters of the final state from the exact conservation laws , is dropped with the argument that both before and after the transition the matter has the same constituents and the same eos , thus the all extensive and intensive thermodynamical parameters as well as the flow velocity must remain the same . then , using the intensive parameters the final particle distributions in the cooper - frye formula , eq.([cf - f ] ) , can be directly evaluated in a straightforward way . this procedure is correct , but only if all features of the two states of the matter and their eos are identical . in some cases the pre transition eos assumes effective hadron masses depending on the matter density , while the final eos is that of a hadron ideal gas mixture , but with fixed vacuum masses . this leads to a difference in the eos , thus the above procedure is approximate . in such cases , the method can be used , but the accurate conservation laws can be enforced by a final adjustment step described in the next subsection . the situation is similar if the constituents and the eos are almost identical before and after the transition , but before the transition a weak or weakening mean field potential or compression energy is taken into account . in addition to the above mentioned approximate methods , even for really identical eos - s across the transition or with generating the final eos parameter based on conservation laws for the final eos , inaccuracies may arise due to other reasons : during the random generation of the initial constituent particles of the md simulation , the exact conservation laws may be violated , due to finite number effects . however , the energy and particle number conservations are usually enforced during the random generation of particles , even if the above procedure of solving the conservation laws beforehand is not fully followed . this is usually the consequence of the fact that the eos of the md model is not necessarily known if the model has complex constituents and laws of motion . in any case to remedy this random error and make the conservation laws exactly satisfied a final correction step is advisable , and it is not always performed . if the energy and particle number conservations are enforced then , the last variable to balance is the momentum conservation . this regulates the flow velocity of the matter after the transition initiating the md simulation . the energy momentum tensor and baryon current for the generated random set of particle species , " @xmath87 , for each fluid cell ( or group of cells if the multiplicity in a single cell is too low ) can be calculated from the kinetic definition : @xmath88 which , yield the resulting momentum and flow velocity of the matter . this can be used to adjust the flow velocity to achieve exact conservation of momentum , and modify the velocity of generated particles by the required lorentz boost . the other conserved quantities may then be affected also , but an iterative procedure to eliminate the error completely is not crucial as the error can be given quantitatively . if the randomly generated state is not following a thermal equilibrium phase space distribution , @xmath89 , and thus does not have an eos , the above described scalar equations can not be used to generate the initial configuration of the md model . nevertheless , the second step to check the conservation laws with the kinetic definition , and then correct the parameters of the generated particles can be done . for a required level of accuracy in this case an iterative procedure may be necessary . another , easier way to remedy this problem is to choose the transition hypersurface earlier so that the subsequent matter is still in thermal equilibrium . this can always be done if the requirement of entropy increase is satisfied . we have mentioned that the assumption for having thermal equilibrium in the final state is neither excluded nor required from transport theoretical considerations . however , thermal equilibrium distribution is not possible if we have to describe fo across a spacelike hypersurface ( see the discussion in section [ intro ] . ) in the md model description the final post fo momentum distributions develope a local anisotropy if the fo has locally a preferred direction . unless the unit normal of the fo hypersurface is equal to the local flow velocity of the pre fo matter , there is always a selected spatial direction which is the dominant direction of fo this situation is discussed in several theoretical works , and some general features can be extracted from these studies . in explicit transport models this situation is handled @xcite : starting from an equilibrium jttner distribution and considering a momentum dependent escape probability in the collision term , - which reflected the direction of the fo front and the distance from the front , - an anisotropic distribution was obtained ( i.e. a distribution , which was anisotropic even in its own lr frame ) . this anisotropic distribution could be approximated with analytic distribution functions @xcite : the starting point is the un - cut , isotropic , jttner distribution in the rest frame of the gas ( rfg ) , which is centered around the 4-velocity vector , @xmath90 . this distribution is then cut or cut and smoothed . the resulting distribution has a different new flow velocity , @xmath91 , which is non - zero in rfg , and is pointing in space in the direction of the normal of the fo hypersurface , @xmath92 , labeled by @xmath93 . this @xmath91 defines the local rest ( lr ) frame of the post fo matter . the spatial direction of @xmath92 is not affected by the lorentz transformation from rff to rfg and then to lr , as @xmath92 is the direction of the lorentz transformation from rfg to lr . , must not be parallel to @xmath94 or @xmath95 but these latter velocities can be decomposed to @xmath93 and @xmath96 components with respect to @xmath92 . due to the construction @xcite of the cut- or cancelling - jttner distributions @xmath97 or @xmath98 . ] in the general case the boost in the @xmath99 direction leads to a change of the distribution function in the @xmath100 direction , but does not affect the distribution in the @xmath101 direction , or the procedure of cutting or cancelling the distribution in the @xmath93 direction . ( the illustration in fig . 2a shows the spatial momentum distribution where the boost in the orthogonal direction @xmath99 is already performed . ) in the final lr frame , the matter is characterized by a rather complex energy momentum tensor , inheriting some parameters from the original uncut distribution in rfg , like the temperature and chemical potential , but as the resulting distribution is not a thermal equilibrium distribution , these parameters are not playing any thermodynamical role . one has to determine all parameters numerically from conservation laws ( [ nconserve],[tconserve ] ) , as done in refs . @xcite . interestingly , a simplified numerical kinetic fo model @xcite led to a fo distribution satisfying the condition @xmath25 for spacelike fo with a smooth distribution function , which is anisotropic ( also in its own lr frame ) and has a symmetry axis pointing in the dominant fo direction . this distribution was then approximated with an analytic , `` cancelling - jttner '' distribution @xcite , which can also be used to solve the fo problem . after fo , the symmetry properties of the energy momentum tensor are the same for the cut - jttner and cancelling - jttner cases @xcite . the fo leads to an anisotropic momentum distribution and therefore to an anisotropic pressure tensor . the energy momentum tensor is not diagonal in the rfg frame , there is a non vanishing transport term , @xmath102 @xcite , in the 2-dimensional plane spanned by the 4-vectors , @xmath90 and @xmath103 . one can , however , diagonalize the energy momentum tensor by making a lorentz boost into the lr frame using landau s definition for the 4-velocity , @xmath91 . in this frame then the energy momentum tensor becomes diagonal , but the pressure terms are not identical , due to the anisotropy of the distribution : @xmath104 here the energy density , @xmath5 , of course must not be the same as in the case of an isotropic , thermal equilibrium post fo momentum distribution . this can be seen from the kinetic definition of the energy momentum tensor as shown in refs . @xcite . we need the complete post fo momentum distribution and the corresponding energy momentum tensor to determine final observables . this depends on the transport processes at fo , and can not be given in general ; however , due to the symmetries of the collision integral , the symmetries of the energy momentum tensor are the same irrespectively of the ansatz used ( e.g. cut - jttner , cancelling - jttner or some other distribution ) . in kinetic transport approaches the microscopic escape probability @xcite is peaking in the direction of @xmath103 , which yields a distribution peaking in this direction , i.e. yielding the same symmetry properties as the previously mentioned analytic ansatzes . the energy momentum tensor in general takes the form @xmath105 where @xmath106 is the orthogonal projector to @xmath107 , and @xmath108 is the unit 4-vector projection of @xmath109 in the direction orthogonal to @xmath107 , i.e. @xmath110 , where @xmath111 ensures normalization to -1 . in the landau lr frame this returns expression ( [ tdiag ] ) the 4-velocity , @xmath107 , and the other parameters of the post fo state of matter , should be determined from the conservation laws ( [ nconserve],[tconserve ] ) . the schematic diagram of the asymmetric distributions and the different reference frames can be seen in fig . [ fab ] . the fo problem was solved for these configurations and ansatzes , by satisfying the conservation laws explicitly for the full energy momentum tensor . we do not have a general eos(s ) that would characterize the connection among @xmath5 , @xmath112 and @xmath113 , furthermore the relation connecting these quantities depends on the 4-vectors @xmath114 and @xmath115 . in addition this connection depends on the details or assumptions of the transport model . the simple models @xcite provide examples for such a dependence . if @xmath116 is known , then for baryon free matter we can determine four unknowns : @xmath115 and , an additional parameter of the post fo distribution , from eq . ( [ tconserve ] ) . ( due to normalization only 3 components of @xmath115 are unknowns . ) for baryon - rich matter we can determine one more unknown parameter , since we have one additional equation , the conservation of baryon charge from eq . ( [ nconserve ] ) . the first step of solution can be done similarly to the isotropic case . then in eq . ( [ em - curr ] ) the enthalpy will change as @xmath117 , and @xmath118 , plus an additive term will appear , @xmath119 . furthermore , eqs . ( [ eq0]-[eq3 ] ) remain of the same form , with @xmath120 , and @xmath113 , plus the additive term @xmath119 will appear in the r.h.s . of eqs.([eq0]-[eq3 ] ) . this additive term will also appear in the expression of @xmath121 after eq . ( [ eqg2a ] ) and in the denominator of eq . ( [ eqg2b ] ) also . the additional term , @xmath119 , in eq . ( [ em - curr ] ) is orthogonal to @xmath27 ( by definition of @xmath122 ) , so when we calculate the scalar product ( [ aa ] ) their cross term vanishes , so @xmath123 now one can express @xmath124 from eq . ( [ ads - p ] ) and inserting it to eq . ( [ aa - p ] ) , we obtain that @xmath125 where this equation is not a scalar equation as it dependes on @xmath126 , where the projector is dependent on @xmath27 . these equations are similar to the ones obtained for the isotropic case , however , to solve this last equation we need a more complex relation among @xmath5 , @xmath112 , @xmath113 . as these arise from the collision integral in the bte approach the needed relation may depend on @xmath127 and @xmath109 . on the other hand , the escape probability may be simple , or may be approximated in a way , which yields an ansatz for this relation with adjustable parameters , and then the problem is solvable . this was the case in refs . @xcite . the recent covariant formulation of the kinetic freeze - out description @xcite indicates that the relation among the different parameters of the anisotropic energy momentum tensor , should be possible to express in terms of invariant scalars , which may facilitate the solution of the anisotropic fo problem . when the adjustable parameters of the post fo matter are determined in this way from the conservation laws , we still need the underlying anisotropic momentum distribution of the emitted particles in order to evaluate the final particle spectra using the cooper - frye formula with this anisotropic distribution function . once again , when all conservation law equations are solved we have to check the non - decreasing entropy condition to see whether the selected fo hypersurface is realistic . in case of an anisotropic final state , due to the increased number of parameters and their more involved relations , the covariant treatment of the problem may not provide a simplification , compared to the direct solution of conservation laws for each component of the energy momentum tensor ( e.g. @xcite ) . recent viscous fluid dynamical calculations evaluate the anisotropy of the momentum distribution is in the pre fo viscous flow ( see e.g. @xcite . ) this anisotropy is governed by the spacetime direction of the viscous transport . the pre and post fo matter may still be different , e.g. the pre fo state may be viscous qgp with current quarks and perturbative vacuum , while post fo we may have a hadron gas or constituent quark gas . the final state will also be anisotropic , not only because of the initial anisotropy but also due to freeze - out . the two physical processes leading to anisotropy are independent , so their dominant directions are in general different . in this case the general symmetries are uncorrelated and can not be exploited to simplify the description of the transition . due to the change of the matter properties , the conservation laws , eqs . ( [ nconserve]-[entropy ] ) , are needed to determine the parameters of the post fo matter before the cooper - frye formula with non - equilibrium post fo distribution is applied to evaluate observables . in this work a new simple covariant treatment is presented for solving the conservation laws across a transition hypersurface . this leads to a significant simplification of the calculation if both the initial and final states are in thermal equilibrium . the same method can also be used for the more complicated anisotropic final state , however , this method is only advantageous if the more involved relations among the parameters of the post fo distribution and the distribution itself is given in covariant form , preferably through invariant scalars . e. molnar , l. p. csernai , v. k. magas , a. nyiri , k. tamosiunas , phys . rev . * c74 * , 024907 ( 2006 ) ; j. phys . g 34 ( 2007 ) 1901 ; e. molnar , l. p. csernai and v. k. magas , acta phys . hung . a * 27 * , 359 ( 2006 ) ; v. k. magas , l. p. csernai and e. molnar , acta phys . hung a * 27 * , 351 ( 2006 ) . v. k. magas , l. p. csernai and e. molnar , eur . phys . j. a * 31 * , 854 ( 2007 ) ; int . j. mod . e * 16 * , 1890 ( 2007 ) ; v. k. magas and l. p. csernai , phys . b * 663 * , 191 ( 2008);l.p . csernai , v.k . magas , e. molnar et al . , j. c 25 , 65 ( 2005);v.k . magas , l.p . csernai , e. molnar et al . phys . a 749 , ( 2005 ) . note , that we actually use for hypersurface construction in 1 + 1 , 2 + 1 , and 3 + 1 dimensional numerical simulations the corresponding computer codes , i.e. , diconex , vesta , and steve , respectively . @xcite explains in great detail the extraction of an oriented fo contour which is represented by a set of contra - variant ( so - called `` diconex iso - contour '' ) vectors . in 2d , the simplices which represent a hypersurface best are line segments , whereas in 3d and 4d they are triangles @xcite and tetrahedrons @xcite , respectively . in particular , the contra - variant 2d fo contour vectors are oriented counter - clockwise around the enclosed spacetime regions . the co - variant normals of the contra - variant simplices are obtained from calculating the mathematical duals of these simplices with respect to a geometric product ( _ cf . _ , e.g. , ref . @xcite ) within the n - dimensional multi - linear space under consideration . note , that the co - variant normal vectors do not depend on any given metric tensor , whereas the contra - variant normal vectors do @xcite . bass , a. dumitru , m. bleicher et al . c60 , 021902 ( 1999 ) ; d. teaney , j. lauret and e.v . shuryak , nucl . phys . a 698 , 479 ( 2002 ) ; s.a . bass , t. renk , j. ruppert et al . , j. phys . g 34 , s979 ( 2007 ) ; c. nonaka , m. asakawa and s.a . bass , j. phys . g 35 , 104099 ( 2008 ) ; h. petersen , j. steinheimer , g. burau , m. bleicher , h. stcker , phys . rev . c 78 ( 2008 ) 044901 ; t. hirano and y. nara , phys . rev . c 79 , 064904 ( 2009 ) .

heavy ion reactions and other collective dynamical processes are frequently described by different theoretical approaches for the different stages of the process , like initial equilibration stage , intermediate locally equilibrated fluid dynamical stage and final freeze - out stage . for the last stage the best known is the cooper - frye description used to generate the phase space distribution of emitted , non - interacting , particles from a fluid dynamical expansion / explosion , assuming a final ideal gas distribution , or ( less frequently ) an out of equilibrium distribution . in this work we do not want to replace the cooper - frye description , rather clarify the ways how to use it and how to choose the parameters of the distribution , eventually how to choose the form of the phase space distribution used in the cooper - frye formula . moreover , the cooper - frye formula is used in connection with the freeze - out problem , while the discussion of transition between different stages of the collision is applicable to other transitions also . more recently hadronization and molecular dynamics models are matched to the end of a fluid dynamical stage to describe hadronization and freeze - out . the stages of the model description can be matched to each other on spacetime hypersurfaces ( just like through the frequently used freeze - out hypersurface ) . this work presents a generalized description of how to match the stages of the description of a reaction to each other , extending the methodology used at freeze - out , in simple covariant form which is easily applicable in its simplest version for most applications .

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Proceed to summarize the following text: the first metal enrichment in the universe was made by the supernova ( sn ) explosions of population ( pop ) iii stars . despite the importance of pop iii stars in the evolution of the early universe , their properties are still uncovered . the main issues is the typical masses of pop iii stars . some studies have suggested that the initial mass function ( imf ) differs from the present day imf ( e.g. , top - heavy imf ; @xcite ) and that a large number of stars might be so massive as to explode as pair - instability sne ( e.g. , @xcite ) . on the other hand , @xcite suggested an imf that is peaked in the range of massive stars that exploded as core - collapse sne . in the early universe , the enrichment by a single sn can dominate the preexisting metal contents ( e.g. , @xcite ) . the pop iii sn shock compresses the sn ejecta consisting of heavy elements , e.g. , o , mg , si , and fe , and the circumstellar materials consisting of h and he , and thus the abundance pattern of the enriched gas may reflect nucleosynthesis in the sn . the sn compression will initiate a sn - induced star formation ( e.g. , @xcite ) and the second - generation stars will be formed from the enriched gas . among the second generation stars , low mass ( @xmath9 ) stars have long life - times and might be observed as extremely metal - poor ( emp ) stars with [ fe / h ] @xmath7 @xcite . ( here [ a / b ] @xmath10 , where the subscript @xmath11 refers to the solar value and @xmath12 and @xmath13 are the abundances of elements a and b , respectively . ) therefore the emp stars should conserve the nucleosynthetic results of the pop iii sn and can constrain the yields of the sn . the elements ejected by various sne are gradually mixed and the abundance patterns of the galaxy becomes homogeneous with time . the abundance patterns of the newly formed stars reflect averaged nucleosynthesis over various sne . it is important to know when the transition from inhomogeneous to homogeneous mixing occurs . the timing of this transition can be informed from chemical evolution calculations with hierarchical models ; @xcite has suggested that a halo ism is unmixed and inhomogeneous at [ fe / h ] @xmath14 , intermediate between unmixed and well mixed at @xmath15 [ fe / h ] @xmath16 , and well mixed at [ fe / h ] @xmath17 ; @xcite has suggested that the mean number of reflected sne is 10 at @xmath18 } \sim -2.8 $ ] . the previous observations ( e.g. , mcwilliam et al . 1995a , b ; @xcite ) provide the abundance patterns of the emp stars that show interesting trends of elemental abundance ratios [ cr / fe ] , [ mn / fe ] , [ co / fe ] , [ zn / fe ] with decreasing [ fe / h ] , although dispersions are rather large . these trends , except for the absolute values of some elements , can be explained by the differences of the progenitors masses and the explosion energies assuming the sn - induced star formation ( @xcite ; umeda & nomoto 2002a , 2005 , hereafter @xcite , @xcite ) . recent observations for @xmath19 } \lsim -2 $ ] by cayrel et al.(2004 , hereafter @xcite ) confirmed these trends shown by the previous studies with much smaller dispersions ( see , however honda et al . 2004 , hereafter @xcite , for the difference in [ cr / fe ] at @xmath20 } \lsim -2 $ ] ) , except for much flatter trends of [ mg / fe ] and [ mn / fe ] than the previous studies . @xcite and @xcite suggested the following interpretation of the observed small dispersions : the elements have been already mixed homogeneously in the halo even below [ fe / h ] @xmath7 and the trends are due to the difference of the lifetime of progenitors with different masses . homogeneous mixing is required because previous sn yields that have been used [ e.g. , woosley & weaver ( 1995 , hereafter @xcite ) ; @xcite ; @xcite ; and chieffi & limongi ( 2002 , hereafter @xcite ) ] show a large scatter in [ @xmath21/fe ] ( where @xmath21 represents @xmath21-elements , for example , o , ne , mg , si , e.g. , @xcite ) . however , this interpretation may not be consistent with the galactic chemical evolution models that suggest inhomogeneous mixing in such early phases ( e.g. , @xcite ) . also , @xmath22-process nuclei observed in the emp stars show too large scatters @xcite to be reproduced by the homogeneous mixing model @xcite , unless there exist a major site of @xmath22-process synthesis other than sn explosions ( see @xcite , who concluded core - collapse sne are more preferable sites of @xmath22-process elements than neutron - star mergers ) . in the regime of inhomogeneous mixing , @xcite have succeeded to reproduce the observed trends of the ratios , [ cr / fe ] , [ mn / fe ] , [ co / fe ] , and [ zn / fe ] , as a result of chemical enrichment of various sn models including hyper - energetic explosions ( @xmath23 : hypernovae , hereafter hne ) . in their approach , variation of @xmath6 and the mixing - fallback process are important @xcite . the mixing - fallback model can solve the disagreement between [ @xmath21/fe ] and [ ( fe - peak element)/fe ] ( e.g. , @xcite ) . traditionally , core - collapse sne were considered to explode with @xmath24 as sn 1987a @xcite , sn 1993j @xcite , and sn 1994i @xcite before the discoveries of hne sn 1997ef ( @xcite ) and sn 1998bw ( patat er al . 2001 ; @xcite ) . after these discoveries , the number of pop i hne has been increasing , and the association with gamma - ray bursts ( grbs ) has been established as grb 980425/sn 1998bw ( galama et al . 1998 ; iwamoto et al . 1998 ; woosley et al . 1999 ; nakamura et al . 2001a ) , grb 030329/sn 2003dh ( stanek et al . 2003 ; hjorth et al . 2003 ; matheson et al . 2003 ; mazzali et al . 2003 ; lipkin et al . 2004 ; deng et al . 2005 ) , and grb 031203/sn 2003lw ( thomsen et al . 2004 ; gal - yam et al . 2004 ; cobb et al . 2004 ; malesani et al . 2004 ; mazzali et al . though it is an interesting issue how much fraction of the core - collapse sne explode as hne ( e.g. , podsiadlowski et al . 2004 ) , non - negligible number of hne occurred at least at present days . nucleosynthesis in hne is characterized by the large amount of @xmath25ni production ( @xmath26 , e.g. , @xcite ) . two sites of @xmath25ni synthesis have been suggested : the shocked stellar core ( e.g. , @xcite ) and the accretion disk surrounding the central black hole ( e.g. , macfadyen & woosley 1999 ) . we investigate the former site because the light curve and spectra of sn 1998bw favor the @xmath25ni synthesis in the shocked stellar core @xcite . in this paper , we construct core - collapse sne models of the pop iii 13 50 @xmath1 stars for various explosion energies of @xmath27 ergs . by applying the mixing - fallback model to the hn models , we show that the yields of these pop iii sne and hne are in good agreements with the observed abundance patterns and trends with reasonably small dispersions @xcite . we do not consider pair - instability sne because previous studies ( @xcite ; umeda & nomoto 2003 , hereafter @xcite ; @xcite ) found that there has been no evidence of the pair - instability sn abundance patterns in the emp stars , although they might have existed before our galaxy become metal - rich - enough to form low - mass stars ( see also @xcite for core - collapse very massive stars ) . in [ sec : model ] , we describe our progenitor and explosion models . in [ sec : individual ] , we show that the abundance patterns of the emp stars are reproduced by hn models and that the abundance patterns of the vmp stars are reproduced by normal sn models or an imf - integrated model . in [ sec : trend ] , we show that the trends with small dispersions can be reproduced by pop iii sn models with various progenitors masses and explosion energies assuming a sn - induce star formation . the scatters in our models are almost consistent with the observed ones . in [ sec : improve ] , we examine how the agreements of sc / fe , ti / fe , mn / fe , and co / fe are improved by modifying the distributions of the neutron - proton ( n / p ) ratio and the density in the presupernova models . in [ sec : discuss ] , summary and discussion are given . the calculation method and other assumptions are the same as described in umeda et al . ( 2000 , hereafter @xcite ) , @xcite , and @xcite . the isotopes in the reaction network for explosive nuclear burning include 280 species up to @xmath28br as in @xcite . we calculate the evolutions of pop iii progenitors , whose main - sequence masses are @xmath29 13 , 15 , 18 , 20 , 25 , 30 , 40 , 50 @xmath1 , and their sn explosions as summarized in table [ tab : models ] . the mass loss rate from stars with metallicity @xmath30 is assumed to be proportional to @xmath31 @xcite , so that pop iii @xmath32 models experience no mass loss . sn hydrodynamical calculations include nuclear energy generation with the @xmath21-network . the yields are obtained by detailed nucleosynthesis calculations as a post - processing . there are various ways to simulate the explosion . for example , @xcite assumed an approximate analytic formula to describe the radiation - dominated shock , although their recent modeling applied full hydrodynamics ( @xcite ) . @xcite and @xcite injected energy as a piston . their piston model could mimic the time delay until the deposited energy by neutrino reaches a critical value . however , since the explosion mechanism of core - collapse sne have not been well uncovered , one still does not know what is the most realistic way to inject explosion energy . further , the yields do not strongly depend on how to generate the shock @xcite . therefore , in our calculation , the explosion is initiated as a thermal bomb with an arbitrary explosion energy , i.e. , we elevate temperatures of an inner most region of the progenitor . we determine the explosion energy with referring to the relation between the main - sequence mass and the explosion energy ( @xmath33 ) as obtained from observations and models of sne ( fig . [ fig : ccsn]a ) . this relation is obtained from pop i sne , but we assume that the same @xmath34 relation holds for pop iii sne because the fe core masses are roughly the same between pop i and pop iii stars @xcite . according to the observed relation , the massive stars with @xmath35 are assumed to explode as hypernovae ( hne ) , which are hyper - energetic explosions and expected to leave black holes behind @xcite , and the explosion energies of the models for @xmath29 20 , 25 , 30 , 40 , 50 @xmath1 are @xmath36 10 , 10 , 20 , 30 , 40 , respectively . the stars smaller than 20 @xmath1 are considered to explode as normal sne with @xmath37 . the explosion energy of normal sn models ( @xmath37 ) is consistent with that of sn 1987a ( @xmath38 , @xcite ) . figures [ fig : snhn]a and [ fig : snhn]b show the abundance distributions in the ejecta of the @xmath39 normal sn ( @xmath40 ) and hn ( @xmath41 ) models , respectively . nucleosynthesis in hne with large explosion energies takes place under high entropies and show the following characteristics @xcite . \(1 ) both complete and incomplete si - burning regions shift outward in mass compared with normal supernovae . as seen in figures [ fig : snhn]a and [ fig : snhn]b , the mass in the complete si - burning region becomes larger , while the incomplete si - burning region does not change much . as a result , higher energy explosions produce larger [ ( zn , co , v)/fe ] and smaller [ ( mn , cr)/fe ] . \(2 ) in the complete si - burning region of hypernovae , elements produced by @xmath21-rich freezeout are enhanced because of high entropies . hence , elements synthesized through the @xmath21-capture process , such as @xmath42ti , @xmath43cr , and @xmath44ge are more abundant . these species decay into @xmath42ca , @xmath43ti , and @xmath44zn , respectively . \(3 ) oxygen burning takes place in more extended regions for the larger explosion energy . then more o , c , al are burned to produce a larger amount of burning products such as si , s , and ar . as a result , hypernova nucleosynthesis is characterized by large abundance ratios of [ ( si , s)/o ] ( @xcite ) . the mass cut is defined to be a boundary between the central remnant and the sn ejecta and thus corresponds to the mass of the compact star remnant . the sn yield is an integration over the ejecta outside the mass cut . in 1d spherically symmetric models , the mass cut can be determined hydrodynamically as a function of the explosion energy @xcite . however , such 1d hydrodynamical determinations may not be relevant because sn explosions are found to be aspherical ( e.g. , @xcite ) . we take into account approximately the aspherical effects with a mixing - fallback model ( see appendix for detail ) parameterising the aspherical sn explosions with three parameters , i.e. , the initial mass cut @xmath45 , the outer boundary of the mixing region @xmath46 and the ejection factor @xmath47 . the initial mass of the central remnant is represented by @xmath45 . during the explosion , an inversion of the abundance distribution and a fallback of the materials above @xmath45 onto the central remnant might occur in the aspherical explosions ( e.g. , @xcite ) , in contrast to the spherical explosions . the inversion is represented by the mixing of the materials between @xmath45 and @xmath46 and the fraction of materials ejected from the mixing region is parameterized by the ejection factor @xmath47 . as a consequence , the final mass of the central remnant @xmath48 is derived with equation ( 7 ) in appendix . the parameters of the mixing - fallback model should essentially be derived from the mechanisms of sn explosions , e.g. , the rotation , the asphericity , and the way to inject energy from the central region . however , the explosion mechanisms of sne have not been clarified . thus the parameters are constrained from the observed fe ( @xmath25ni ) mass or from the comparison between the yield and the observed abundance pattern . for normal sn models , we determine the mass cuts to yield @xmath49 because the ejected @xmath25ni mass of the observed normal sne are clustered around this value ( fig . [ fig : ccsn]b , e.g. , sn 1987a , @xcite ; sn 1993j , @xcite ; sn 1994i , @xcite ) . we also assume that the whole materials above the mass cut are ejected without the mixing - fallback process . this corresponds to the ejection factor @xmath50 in the mixing - fallback model ( see appendix ) . for hn models , we apply the mixing - fallback model and determine parameters to yield [ o / fe ] @xmath51 ( case a ) for all hn models or [ mg / fe ] @xmath52 ( case b ) for massive hn models ( @xmath53 ) as described in appendix . while case a is similar to @xcite , case b is to reproduce [ mg / fe ] @xmath54 plateau for [ fe / h ] @xmath55 in @xcite . the plateau had not been observed by the previous studies . since case b has larger amount of fallback , i.e. , larger @xmath46 and smaller @xmath47 than case a , both of small [ mg / fe ] and [ fe / h ] is realized in case b. the applied mixing - fallback parameters are summarised in tables [ tab : models ] , [ tab : modelsye ] , and [ tab : modelsyelow ] and the final yields are summarised in tables [ tab : yield ] , [ tab : yield2 ] , [ tab : yieldye ] , [ tab : yield2ye ] , [ tab : yieldyelow ] , and [ tab : yield2yelow ] . we assume that [ fe / h ] of a star formed by the sn shock compression is determined by the ratio between the ejected fe mass @xmath56(fe ) and the swept - up h mass @xmath56(h ) @xcite . according to @xcite , the swept - up h mass is given as @xmath57 where @xmath58 is the mass fraction of h in the primordial gases , @xmath59 is the total amount of the swept - up materials , and @xmath60 is the number density of the circumstellar medium . here we adopt @xmath61 as obtained from _ wilkinson microwave anisotropy probe _ @xcite and standard big bang nucleosynthesis @xcite . [ fe / h ] of the star is approximated as @xmath62 } & = & \log_{10}(m{\rm ( fe)}/m{\rm ( h ) } ) \\ \nonumber & & -\log_{10}{(x{\rm ( fe)}/x{\rm ( h)})_\odot } \\ & \simeq & \log_{10}\left({m({\rm fe})/{{m_{\odot}}}\over{e_{51}^{6/7}}}\right)-c . \label{eq : snsf}\end{aligned}\ ] ] here @xmath63 is the solar abundance ratio in mass between fe and h @xcite and @xmath64 is assumed to be a constant value of 1.4 , which corresponds to @xmath65 @xmath66 . resulting values of [ fe / h ] are summarised in tables [ tab : models ] , [ tab : modelsye ] , and [ tab : modelsyelow ] . [ fe / h ] of stars formed from the ejecta of hne and normal sne can be consistent with the observed [ fe / h ] of the emp and vmp stars , respectively . @xcite provided abundance patterns of 35 metal - poor stars with small error bars for @xmath67}\lsim-2 $ ] . each emp star may be formed from the ejecta of a single pop iii sn , although some of them might be the second or later generation stars . the yields of sne with the progenitors of [ fe / h ] @xmath68 can be well - approximated by those of pop iii sne since they are similar @xcite . in this subsection , the theoretical yields are compared with the averaged abundance pattern of four emp stars , cs 22189 - 009 , cd-38:245 , cs 22172 - 002 and cs 22885 - 096 , which have low metallicities ( @xmath69}<-3.5 $ ] ) and normal [ c / fe ] @xmath70 . stars with large [ c / fe ] ( @xmath71 ) , called c - rich emp stars , are discussed in @xcite , @xcite , and @xcite . the origin of some of those stars may be a faint sn , being different from those of [ c / fe ] @xmath70 stars ( see [ sec : c ] ) . also some of the c - rich emp stars might be originated from the mass transfer from the c - rich companion in close binaries ( e.g. , @xcite ) . comparisons between the hn yields and the abundance pattern of the emp stars are made in figures [ fig : emp]a-[fig : emp]e . in the mixing - fallback model , both [ ( fe - peak elements)/fe ] and [ @xmath21/fe ] give good agreements with the observations , except for some elements , e.g. , k , sc , ti , cr , mn , and co. possible ways to improve these elements are discussed in [ sec : improve ] . @xcite also provided the abundance patterns of the vmp stars whose metallicities ( [ fe / h ] @xmath72 ) are higher than the emp stars . the observed abundance pattern is represented by the averaged abundance pattern of five stars bd+17:3248 , hd 2796 , hd 186478 , cs 22966 - 057 and cs 22896 - 154 , which have relatively high metallicities ( @xmath73}<-2.0 $ ] ) . since these metallicities correspond to those of normal sn models according to equation ( [ eq : snsf ] ) for the sn - induced star formation model , we first compare the observations with normal sn yields ( figs . [ fig : vmp]abc ) . the mass cuts of normal sn models are determined so that the ejected fe ( @xmath25ni ) mass is @xmath74 . on the other hand , most vmp stars are considered to have the abundance pattern averaged over imf and metallicity of the progenitors , thus we also compare with the imf - integrated yield ( fig . [ fig : imf ] ) . since the yields of sne with the progenitors of [ fe / h ] @xmath68 are quite similar to those of pop iii sne @xcite , we use the pop iii yields for these stars as well . the imf integration is performed from 10 @xmath1 to 50 @xmath1 with 8 models , 13 , 15 , 18 , 20 , 25 , 30 , 40 , 50 @xmath1 and the extrapolations . we make the power - law imf integrations as follows : @xmath75 where @xmath76 is the number of stars within the mass range of [ @xmath77 , @xmath78 is a normalization constant , and @xmath79 is an integration index ( salpeter imf has @xmath80 ) . the integration is performed and normalized by the total amount of gases forming stars as follows : @xmath81 where @xmath82 is an integrated mass fraction of an element , a , @xmath83 is mass fraction of a in a model interpolated between the nearest models , and @xmath84 is an ejected mass interpolated between the nearest models or the nearest model and the edge of the imf - integrated mass range . here we assume @xmath85 and @xmath86 stars do not yield any materials as type ii sne or hne , i.e. , @xmath87 . comparing the integrated yield with the salpeter s imf with the abundance pattern of the vmp stars ( fig . [ fig : imf ] ) , most elements show reasonable agreements , except for n , k , sc , ti , and mn . the integrated yields are summarised in table [ tab : imf ] . figures [ fig : vmp]a-[fig : vmp]c and [ fig : imf ] show that n is underproduced in our models . there are two possible explanations ( 1 ) and ( 2 ) for this discrepancy : \(1 ) n was actually underproduced in the pop iii sn as in our models , but was enhanced as observed during the first dredge - up in the low - mass red - giant emp stars ( e.g. , @xcite ) . observationally , the emp and vmp stars in @xcite are giants and some of them show evidences of the deep mixing , i.e. , the dilution of li and the low @xmath88c/@xmath89c ratio @xcite . on the other hand , the emp and vmp stars with no evidence of the deep mixing show relatively small [ n / fe ] ( @xmath90 , @xcite ) . however , [ n / fe ] in our models are even smaller than the smallest [ n / fe ] observed in the emp and vmp stars @xcite . thus the following mechanism might be important . \(2 ) n was enhanced in massive progenitor stars before the sn explosion . n is mainly synthesized by the mixing between the he convective shell and the h - rich envelope ( e.g. , @xcite ) . the mixing can be enhanced by rotation @xcite . suppose that the pop iii sn progenitors were rotating faster than more metal - rich stars because of smaller mass loss , then [ n / fe ] was enhanced as observed in the emp stars . [ o / fe ] of the 18 @xmath1 and the imf - integrated model are in good agreement with the observations ( figs . [ fig : vmp]c and [ fig : imf ] ) , while [ o / fe ] of the 13 and 15 @xmath1 models are lower than the observations ( figs . [ fig : vmp]a and [ fig : vmp]b ) . in the abundance determination of o , however , uncertain hydrodynamical ( 3d ) effects are important @xcite and @xcite applied the 3d correction for dwarfs to the metal - deficient giants . if the observed values of [ o / fe ] in the figures are correct , this may indicate that the contribution of a single normal sn from a small - mass progenitor to the chemical enrichment in the vmp stars is small . we assumed all massive stars with @xmath91 explode as hne . however , there is a suggestion that the fraction of hne to whole sne ( @xmath92 ) is @xmath93 ( @xcite ) . if @xmath93 , [ ( c , n , o)/fe ] and [ zn / fe ] are slightly larger and smaller , respectively , than the case with @xmath94 , but these are still in good agreement with the observation ( fig . 12 in @xcite ) . on the other hand , if the contribution of the faint sne ( @xmath91 ) to reproduce the abundance patterns of the c - rich emp stars is large enough ( see [ sec : c ] ; @xcite ) , [ ( c , n , o)/fe ] is enhanced . this is because the faint sne produce large [ ( c , n , o)/fe ] due to a small amount of fe ejection . the contribution of the faint sne , however , might be small , since [ mg / fe ] is close to the upper limit of the observations without the contribution of the faint sne . in order to estimate the ratios of hne and faint sne relative to all core - collapse sne , further investigations are necessary . in [ sec : emp ] and [ sec : vmp ] , we show that the observed abundance patterns can be reasonably reproduced by the mixing - fallback model . @xcite showed not only the abundance patterns of individual stars but also the existence of certain trends of the abundance ratios with respect to [ fe / h ] . in this section , we compare the observed trends with our models in tables [ tab : models]-[tab : yield2 ] . in figure [ fig : trend ] , the observed abundance ratios [ x / fe ] against [ fe / h ] are compared with yields of individual sn models in table [ tab : models ] and the imf - integrated yield described in [ sec : vmp ] . here [ fe / h ] of individual sn models are determined by equation ( [ eq : snsf ] ) , while [ fe / h ] of the imf - integrated abundance ratios are assumed to be same as normal sn models ( [ fe / h ] @xmath95 ) . we note that the observed abundance ratios of most elements are roughly constant for @xmath96 [ fe / h ] @xmath97 . this can be interpreted that the sn ejecta had been mixed homogeneously in the halo at @xmath96 [ fe / h ] . this is consistent with the chemical evolution models in @xcite , @xcite , @xcite , @xcite , and @xcite . according to the sn - induced star formation model ( eq . [ eq : snsf ] ) , our models cluster around [ fe / h ] @xmath98 and only a few model exists around [ fe / h ] @xmath99 , because we applied only one explosion energy for each mass . in reality , the explosion energies of hne may depend , e.g. , on the rotation of the progenitors , even if the progenitors masses of hne are similar . the progenitors of sne 1997ef and 2003dh , for instance , have similar masses , but the explosion energies are @xmath100 ergs and @xmath101 ergs , respectively @xcite . variations of the explosion energy for the same stellar mass lead to variations of [ fe / h ] . in order to produce a model with @xmath18}\sim-3 $ ] , therefore , we add the @xmath39 model with @xmath102 ( see fig . [ fig:25e5 ] for its abundance pattern ) . the trends of most elements can be well reproduced by our models , except for k , sc , ti , cr , mn , and co. in the followings , we discuss the trend of each element in more detail . the ratio c / fe in our models are clustered around [ c / fe ] @xmath70 , while the observed [ c / fe ] of the emp stars are scattered ( see fig . [ fig : trend ] ) . the large [ c / fe ] ( @xmath103 ) can be interpreted as originated from the faint sne that are characterised by a small ejection factor @xmath47 and the resultant large [ ( c , n , o)/fe ] @xcite . in fact , figure [ fig : faint ] shows that the abundance pattern of the c - rich emp stars ( cs 29498043 : @xcite ) can be reproduced by our 25 @xmath1 faint sn model with a normal explosion energy @xmath37 and small @xmath104 ( model 25f in tables [ tab : models]-[tab : yield2 ] ) . in this faint sn model , n / fe is too small but can be enhanced as described in [ sec : no ] . we should note that the large [ co / fe ] of cs 29498043 is difficult to be reproduced by the faint sn model of @xmath37 . the parent sn of cs 29498043 might be a faint sn with high - entropy jets @xcite . alternative explanation is that the production of c in other sites is important , that is , c is produced not only from sne but also from mass - losing wolf - rayet ( wr ) stars . fast rotating stars may undergo strong mass - loss and enter the wr phase , even if their metallicities are considerably low @xcite . also , the c - enhancement can be realized by transferring mass from the c - rich binary companion ( e.g. , @xcite ) . the ratio mg / fe in @xcite is less - scattered around [ mg / fe ] @xmath105 which is smaller than [ mg / fe ] @xmath106 obtained by @xcite and other previous studies . case a models , whose ejection factor @xmath47 is set to produce [ o / fe ] @xmath51 , appear around @xmath107 [ fe / h ] @xmath108 and @xmath109 [ mg / fe ] @xmath110 . these are larger by 0.1 dex than observed [ mg / fe ] in @xcite but consistent with these in the previous studies . we also consider case b models to reproduce the [ mg / fe ] @xmath105 plateau in @xcite at [ fe / h ] @xmath111 , although the plateau had not been observed by the previous studies . [ ( si , ca)/fe ] in @xcite and @xcite are slightly less - scattered than in previous studies . [ ( si , ca)/fe ] in our models are in good agreements with the observations and the widths of the scatter in our models are consistent with the observations . the small scatters of [ ( mg , si , ca)/fe ] in our present model are different from previous yields ( @xcite ; thielemann et al . this difference stems from the following different assumption . the previous yields assumed the ejected fe mass ( @xmath112 ) depends weakly on the progenitors masses . therefore [ @xmath21/fe ] in massive star models depends mainly on the @xmath21-element abundances , which strongly depend on the stellar mass . this leads to the large scatters . in the present stars , the emp stars with [ fe / h ] @xmath7 are produced by hne only ( @xmath113 ) , and those hn models produce fe much more than @xmath114 . instead , we assume [ o / fe ] @xmath51 or [ mg / fe ] @xmath52 for @xmath113 models . our approach suggests that the observed small scatter of [ @xmath21/fe ] implies that larger amount of fe is produced in more massive stars . this is consistent with the observations that typical hne eject larger amount of fe than normal sne ( fig . [ fig : ccsn]b ) . [ ti / fe ] in our models is smaller than the observations . there are no clear trend in [ ti / fe ] in our models as in the observations and the scatter is similar to the observations . [ ti / fe ] may be enhanced by nucleosynthesis in high - entropy environments as will be discussed in [ sec : improve ] ( a `` low - density '' modification , see also @xcite ) or in a jet - like explosion @xcite . the abundances of na and al in @xcite are different from previous studies . this is partly because @xcite took into account non - lte ( nlte ) effects ( @xmath115 dex and @xmath116 dex ) . the @xcite result shows a trend in [ na / fe ] against [ fe / h ] . in figure [ fig : trend ] , our results also show such a trend , although na / fe is slightly smaller than the observations in the hn models , especially in @xmath117 models . because all our models are pop iii models , the trend is not due to the metallicity effect but due to the combination of the progenitors masses and the explosion energies . the trend stems from the fact that more massive stars produce smaller @xmath118(na ) and that more energetic explosions burn more na . [ al / fe ] is also smaller by 0.5 dex than in @xcite , especially in the hn model . however our models are in good agreements with [ al / fe ] in @xcite . the difference in the observed [ al / fe ] between @xcite and @xcite may be mainly because @xcite does not take into account nlte effects . the larger @xmath46 in case b enhances na / fe but reduces al / fe . this is because na and al are mainly synthesized near the outer and the inner edges of the o - rich region , respectively . only [ al / fe ] of model 40b is in reasonable agreement with @xcite because the 40 @xmath1 progenitor star produce large @xmath118(al ) . both na and al are mostly synthesized in c shell - burning , and the produced amount depends on the overshooting at the edge of the convective c - burning shell @xcite . in the present presupernova evolution models , no overshooting is included . if the nlte correction of al is correct , it might be needed to consider convective overshooting in the c - burning shell . this could enhance na / mg and al / mg , thus weakening the odd - even effect in the abundance patterns and leading to a better agreement with @xcite . [ k / fe ] and [ sc / fe ] in our models are much smaller than in @xcite and @xcite . possible improvements are discussed in detail in [ sec : improve ] . k / fe is slightly enhanced by the `` low - density '' modification as described in [ sec : improve ] , but still not large enough . iwamoto et al . ( in preparation ) suggests that the model with large @xmath8 ( @xmath119 ) in the inner region can produce enough k. sc / fe can be enhanced by the `` low - density '' modification ( see [ sec : improve ] ; @xcite ) . further enhancement can be realized if @xmath120 . recently , @xcite and @xcite calculated nucleosynthesis based on the core - collapse sn simulations with neutrino transport . in their models , neutrino absorption enhances @xmath8 near the mass cut of the ejecta . according to their results , sc / fe in the normal sn model is enhanced to [ sc / fe ] @xmath70 due to the large @xmath8 ( @xmath121 ) and high - entropy ( @xmath122 , where @xmath123 denotes the the entropy per nucleon and @xmath124 is the boltzmann constant ) . [ cr / fe ] in our models is larger than @xcite but consistent with @xcite . the difference between @xcite and @xcite stems from the use of the different cr lines (: @xcite and : @xcite ) in obtaining [ cr / fe ] . it is still uncertain which line is better to use in estimating [ cr / fe ] . if gives reliable abundance , cr in our models is overproduced , although the trend in our models is similar to the observations . since cr is mostly produced in the incomplete si - burning region , the size of this region relative to the complete si - burning region should be smaller than the present model in order to produce smaller cr / fe . we also need to examine the nuclear reaction rates related to the synthesis of @xmath125 that decays into @xmath126 . [ mn / fe ] and [ co / fe ] in our models are smaller than the observations , although the trend of [ co / fe ] in our models is similar to the observations . the negligibly small dependence of [ mn / fe ] on [ fe / h ] in @xcite is different from previous observations that [ mn / fe ] significantly decreases toward smaller [ fe / h ] @xcite . they can be improved by the n / p modification as discussed in [ sec : ye ] . also , mn can be efficiently enhanced by a neutrino process ( see [ sec : yelowtrend ] ; @xcite ; t. yoshida et al . , in preparation ) . therefore the mn / fe ratio is important to constrain the physical processes during the explosion . [ ni / fe ] and [ zn / fe ] in our models are in good agreement with the observations , although [ ni / fe ] is slightly smaller than the observation . ni / fe is higher if @xmath127 is smaller ( i.e. , the mass cut is deeper ) because @xmath128ni , a main isotope of stable ni , is mainly synthesized in a deep region with @xmath129 . however , a smaller @xmath127 tends to suppress zn / fe ( see appendix ) , thus requiring more energetic explosions to produce large enough zn / fe . the good agreement of [ zn / fe ] with observations strongly supports the sn - induced star formation model and suggests that the emp stars with smaller [ fe / h ] are made from the ejecta of hne with higher explosion energies and larger progenitor s masses . this is because higher explosion energies lead to larger [ zn / fe ] and smaller [ fe / h ] . other possible production sites of zn include the neutrino - driven wind from a proto - neutron star ( e.g. , @xcite ) and the accretion disk of a black hole ( e.g. , @xcite ) . further studies are needed to see how large the contributions of these sites to the zn production are . in order to reproduce the trend and small scatter of [ zn / fe ] , there must be some `` hidden '' relations between the explosion energy and nucleosynthesis in the neutrino - driven wind or the accretion disk models . if the vmp stars with [ fe / h ] @xmath72 is made from normal sne with @xmath130 , zn is underproduced in our models . nucleosynthesis studies with neutrino transport @xcite suggested that zn in the normal sn model is enhanced to [ zn / fe ] @xmath70 . however , the enhancement is not large enough to explain the large [ zn / fe ] ( @xmath103 ) in the emp stars . in this section , we present the models based on the modified presupernova distributions of the `` n / p ratio ( i.e. , @xmath8 ) '' and the `` density '' . we then discuss how these modifications improve sc , ti , mn , and co. the imf - integrated yields , the parameters , and the yields of the individual sne are summarised in tables [ tab : imf]-[tab : yield2yelow ] . in the above discussion , we assume that @xmath8 in the presupernova model is kept almost constant during the explosion . however , recent studies ( e.g. , @xcite ) have suggested that @xmath8 may be significantly varied by the neutrino process during explosion . further , the region , where the neutrino absorption and @xmath8 variation occur , is rayleigh - taylor unstable because of neutrino heating , so that there exists a large uncertainty in @xmath8 and its distribution . we apply the following @xmath8 profile that was found to produce reasonable results in @xcite , i.e. , @xmath131 in the complete si burning region and @xmath132 in the incomplete si burning region . the @xmath8 profile is modified by adjusting the isotope ratios of si . according to the recent explosion calculations with the neutrino effect ( e.g. , @xcite ) , materials with large @xmath8 ( @xmath133 ) may be ejected . however , @xmath8 might be diluted by mixing , and our @xmath8 profile might mimic such dilution . the adopted high @xmath8 in the complete si - burning region and low @xmath8 in the incomplete si - burning region lead to large co / fe and mn / fe , respectively . figures [ fig : ye]a-[fig : ye]j show better agreements between the models and the observations . here we applied the same @xmath45 as in the models without the @xmath8 modification . the imf - integrated yield , the parameters , and the yields of the individual sne are summarised in tables [ tab : imf]-[tab : yield2ye ] . since the variation of @xmath8 is considered to be due to the @xmath134-process , the @xmath8 modification might be applied to any core - collapse sne . in the `` low - density '' modification , the density of the presupernova progenitor is artificially reduced without changing the total mass . @xcite assumed that such a low density would be realized if the explosion is induced by multiple jets consisted of the primary weak jets and the main strong sne jets . the primary weak jets expand the interior of the progenitor before the sn explosions driven by the main jets ( as described in appendix in @xcite ) . alternate explosion mechanism that realizes `` low - density '' is a delayed explosion ( e.g. , @xcite , who investigated explosive nucleosynthesis induced by a black hole forming collapse comparing a direct collapse and a delayed collapse caused by fallback ) . the `` low - density '' is presumed to be realized in the jet - like or delayed explosions involving fallback , thus being applied only to the hn models but not to the normal sn models . other mechanism to realize `` low - density '' was suggested by @xcite , i.e. , the l model using `` low '' @xmath88c(@xmath21 , @xmath135)@xmath136o rate @xcite . they showed that the `` low '' @xmath88c(@xmath21 , @xmath135)@xmath136o rate leads the higher c abundance ( i.e. , larger c / o ratio ) and more active c - shell burning . as a consequence , the average density in the complete and incomplete si - burning region is lower than the `` high '' @xmath88c(@xmath21 , @xmath135)@xmath136o rate case @xcite . if this mechanism is valid , the `` low - density '' will be realized in whole sne . however , the large c / o ratio leads to overproduce ne and na at the solar metallicity @xcite . explosive nucleosynthesis in the model with the `` low - density '' modification takes place at higher entropy ( @xmath137 ) than in the model without modification ( @xmath138 ) , thus enhancing the @xmath21-rich freeze - out compared with the standard model . as a result , the sc / fe and ti / fe ratios are particularly enhanced . figure [ fig : yelow]a shows that the sc / fe and ti / fe ratios in the emp stars can be better reproduced by the model of @xmath139 and @xmath140 with the `` low - density '' modification , whose presupernova density is reduced by a factor of 2 . here we applied the @xmath8 modification and the same @xmath45 as in the models without the @xmath8 modification . the other hn models with both the @xmath8 and `` low - density '' modifications are also in good agreement with the abundance pattern of the emp stars ( figs . [ fig : yelow]b-[fig : yelow]g ) . because the degree of `` low - density '' is likely to be different in each explosion , we assume the `` low - density '' factor so that each model reproduces [ sc / fe ] of the emp stars as given in table [ tab : modelsyelow ] . the imf - integrated yield , the parameters , and the yields of the individual sne are summarised in tables [ tab : imf ] and [ tab : modelsyelow]-[tab : yield2yelow ] . the good agreements suggest that the @xmath8 and `` low - density '' modifications might actually be realized in the sn explosions . in the present study , we consider the global enhancement of entropy due to the jet - like or delayed explosion , but such high - entropy region is also realized locally in the neutrino - driven wind or the accretion disk . neutrino - driven wind nucleosynthesis @xcite for @xmath141 and @xmath142 produces [ sc / fe ] ( @xmath143 ) being consistent with that of the emp stars but smaller [ co / fe ] and [ zn / fe ] than those of the emp stars ; for @xmath144 and @xmath145 , smaller [ sc / fe ] ( @xmath90 ) is produced than that of the emp stars . accretion disk nucleosynthesis ( @xmath146 and @xmath147 , @xcite ) produces much larger [ sc / fe ] ( @xmath148 ) than that of the emp stars and does not produce simultaneously [ co / fe ] @xmath106 and [ zn / fe ] @xmath106 as observed in the emp stars . the above models show that it seems difficult to reproduce the overall abundance pattern of the emp stars only with nucleosynthesis in the neutrino - driven wind or the accretion disk . however , if some contributions from these sites can be added to our models , it could enhance sc / fe . in order to obtain [ sc / fe ] @xmath149 , the sn ejecta should have @xmath150 @xcite . when about @xmath151 of fe and little sc are ejected as in our model of @xmath152 and @xmath140 , [ sc / fe ] @xmath149 is obtained by the ejection of extra @xmath153 from either the neutrino - driven wind or the accretion disk . figure [ fig : trendyelow ] shows the trends resulting from the @xmath8 modification made for all models and the `` low - density '' modification applied only for hn models . each comparison between the model and the observation is shown in figures [ fig : ye]a-[fig : ye]c , and [ fig : yelow]a-[fig : yelow]g . while [ ( k , cr)/fe ] still do not explain the observations well , the trends of [ ( sc , ti , mn , co)/fe ] are in much better agreement with the observations than the models without the modifications . other elements also show good agreements and small scatters . the sc / fe and ti / fe ratios in normal sn models are lower than the observations . this is partly because the `` low - density '' modification is not applied to normal sne . to reproduce the observed high sc / fe and ti / fe ratios , there might be some mechanisms to realize the high entropy ( @xmath137 ) even for normal sne . in this connection , it is interesting to note that the recent x - ray flash grb 060218 was found to be associated with sn 2006aj ( e.g. , @xcite ) . the properties of sn 2006aj are close to normal sne ; the progenitor s main - sequence mass and the explosion energy were estimated to be @xmath154 and @xmath155 @xcite . this suggests that the `` low - density '' modification could be realized in some normal sne . such explosions might contribute to enhance sc / fe and ti / fe . [ cr / fe ] and [ mn / fe ] in our models are larger and smaller than @xcite , respectively , while [ co / fe ] is in good agreement with @xcite . [ mn / fe ] could also be enhanced by the following @xmath134-process in the complete si - burning region and the subsequent radioactive decay : @xmath156 ( @xcite ; t. yoshida et al . , in preparation ) . in this paper we performed the hydrodynamical and nucleosynthesis calculations of pop iii 13 50 @xmath1 core - collapse sne and provided the yields by adopting the mixing - fallback model . we show that our yields are consistent with the observed abundance patterns of the emp and vmp stars @xcite . the trends of [ x / fe ] vs. [ fe / h ] with small scatters can be reproduced by our models as a sequence resulting from the combination of different progenitors masses ( @xmath5 ) and explosion energies ( @xmath6 ) . this is because we adopt the empirical relation that a larger amount of fe is ejected by massive hne ( @xmath113 ) than normal sne . this indicates that the observed trends with small scatters do not necessarily mean the homogeneous mixing in the interstellar medium , but can be reproduced by the `` inhomogeneous '' chemical evolution model , in which the emp stars are enriched by the individual sne with different ( @xmath5 , @xmath6 ) . in our model , yields of more massive hne correspond to the emp stars with smaller [ fe / h ] ( @xmath7 ) . we should stress that this is not because hne were dominant among pop iii sne , but because the second generation stars produced by pop iii core - collapse sne with higher explosion energies tend to have smaller [ fe / h ] as @xmath18 } \simeq \log_{10}\left({m({\rm fe})\over{{m_{\odot}}}}/{e_{\rm 51}^{6/7}}\right ) -c$ ] ( eq . [ eq : snsf ] ) . almost all stars in @xcite and @xcite can be reasonably well reproduced by core - collapse sne yields , but none by the pure pair - instability sn yields ( e.g. , @xcite ) . in other words , the emp stars in @xcite show no clear features of top - heavy imf . further , the vmp stars ( [ fe / h ] @xmath72 ) can be well - reproduced by integrating the yields of pop iii sne over the salpeter s imf . this also implies that the imf of pop iii stars was not top - heavy , but approximately salpeter s . to constrain the imf of pop iii stars , we need more complete set of the observed data as well as inhomogeneous chemical evolution models which properly take into account our model , especially the @xmath157-[fe / h ] relation ( see , e.g. , @xcite ) . we also investigate the yields of the models with the @xmath8 and `` low - density '' modifications suggested in @xcite and show that the yields are in better agreement with the observed abundance patterns than those without such modifications . the good agreements suggest that such modifications might be realized in the actual core - collapse sn explosions . we suggest that the @xmath8 and `` low - density '' modifications are actualized by the neutrino effects and the jet - like or delayed explosions , respectively . the neutrino - driven wind and the accretion disk are other possible nucleosynthesis sites for sc because they can actualize the @xmath8 ( @xmath121 ) and `` low - density '' modifications . however , nucleosynthesis in the neutrino - driven wind and the accretion disk might not be dominant sites of fe synthesis because they can not reproduce the abundance ratios among the fe - peak elements of the emp stars . we thus suggest that the progenitor s presupernova nucleosynthesis and explosive nucleosynthesis are predominant synthesis sites of most elements , while some elements ( e.g. , sc , ti , mn , and co ) can be enhanced by the @xmath8 and `` low - density '' modifications . we would like to thank c. kobayashi for valuable discussion . the author n.t . is supported through the jsps ( japan society for the promotion of science ) research fellowship for young scientists . this work has been supported in part by the grant - in - aid for scientific research ( 17030005 , 17033002 , 18104003 , 18540231 ) and the 21st century coe program ( quest ) from the jsps and mext of japan . the mixing - fallback model proposed by @xcite and @xcite can successfully reproduce the abundance patterns of the hyper metal - poor ( he01075240 : @xcite , he13272326 : @xcite ) and emp @xcite stars . the mixing - fallback model assumes the following situation : first , inner materials are mixed by some mixing process ( e.g. , rayleigh - taylor instabilities and/or aspherical explosions ) during the shockwave propagations in the star ( e.g. , @xcite ) . later , some fraction of materials in the mixing region undergoes fallback onto the central remnant by gravity ( e.g. , @xcite ) , and the rests are ejected into interstellar space . the fallback mass depends on the explosion energy , the gravitational potential , and asphericity . the mixing - fallback model can solve a problem associated with the ratios between the fe - peak elements and fe in the emp stars , [ ( fe - peak)/fe ] . the large zn / fe ratio implies an energetic explosion as a hn and a deep mass cut , but these lead to eject too large amount of fe to reproduce small [ fe / h ] and the large enough [ @xmath21/fe ] ratio , if ones assume that the whole material above the mass cut are ejected . however , one can realize both the deep mass cut and the small amount of ejected fe with the mixing - fallback model . in the spherical models , @xcite found that the fallback takes place if the explosion energy is less than @xmath158 for the 25 @xmath1 star and obtained a relation between @xmath159 and the final remnant mass , @xmath48 , i.e. , smaller @xmath159 leads to a larger @xmath48 . for example , @xmath160 and @xmath161 for the 25 @xmath1 star lead to the final remnant mass @xmath162 and @xmath163 , respectively . it is difficult for a spherical hn explosion to initiate a fallback , although for a larger star the fallback can occur even for a larger explosion energy because of a deeper gravitational potential . for instance , the fallback is found to occur for @xmath164 in the 50 @xmath1 star . however , fallback can take place not only for relatively low energy explosions but for very energetic jet - like explosions @xcite . in fact , @xcite have simulated jet - induced explosions and showed that the resultant yields can reproduce the abundance patterns of the emp stars as the mixing - fallback model . the mixing - fallback model mimics such aspherical explosions , although the spherical model tends to require larger explosion energies than the jet model to obtain similar yields @xcite . * @xmath45 : initial mass cut , which is corresponding to the inner boundary of the mixing region . * @xmath46 : outer boundary of the mixing region . * @xmath47 : a fraction of matter ejected from the mixed region . it determines [ @xmath21/fe ] . figures [ fig : mf]ab illustrate these parameters for spherical and aspherical models . the final remnant mass , @xmath48 , is determined by the above three parameters @xmath165 in this paper , we determine these parameters as follows . * @xmath45 : the initial mass cut is adopted so that [ zn / fe ] attains maximum , thus locates at the bottom of the @xmath166 layer ( fig . [ fig:25e20ad ] ) where is very close to the surface of fe core of the progenitors . figure [ fig:25e20ad ] shows the abundance distribution of the 25 @xmath1 star with @xmath140 around the complete si burning region . for population iii sne , the dominant isotope of zn is @xmath44zn , which is the decay product of @xmath44ge . @xmath44zn is mostly produced in the complete si burning region where @xmath167 and the zn / fe ratio decreases for lower @xmath8 ( see fig . 4 in @xcite ) . therefore , as the mass cut decreases , [ zn / fe ] in the ejecta first increases and then decreases . this choice of @xmath45 tends to give a smaller estimate of @xmath159 to fit to the observed zn / fe , because zn / fe is larger for larger @xmath159 . * @xmath46 and @xmath47 : we study two case as follows . case b is an additional model for massive star models . the abundance distribution of the 30 @xmath1 , @xmath168 model is shown in figure [ fig:30e20ad ] illustrating the mixing region in both cases . + case `` a '' : : : we assume that the mixing occurs in the si burning region and thus @xmath46 is where @xmath169 . @xmath47 is chosen to yield [ o / fe ] @xmath51 . for most case , @xmath170 . case `` b '' : : : for the @xmath171 models , we consider another case . in this case , we assume that @xmath46 is 2/3 of the o - rich layer , and determine @xmath47 so that [ mg / fe ] @xmath52 to be consistent with the [ fe / h ] @xmath55 stars in @xcite . in this paper , we consider two cases a and b for @xmath172 stars . [ mg / fe ] in case b is smaller than in case a but the ejected fe mass in case b is smaller than in case a ( tables [ tab : models ] , [ tab : modelsye ] , and [ tab : modelsyelow ] ) . this is because case b has larger amount of fallback , i.e. , larger @xmath46 and smaller @xmath47 , than case a. such large fallback which reaches the mg - rich layer decreases the amount of ejected mg and leads to a smaller mg / fe for case b than case a in spite of a smaller amount of ejected fe . the observed hne have variations of the ejected fe mass . case a corresponds to typical hne like sn 1998bw that ejected @xmath173 @xcite and case b corresponds to hne like sn 1997ef that ejected @xmath174 @xcite . alternative interpretation of [ mg / fe ] @xmath54 is to eject a larger amount of fe than case a for the same amount of ejected mg . in this case , because of larger fe , larger explosion energies ( @xmath175 ) are needed for this model to reproduce the small [ fe / h ] ( @xmath176 ) , as long as [ fe / h ] is determined by equation ( [ eq : snsf ] ) ( i.e. , the same constant @xmath64 ) of the sn - induced star formation model . the explosion energies are considerably larger than those of pop i sne . we thus do not consider such a possibility in this paper . cccccccccccc 13 & 1 & 0.07 & 0.09 & @xmath1772.55 & @xmath1770.08 & 0.37 & 1.57 & & & 1 & fig . [ fig : vmp]a + 15 & 1 & 0.07 & 0.07 & @xmath1772.55 & 0.15 & 0.27 & 1.48 & & & 1 & fig . [ fig : vmp]b + 18 & 1 & 0.07 & 0.16 & @xmath1772.55 & 0.41 & 0.63 & 1.65 & & & 1 & fig . [ fig : vmp]c + 20 & 10 & 0.08 & 0.17 & @xmath1773.34 & 0.50 & 0.58 & 1.90 & 1.52 & 2.05 & 0.28 & fig . [ fig : emp]a + 25 & 5 & 0.11 & 0.14 & @xmath1772.97 & 0.50 & 0.41 & 2.45 & 1.79 & 2.69 & 0.26 & fig . [ fig:25e5 ] + 25 & 10 & 0.10 & 0.15 & @xmath1773.27 & 0.50 & 0.48 & 2.86 & 1.79 & 3.07 & 0.16 & fig . [ fig : emp]b + 25f & 1 & 0.0008 & 0.03 & @xmath1774.50 & 2.40 & 1.81 & 4.20 & 1.79 & 4.21 & 0.004 & fig . [ fig : faint ] + 30a & 20 & 0.16 & 0.22 & @xmath1773.31 & 0.50 & 0.41 & 3.27 & 1.65 & 3.59 & 0.17 & fig . [ fig : emp]c + 30b & 20 & 0.05 & 0.04 & @xmath1773.81 & 0.55 & 0.20 & 6.73 & 1.65 & 7.01 & 0.05 & fig . [ fig : emp]c + 40a & 30 & 0.26 & 0.34 & @xmath1773.25 & 0.50 & 0.39 & 5.53 & 2.24 & 6.00 & 0.12 & fig . [ fig : emp]d + 40b & 30 & 0.11 & 0.09 & @xmath1773.63 & 0.46 & 0.20 & 10.70 & 2.24 & 11.16 & 0.05 & fig . [ fig : emp]d + 50a & 40 & 0.36 & 0.57 & @xmath1773.22 & 0.50 & 0.47 & 3.76 & 1.89 & 4.49 & 0.28 & fig . [ fig : emp]e + 50b & 40 & 0.23 & 0.20 & @xmath1773.40 & 0.35 & 0.20 & 11.01 & 1.89 & 13.05 & 0.18 & fig . [ fig : emp]e + crrrrrrrrrrrrr p & 6.59e+00 & 7.58e+00 & 8.43e+00 & 8.77e+00 & 1.06e+01 & 1.06e+01 & 1.06e+01 & 1.17e+01 & 1.17e+01 & 1.40e+01 & 1.40e+01 & 1.63e+01 & 1.63e+01 + d & 1.49e@xmath17716 & 1.69e@xmath17716 & 1.28e@xmath17716 & 8.66e@xmath17717 & 2.04e@xmath17716 & 2.06e@xmath17716 & 2.02e@xmath17716 & 1.09e@xmath17714 & 1.09e@xmath17714 & 1.66e@xmath17714 & 1.66e@xmath17714 & 1.24e@xmath17715 & 1.24e@xmath17715 + @xmath178he & 4.12e@xmath17705 & 4.09e@xmath17705 & 3.33e@xmath17705 & 4.76e@xmath17705 & 2.11e@xmath17704 & 2.11e@xmath17704 & 2.11e@xmath17704 & 2.06e@xmath17704 & 2.06e@xmath17704 & 2.56e@xmath17705 & 2.56e@xmath17705 & 2.86e@xmath17705 & 2.86e@xmath17705 + @xmath179he & 4.01e+00 & 4.40e+00 & 5.42e+00 & 5.96e+00 & 8.03e+00 & 8.02e+00 & 8.03e+00 & 9.54e+00 & 9.51e+00 & 1.18e+01 & 1.18e+01 & 1.56e+01 & 1.55e+01 + @xmath180li & 3.29e@xmath17723 & 1.28e@xmath17722 & 4.52e@xmath17723 & 1.36e@xmath17722 & 2.28e@xmath17720 & 4.26e@xmath17720 & 2.68e@xmath17721 & 3.50e@xmath17717 & 3.50e@xmath17717 & 5.39e@xmath17717 & 5.39e@xmath17717 & 2.02e@xmath17718 & 2.02e@xmath17718 + @xmath181li & 2.17e@xmath17710 & 2.94e@xmath17710 & 7.34e@xmath17711 & 2.79e@xmath17710 & 5.68e@xmath17709 & 5.68e@xmath17709 & 5.68e@xmath17709 & 2.36e@xmath17708 & 2.36e@xmath17708 & 3.42e@xmath17711 & 3.42e@xmath17711 & 8.78e@xmath17712 & 8.78e@xmath17712 + @xmath182be & 1.77e@xmath17720 & 3.22e@xmath17722 & 1.05e@xmath17722 & 4.83e@xmath17720 & 5.00e@xmath17717 & 3.69e@xmath17717 & 1.24e@xmath17717 & 3.09e@xmath17718 & 3.09e@xmath17718 & 9.03e@xmath17718 & 9.03e@xmath17718 & 6.04e@xmath17717 & 6.04e@xmath17717 + @xmath183b & 3.16e@xmath17721 & 3.92e@xmath17720 & 2.86e@xmath17720 & 2.88e@xmath17719 & 2.79e@xmath17715 & 2.05e@xmath17713 & 2.67e@xmath17718 & 1.05e@xmath17714 & 1.05e@xmath17714 & 9.41e@xmath17715 & 9.41e@xmath17715 & 1.72e@xmath17717 & 7.85e@xmath17718 + @xmath184b & 2.89e@xmath17716 & 3.55e@xmath17716 & 1.59e@xmath17714 & 1.19e@xmath17715 & 6.09e@xmath17714 & 5.27e@xmath17713 & 1.90e@xmath17716 & 9.61e@xmath17714 & 9.16e@xmath17714 & 9.41e@xmath17713 & 9.40e@xmath17713 & 2.49e@xmath17711 & 1.62e@xmath17711 + @xmath88c & 7.41e@xmath17702 & 1.72e@xmath17701 & 2.18e@xmath17701 & 1.90e@xmath17701 & 2.79e@xmath17701 & 2.67e@xmath17701 & 2.74e@xmath17701 & 3.16e@xmath17701 & 2.86e@xmath17701 & 3.72e@xmath17701 & 3.51e@xmath17701 & 1.56e+00 & 1.17e+00 + @xmath89c & 8.39e@xmath17708 & 6.21e@xmath17708 & 2.63e@xmath17709 & 1.18e@xmath17708 & 2.90e@xmath17708 & 6.94e@xmath17708 & 1.05e@xmath17708 & 6.32e@xmath17708 & 6.03e@xmath17708 & 8.18e@xmath17708 & 7.84e@xmath17708 & 2.38e@xmath17707 & 1.70e@xmath17707 + @xmath185n & 1.83e@xmath17703 & 1.86e@xmath17703 & 1.89e@xmath17704 & 5.42e@xmath17705 & 5.92e@xmath17704 & 5.96e@xmath17704 & 5.91e@xmath17704 & 4.18e@xmath17705 & 4.17e@xmath17705 & 3.39e@xmath17706 & 3.33e@xmath17706 & 4.38e@xmath17704 & 4.37e@xmath17704 + @xmath186n & 6.38e@xmath17708 & 6.86e@xmath17708 & 2.40e@xmath17708 & 2.96e@xmath17708 & 1.64e@xmath17707 & 1.75e@xmath17707 & 7.17e@xmath17708 & 2.20e@xmath17707 & 1.16e@xmath17707 & 6.54e@xmath17707 & 6.51e@xmath17707 & 3.33e@xmath17707 & 8.27e@xmath17708 + @xmath136o & 4.50e@xmath17701 & 7.73e@xmath17701 & 1.38e+00 & 2.03e+00 & 2.60e+00 & 2.37e+00 & 1.52e+00 & 3.92e+00 & 1.37e+00 & 6.32e+00 & 2.40e+00 & 8.81e+00 & 4.02e+00 + @xmath187o & 1.69e@xmath17706 & 1.57e@xmath17706 & 2.79e@xmath17707 & 7.13e@xmath17708 & 1.49e@xmath17706 & 1.49e@xmath17706 & 1.48e@xmath17706 & 3.81e@xmath17708 & 2.64e@xmath17708 & 1.23e@xmath17708 & 8.65e@xmath17709 & 8.60e@xmath17706 & 8.56e@xmath17706 + @xmath188o & 5.79e@xmath17708 & 4.89e@xmath17706 & 4.63e@xmath17706 & 2.33e@xmath17708 & 4.66e@xmath17707 & 3.87e@xmath17707 & 6.73e@xmath17707 & 5.03e@xmath17707 & 5.02e@xmath17707 & 2.93e@xmath17707 & 2.92e@xmath17707 & 6.00e@xmath17706 & 5.99e@xmath17706 + @xmath189f & 1.17e@xmath17710 & 1.97e@xmath17709 & 7.91e@xmath17709 & 2.12e@xmath17709 & 1.43e@xmath17709 & 1.67e@xmath17709 & 1.01e@xmath17709 & 7.88e@xmath17709 & 6.41e@xmath17709 & 1.17e@xmath17707 & 1.16e@xmath17707 & 8.49e@xmath17709 & 4.06e@xmath17709 + @xmath190ne & 1.53e@xmath17702 & 3.27e@xmath17701 & 4.94e@xmath17701 & 7.49e@xmath17701 & 3.91e@xmath17701 & 2.85e@xmath17701 & 3.16e@xmath17701 & 5.20e@xmath17701 & 2.60e@xmath17701 & 2.64e@xmath17701 & 2.19e@xmath17701 & 2.48e+00 & 1.30e+00 + @xmath191ne & 5.42e@xmath17707 & 3.76e@xmath17705 & 9.12e@xmath17705 & 3.58e@xmath17705 & 1.47e@xmath17705 & 1.22e@xmath17705 & 3.05e@xmath17706 & 3.51e@xmath17705 & 2.26e@xmath17705 & 1.41e@xmath17705 & 1.07e@xmath17705 & 2.40e@xmath17704 & 1.14e@xmath17704 + @xmath192ne & 1.98e@xmath17707 & 1.61e@xmath17705 & 2.57e@xmath17705 & 5.51e@xmath17705 & 1.30e@xmath17705 & 8.62e@xmath17706 & 1.27e@xmath17705 & 3.52e@xmath17705 & 1.98e@xmath17705 & 1.66e@xmath17705 & 1.45e@xmath17705 & 2.10e@xmath17704 & 1.08e@xmath17704 + @xmath193na & 1.44e@xmath17704 & 2.45e@xmath17703 & 2.08e@xmath17703 & 2.31e@xmath17703 & 6.74e@xmath17704 & 4.42e@xmath17704 & 6.61e@xmath17704 & 7.36e@xmath17704 & 3.24e@xmath17704 & 3.28e@xmath17704 & 1.66e@xmath17704 & 4.27e@xmath17703 & 1.99e@xmath17703 + @xmath194 mg & 8.62e@xmath17702 & 6.82e@xmath17702 & 1.57e@xmath17701 & 1.65e@xmath17701 & 1.43e@xmath17701 & 1.53e@xmath17701 & 2.70e@xmath17702 & 2.17e@xmath17701 & 4.20e@xmath17702 & 3.37e@xmath17701 & 8.99e@xmath17702 & 5.70e@xmath17701 & 1.97e@xmath17701 + @xmath195 mg & 1.56e@xmath17704 & 2.98e@xmath17704 & 5.83e@xmath17704 & 1.07e@xmath17704 & 4.52e@xmath17705 & 4.57e@xmath17705 & 1.29e@xmath17705 & 1.45e@xmath17704 & 9.00e@xmath17705 & 5.95e@xmath17704 & 4.11e@xmath17704 & 3.10e@xmath17704 & 1.16e@xmath17704 + @xmath196 mg & 7.07e@xmath17705 & 3.98e@xmath17704 & 8.73e@xmath17704 & 2.09e@xmath17704 & 4.31e@xmath17705 & 3.89e@xmath17705 & 1.48e@xmath17705 & 8.00e@xmath17705 & 2.62e@xmath17705 & 6.90e@xmath17705 & 1.22e@xmath17705 & 2.32e@xmath17704 & 9.70e@xmath17705 + @xmath196al & 9.92e@xmath17707 & 1.11e@xmath17706 & 3.33e@xmath17706 & 1.02e@xmath17706 & 1.21e@xmath17706 & 1.28e@xmath17706 & 5.34e@xmath17709 & 2.92e@xmath17706 & 1.33e@xmath17706 & 3.80e@xmath17705 & 3.43e@xmath17705 & 5.46e@xmath17706 & 1.40e@xmath17706 + @xmath197al & 3.78e@xmath17703 & 1.37e@xmath17703 & 3.14e@xmath17703 & 1.50e@xmath17703 & 8.59e@xmath17704 & 8.93e@xmath17704 & 9.78e@xmath17705 & 1.55e@xmath17703 & 1.59e@xmath17704 & 7.52e@xmath17703 & 1.34e@xmath17703 & 3.29e@xmath17703 & 8.07e@xmath17704 + @xmath198si & 8.08e@xmath17702 & 7.18e@xmath17702 & 1.07e@xmath17701 & 9.15e@xmath17702 & 2.21e@xmath17701 & 1.97e@xmath17701 & 2.07e@xmath17703 & 2.44e@xmath17701 & 3.11e@xmath17702 & 7.17e@xmath17701 & 2.28e@xmath17701 & 3.61e@xmath17701 & 1.18e@xmath17701 + @xmath199si & 7.50e@xmath17704 & 2.38e@xmath17704 & 4.36e@xmath17704 & 2.89e@xmath17704 & 5.12e@xmath17704 & 5.23e@xmath17704 & 3.96e@xmath17706 & 8.85e@xmath17704 & 4.89e@xmath17705 & 3.72e@xmath17703 & 1.29e@xmath17703 & 9.32e@xmath17704 & 1.80e@xmath17704 + @xmath200si & 1.42e@xmath17703 & 1.49e@xmath17704 & 3.50e@xmath17704 & 1.12e@xmath17704 & 5.90e@xmath17705 & 6.13e@xmath17705 & 2.66e@xmath17706 & 1.47e@xmath17704 & 9.97e@xmath17706 & 2.82e@xmath17703 & 4.83e@xmath17704 & 2.10e@xmath17704 & 5.04e@xmath17705 + @xmath201p & 4.88e@xmath17704 & 5.66e@xmath17705 & 1.31e@xmath17704 & 7.30e@xmath17705 & 5.44e@xmath17705 & 5.03e@xmath17705 & 1.68e@xmath17706 & 1.18e@xmath17704 & 7.87e@xmath17706 & 1.01e@xmath17703 & 2.24e@xmath17704 & 1.75e@xmath17704 & 4.26e@xmath17705 + @xmath202s & 2.36e@xmath17702 & 3.24e@xmath17702 & 4.45e@xmath17702 & 3.71e@xmath17702 & 1.00e@xmath17701 & 7.52e@xmath17702 & 8.10e@xmath17704 & 8.49e@xmath17702 & 1.49e@xmath17702 & 2.60e@xmath17701 & 6.11e@xmath17702 & 1.41e@xmath17701 & 6.23e@xmath17702 + @xmath203s & 9.00e@xmath17705 & 7.48e@xmath17705 & 9.49e@xmath17705 & 1.31e@xmath17704 & 2.06e@xmath17704 & 1.99e@xmath17704 & 9.32e@xmath17707 & 3.02e@xmath17704 & 1.59e@xmath17705 & 8.44e@xmath17704 & 1.29e@xmath17704 & 3.89e@xmath17704 & 7.15e@xmath17705 + @xmath204s & 2.79e@xmath17704 & 2.01e@xmath17704 & 2.59e@xmath17704 & 1.46e@xmath17704 & 5.13e@xmath17705 & 2.47e@xmath17705 & 5.83e@xmath17707 & 2.70e@xmath17704 & 1.78e@xmath17705 & 2.08e@xmath17703 & 1.32e@xmath17704 & 6.60e@xmath17705 & 2.92e@xmath17705 + @xmath205s & 1.48e@xmath17708 & 1.43e@xmath17709 & 5.34e@xmath17709 & 8.33e@xmath17710 & 6.69e@xmath17711 & 5.25e@xmath17711 & 1.35e@xmath17712 & 1.41e@xmath17709 & 7.70e@xmath17711 & 5.04e@xmath17708 & 2.63e@xmath17709 & 4.52e@xmath17711 & 8.39e@xmath17712 + @xmath206cl & 5.48e@xmath17705 & 1.44e@xmath17705 & 2.42e@xmath17705 & 3.28e@xmath17705 & 2.01e@xmath17705 & 1.68e@xmath17705 & 1.83e@xmath17707 & 4.51e@xmath17705 & 6.11e@xmath17706 & 1.80e@xmath17704 & 2.46e@xmath17705 & 6.89e@xmath17705 & 3.30e@xmath17705 + @xmath207cl & 3.11e@xmath17706 & 5.62e@xmath17706 & 7.80e@xmath17706 & 1.52e@xmath17705 & 2.26e@xmath17705 & 2.02e@xmath17705 & 2.21e@xmath17707 & 2.37e@xmath17705 & 1.29e@xmath17706 & 7.92e@xmath17705 & 4.28e@xmath17706 & 3.29e@xmath17705 & 6.06e@xmath17706 + @xmath205ar & 3.18e@xmath17703 & 5.53e@xmath17703 & 7.26e@xmath17703 & 6.04e@xmath17703 & 1.67e@xmath17702 & 1.15e@xmath17702 & 1.54e@xmath17704 & 1.18e@xmath17702 & 2.71e@xmath17703 & 3.62e@xmath17702 & 6.08e@xmath17703 & 2.29e@xmath17702 & 1.23e@xmath17702 + @xmath208ar & 5.23e@xmath17705 & 6.18e@xmath17705 & 1.45e@xmath17704 & 6.35e@xmath17705 & 4.55e@xmath17705 & 8.35e@xmath17706 & 6.11e@xmath17707 & 9.06e@xmath17705 & 5.58e@xmath17706 & 7.83e@xmath17704 & 4.15e@xmath17705 & 1.81e@xmath17705 & 7.26e@xmath17706 + @xmath209ar & 8.01e@xmath17711 & 1.78e@xmath17711 & 3.95e@xmath17711 & 3.77e@xmath17711 & 5.17e@xmath17712 & 3.77e@xmath17712 & 1.40e@xmath17713 & 1.87e@xmath17711 & 2.69e@xmath17712 & 2.68e@xmath17710 & 1.99e@xmath17711 & 1.38e@xmath17712 & 3.41e@xmath17713 + @xmath210k & 5.14e@xmath17706 & 7.23e@xmath17706 & 1.59e@xmath17705 & 1.06e@xmath17705 & 1.61e@xmath17705 & 8.69e@xmath17706 & 2.08e@xmath17707 & 2.05e@xmath17705 & 2.88e@xmath17706 & 1.31e@xmath17704 & 1.07e@xmath17705 & 2.52e@xmath17705 & 1.47e@xmath17705 + @xmath209k & 1.14e@xmath17709 & 9.07e@xmath17710 & 1.84e@xmath17709 & 3.95e@xmath17709 & 1.82e@xmath17709 & 4.46e@xmath17710 & 2.89e@xmath17711 & 2.52e@xmath17709 & 1.32e@xmath17710 & 1.18e@xmath17708 & 6.14e@xmath17710 & 7.09e@xmath17710 & 1.30e@xmath17710 + @xmath211k & 3.72e@xmath17707 & 7.43e@xmath17707 & 1.51e@xmath17706 & 2.63e@xmath17706 & 5.94e@xmath17706 & 4.00e@xmath17706 & 7.98e@xmath17708 & 3.63e@xmath17706 & 2.04e@xmath17707 & 2.12e@xmath17705 & 1.14e@xmath17706 & 1.72e@xmath17706 & 4.78e@xmath17707 + @xmath209ca & 2.82e@xmath17703 & 4.72e@xmath17703 & 6.14e@xmath17703 & 4.39e@xmath17703 & 1.43e@xmath17702 & 9.20e@xmath17703 & 1.36e@xmath17704 & 8.77e@xmath17703 & 2.45e@xmath17703 & 2.97e@xmath17702 & 5.60e@xmath17703 & 1.87e@xmath17702 & 1.20e@xmath17702 + @xmath212ca & 9.80e@xmath17707 & 1.21e@xmath17706 & 3.15e@xmath17706 & 1.34e@xmath17706 & 1.11e@xmath17706 & 1.76e@xmath17707 & 1.65e@xmath17708 & 1.62e@xmath17706 & 8.48e@xmath17708 & 1.97e@xmath17705 & 1.02e@xmath17706 & 2.29e@xmath17707 & 7.46e@xmath17708 + @xmath213ca & 6.66e@xmath17708 & 4.76e@xmath17708 & 3.48e@xmath17708 & 2.59e@xmath17707 & 4.68e@xmath17708 & 7.12e@xmath17708 & 3.33e@xmath17710 & 1.71e@xmath17707 & 5.33e@xmath17708 & 1.40e@xmath17707 & 5.32e@xmath17708 & 7.31e@xmath17707 & 4.75e@xmath17707 + @xmath42ca & 1.70e@xmath17705 & 2.17e@xmath17705 & 1.40e@xmath17705 & 1.25e@xmath17704 & 4.26e@xmath17705 & 6.82e@xmath17705 & 1.82e@xmath17707 & 1.80e@xmath17704 & 5.67e@xmath17705 & 1.76e@xmath17704 & 7.16e@xmath17705 & 6.08e@xmath17704 & 3.95e@xmath17704 + @xmath214ca & 1.07e@xmath17712 & 1.75e@xmath17712 & 9.26e@xmath17712 & 1.15e@xmath17711 & 2.61e@xmath17712 & 1.80e@xmath17711 & 1.18e@xmath17713 & 9.25e@xmath17712 & 7.64e@xmath17712 & 3.69e@xmath17711 & 2.45e@xmath17711 & 2.66e@xmath17713 & 6.72e@xmath17714 + @xmath43ca & 1.99e@xmath17717 & 4.21e@xmath17714 & 4.21e@xmath17716 & 5.87e@xmath17716 & 8.89e@xmath17712 & 8.91e@xmath17712 & 9.58e@xmath17712 & 8.16e@xmath17713 & 8.15e@xmath17713 & 1.27e@xmath17711 & 1.27e@xmath17711 & 1.56e@xmath17713 & 5.63e@xmath17714 + @xmath215sc & 2.16e@xmath17708 & 3.53e@xmath17708 & 4.82e@xmath17708 & 1.44e@xmath17707 & 1.77e@xmath17707 & 1.24e@xmath17707 & 2.44e@xmath17709 & 6.09e@xmath17708 & 1.06e@xmath17708 & 6.15e@xmath17707 & 4.34e@xmath17708 & 6.31e@xmath17708 & 3.69e@xmath17708 + @xmath214ti & 6.21e@xmath17706 & 2.56e@xmath17706 & 3.41e@xmath17706 & 5.59e@xmath17706 & 2.21e@xmath17706 & 2.88e@xmath17706 & 8.20e@xmath17709 & 6.13e@xmath17706 & 1.80e@xmath17706 & 1.11e@xmath17705 & 1.99e@xmath17706 & 1.40e@xmath17705 & 9.07e@xmath17706 + @xmath216ti & 8.72e@xmath17706 & 3.68e@xmath17706 & 4.68e@xmath17706 & 9.77e@xmath17706 & 8.14e@xmath17706 & 9.89e@xmath17706 & 2.42e@xmath17708 & 2.06e@xmath17705 & 6.47e@xmath17706 & 2.59e@xmath17705 & 1.08e@xmath17705 & 5.85e@xmath17705 & 3.80e@xmath17705 + @xmath43ti & 6.32e@xmath17705 & 8.94e@xmath17705 & 9.35e@xmath17705 & 1.59e@xmath17704 & 1.37e@xmath17704 & 1.41e@xmath17704 & 1.09e@xmath17706 & 2.88e@xmath17704 & 9.07e@xmath17705 & 3.53e@xmath17704 & 1.47e@xmath17704 & 7.84e@xmath17704 & 5.09e@xmath17704 + @xmath217ti & 2.24e@xmath17706 & 2.79e@xmath17706 & 9.83e@xmath17707 & 2.65e@xmath17706 & 1.89e@xmath17706 & 2.22e@xmath17706 & 1.02e@xmath17708 & 3.44e@xmath17706 & 1.08e@xmath17706 & 5.00e@xmath17706 & 1.89e@xmath17706 & 3.93e@xmath17706 & 2.55e@xmath17706 + @xmath218ti & 1.18e@xmath17712 & 9.24e@xmath17713 & 1.62e@xmath17712 & 8.98e@xmath17713 & 1.92e@xmath17712 & 1.96e@xmath17712 & 5.20e@xmath17713 & 2.49e@xmath17712 & 2.09e@xmath17712 & 3.37e@xmath17711 & 2.94e@xmath17711 & 1.45e@xmath17712 & 5.46e@xmath17713 + @xmath218v & 1.36e@xmath17711 & 1.04e@xmath17711 & 3.08e@xmath17711 & 1.44e@xmath17711 & 7.71e@xmath17712 & 2.13e@xmath17712 & 2.17e@xmath17713 & 9.03e@xmath17712 & 7.15e@xmath17713 & 1.20e@xmath17710 & 2.76e@xmath17711 & 1.14e@xmath17712 & 2.83e@xmath17713 + @xmath219v & 1.63e@xmath17705 & 8.53e@xmath17706 & 8.97e@xmath17706 & 2.36e@xmath17705 & 1.01e@xmath17705 & 1.59e@xmath17705 & 1.74e@xmath17708 & 3.24e@xmath17705 & 1.02e@xmath17705 & 3.06e@xmath17705 & 1.26e@xmath17705 & 7.13e@xmath17705 & 4.63e@xmath17705 + @xmath218cr & 1.02e@xmath17705 & 7.43e@xmath17706 & 6.90e@xmath17706 & 8.43e@xmath17706 & 2.89e@xmath17706 & 2.79e@xmath17706 & 1.70e@xmath17708 & 5.18e@xmath17706 & 1.40e@xmath17706 & 2.57e@xmath17705 & 3.29e@xmath17706 & 9.36e@xmath17706 & 6.06e@xmath17706 + @xmath220cr & 8.66e@xmath17704 & 1.17e@xmath17703 & 1.35e@xmath17703 & 6.79e@xmath17704 & 1.55e@xmath17703 & 1.30e@xmath17703 & 1.47e@xmath17705 & 1.61e@xmath17703 & 5.06e@xmath17704 & 2.82e@xmath17703 & 1.18e@xmath17703 & 3.13e@xmath17703 & 2.03e@xmath17703 + @xmath221cr & 4.97e@xmath17705 & 5.66e@xmath17705 & 1.87e@xmath17705 & 3.03e@xmath17705 & 3.18e@xmath17705 & 4.17e@xmath17705 & 1.60e@xmath17707 & 5.58e@xmath17705 & 1.76e@xmath17705 & 9.62e@xmath17705 & 3.94e@xmath17705 & 4.87e@xmath17705 & 3.16e@xmath17705 + @xmath222cr & 2.38e@xmath17710 & 3.84e@xmath17710 & 7.70e@xmath17710 & 1.35e@xmath17710 & 2.15e@xmath17710 & 3.78e@xmath17711 & 1.45e@xmath17712 & 9.63e@xmath17711 & 1.16e@xmath17711 & 3.51e@xmath17709 & 2.09e@xmath17710 & 6.61e@xmath17712 & 1.72e@xmath17712 + @xmath223mn & 1.39e@xmath17704 & 9.01e@xmath17705 & 3.74e@xmath17705 & 4.77e@xmath17705 & 3.70e@xmath17705 & 3.05e@xmath17705 & 1.44e@xmath17707 & 2.77e@xmath17705 & 8.68e@xmath17706 & 1.28e@xmath17704 & 4.80e@xmath17705 & 2.87e@xmath17705 & 1.86e@xmath17705 + @xmath222fe & 7.23e@xmath17704 & 5.00e@xmath17704 & 2.70e@xmath17704 & 1.85e@xmath17704 & 2.14e@xmath17704 & 8.50e@xmath17705 & 4.48e@xmath17707 & 3.94e@xmath17705 & 1.17e@xmath17705 & 1.31e@xmath17703 & 2.42e@xmath17704 & 2.51e@xmath17705 & 1.63e@xmath17705 + @xmath25fe & 7.00e@xmath17702 & 7.00e@xmath17702 & 7.00e@xmath17702 & 8.30e@xmath17702 & 1.07e@xmath17701 & 9.70e@xmath17702 & 7.93e@xmath17704 & 1.60e@xmath17701 & 5.04e@xmath17702 & 2.58e@xmath17701 & 1.08e@xmath17701 & 3.61e@xmath17701 & 2.34e@xmath17701 + @xmath224fe & 1.01e@xmath17703 & 1.08e@xmath17703 & 7.98e@xmath17704 & 1.77e@xmath17703 & 1.54e@xmath17703 & 1.55e@xmath17703 & 9.27e@xmath17706 & 3.03e@xmath17703 & 9.53e@xmath17704 & 4.17e@xmath17703 & 1.74e@xmath17703 & 7.35e@xmath17703 & 4.77e@xmath17703 + @xmath128fe & 5.96e@xmath17711 & 1.56e@xmath17710 & 2.76e@xmath17710 & 1.32e@xmath17710 & 5.82e@xmath17710 & 1.21e@xmath17710 & 3.68e@xmath17712 & 9.17e@xmath17711 & 6.18e@xmath17712 & 3.30e@xmath17709 & 2.33e@xmath17710 & 1.53e@xmath17711 & 4.52e@xmath17712 + @xmath225co & 1.75e@xmath17704 & 1.30e@xmath17704 & 1.49e@xmath17704 & 3.72e@xmath17704 & 1.96e@xmath17704 & 2.42e@xmath17704 & 4.36e@xmath17707 & 5.14e@xmath17704 & 1.62e@xmath17704 & 5.19e@xmath17704 & 2.16e@xmath17704 & 1.34e@xmath17703 & 8.70e@xmath17704 + @xmath128ni & 3.79e@xmath17704 & 3.55e@xmath17704 & 2.92e@xmath17704 & 8.45e@xmath17704 & 4.56e@xmath17704 & 5.44e@xmath17704 & 1.30e@xmath17706 & 1.15e@xmath17703 & 3.62e@xmath17704 & 1.28e@xmath17703 & 4.84e@xmath17704 & 2.76e@xmath17703 & 1.79e@xmath17703 + @xmath226ni & 2.14e@xmath17703 & 1.56e@xmath17703 & 1.44e@xmath17703 & 2.99e@xmath17703 & 2.74e@xmath17703 & 2.78e@xmath17703 & 1.89e@xmath17705 & 5.43e@xmath17703 & 1.71e@xmath17703 & 8.37e@xmath17703 & 3.49e@xmath17703 & 1.31e@xmath17702 & 8.49e@xmath17703 + @xmath227ni & 3.68e@xmath17705 & 3.03e@xmath17705 & 1.95e@xmath17705 & 6.27e@xmath17705 & 3.66e@xmath17705 & 4.19e@xmath17705 & 2.33e@xmath17707 & 8.54e@xmath17705 & 2.69e@xmath17705 & 9.87e@xmath17705 & 4.12e@xmath17705 & 2.18e@xmath17704 & 1.42e@xmath17704 + @xmath228ni & 1.92e@xmath17705 & 1.47e@xmath17705 & 1.23e@xmath17705 & 4.24e@xmath17705 & 2.25e@xmath17705 & 2.86e@xmath17705 & 2.00e@xmath17707 & 5.68e@xmath17705 & 1.79e@xmath17705 & 6.74e@xmath17705 & 2.81e@xmath17705 & 1.43e@xmath17704 & 9.31e@xmath17705 + @xmath44ni & 2.18e@xmath17715 & 2.67e@xmath17713 & 1.35e@xmath17714 & 8.70e@xmath17713 & 3.73e@xmath17712 & 9.60e@xmath17712 & 1.84e@xmath17712 & 3.38e@xmath17712 & 2.41e@xmath17712 & 2.46e@xmath17711 & 2.46e@xmath17711 & 6.72e@xmath17712 & 1.67e@xmath17712 + @xmath229cu & 4.86e@xmath17706 & 3.45e@xmath17706 & 3.58e@xmath17706 & 1.19e@xmath17705 & 6.41e@xmath17706 & 8.18e@xmath17706 & 2.02e@xmath17708 & 1.70e@xmath17705 & 5.34e@xmath17706 & 2.06e@xmath17705 & 8.60e@xmath17706 & 4.44e@xmath17705 & 2.88e@xmath17705 + @xmath230cu & 2.24e@xmath17707 & 2.33e@xmath17707 & 1.50e@xmath17707 & 7.21e@xmath17707 & 4.92e@xmath17707 & 5.77e@xmath17707 & 3.71e@xmath17709 & 1.37e@xmath17706 & 4.32e@xmath17707 & 1.86e@xmath17706 & 7.75e@xmath17707 & 4.37e@xmath17706 & 2.83e@xmath17706 + @xmath44zn & 1.26e@xmath17704 & 1.18e@xmath17704 & 8.72e@xmath17705 & 3.79e@xmath17704 & 2.04e@xmath17704 & 2.58e@xmath17704 & 9.63e@xmath17707 & 5.77e@xmath17704 & 1.82e@xmath17704 & 6.88e@xmath17704 & 2.87e@xmath17704 & 1.56e@xmath17703 & 1.01e@xmath17703 + @xmath231zn & 7.74e@xmath17707 & 1.03e@xmath17706 & 4.67e@xmath17707 & 5.05e@xmath17706 & 1.66e@xmath17706 & 2.44e@xmath17706 & 1.14e@xmath17708 & 6.98e@xmath17706 & 2.20e@xmath17706 & 6.90e@xmath17706 & 2.88e@xmath17706 & 2.29e@xmath17705 & 1.48e@xmath17705 + @xmath232zn & 1.73e@xmath17708 & 2.30e@xmath17708 & 1.20e@xmath17708 & 1.94e@xmath17707 & 2.66e@xmath17708 & 5.83e@xmath17708 & 3.99e@xmath17711 & 1.89e@xmath17707 & 5.95e@xmath17708 & 1.18e@xmath17707 & 4.93e@xmath17708 & 5.60e@xmath17707 & 3.63e@xmath17707 + @xmath233zn & 2.85e@xmath17708 & 3.16e@xmath17708 & 3.66e@xmath17708 & 8.93e@xmath17708 & 1.25e@xmath17707 & 1.21e@xmath17707 & 1.07e@xmath17709 & 2.67e@xmath17707 & 8.40e@xmath17708 & 4.86e@xmath17707 & 2.03e@xmath17707 & 8.45e@xmath17707 & 5.48e@xmath17707 + @xmath234zn & 3.04e@xmath17716 & 3.03e@xmath17714 & 7.45e@xmath17715 & 1.08e@xmath17713 & 2.04e@xmath17713 & 2.90e@xmath17712 & 1.82e@xmath17716 & 8.49e@xmath17713 & 6.00e@xmath17713 & 1.95e@xmath17711 & 1.95e@xmath17711 & 1.00e@xmath17711 & 5.36e@xmath17712 + @xmath235ga & 7.59e@xmath17709 & 5.50e@xmath17709 & 5.55e@xmath17709 & 1.77e@xmath17708 & 1.91e@xmath17708 & 2.12e@xmath17708 & 1.41e@xmath17710 & 4.04e@xmath17708 & 1.27e@xmath17708 & 8.95e@xmath17708 & 3.74e@xmath17708 & 1.08e@xmath17707 & 7.00e@xmath17708 + @xmath236ga & 1.00e@xmath17714 & 8.31e@xmath17714 & 3.35e@xmath17714 & 1.53e@xmath17712 & 1.60e@xmath17712 & 2.95e@xmath17711 & 1.05e@xmath17714 & 3.79e@xmath17712 & 2.38e@xmath17712 & 1.57e@xmath17710 & 1.57e@xmath17710 & 2.30e@xmath17711 & 1.06e@xmath17711 + @xmath234ge & 8.90e@xmath17709 & 6.27e@xmath17709 & 4.35e@xmath17709 & 2.41e@xmath17708 & 1.04e@xmath17708 & 1.41e@xmath17708 & 6.21e@xmath17711 & 2.92e@xmath17708 & 9.20e@xmath17709 & 3.75e@xmath17708 & 1.57e@xmath17708 & 8.09e@xmath17708 & 5.23e@xmath17708 + @xmath237ge & 7.42e@xmath17715 & 7.63e@xmath17713 & 5.90e@xmath17713 & 5.95e@xmath17712 & 2.83e@xmath17712 & 2.09e@xmath17711 & 3.85e@xmath17714 & 7.43e@xmath17712 & 3.96e@xmath17712 & 7.11e@xmath17711 & 7.10e@xmath17711 & 4.76e@xmath17711 & 1.59e@xmath17711 + @xmath238ge & 1.37e@xmath17714 & 1.84e@xmath17713 & 1.36e@xmath17713 & 3.73e@xmath17712 & 5.55e@xmath17712 & 3.95e@xmath17711 & 7.15e@xmath17715 & 6.59e@xmath17712 & 4.98e@xmath17712 & 9.15e@xmath17711 & 9.15e@xmath17711 & 3.12e@xmath17711 & 1.26e@xmath17711 + @xmath239ge & 3.77e@xmath17715 & 5.05e@xmath17714 & 5.80e@xmath17714 & 2.14e@xmath17713 & 1.16e@xmath17712 & 1.45e@xmath17711 & 9.96e@xmath17714 & 2.47e@xmath17712 & 1.68e@xmath17712 & 4.28e@xmath17711 & 4.28e@xmath17711 & 6.22e@xmath17712 & 1.98e@xmath17712 crrrrrrrrrrrrr @xmath192na & 1.51e@xmath17707 & 7.70e@xmath17706 & 1.68e@xmath17705 & 4.65e@xmath17705 & 9.97e@xmath17706 & 6.54e@xmath17706 & 8.98e@xmath17706 & 2.70e@xmath17705 & 1.34e@xmath17705 & 1.37e@xmath17705 & 1.19e@xmath17705 & 1.77e@xmath17704 & 9.12e@xmath17705 + @xmath196al & 9.80e@xmath17707 & 1.10e@xmath17706 & 3.32e@xmath17706 & 8.93e@xmath17707 & 1.21e@xmath17706 & 1.26e@xmath17706 & 5.34e@xmath17709 & 2.87e@xmath17706 & 1.32e@xmath17706 & 2.13e@xmath17705 & 1.76e@xmath17705 & 5.34e@xmath17706 & 1.32e@xmath17706 + @xmath211ca & 3.72e@xmath17707 & 7.42e@xmath17707 & 1.51e@xmath17706 & 2.63e@xmath17706 & 5.94e@xmath17706 & 4.00e@xmath17706 & 7.98e@xmath17708 & 3.63e@xmath17706 & 2.04e@xmath17707 & 2.12e@xmath17705 & 1.14e@xmath17706 & 1.72e@xmath17706 & 4.78e@xmath17707 + @xmath42ti & 1.70e@xmath17705 & 2.17e@xmath17705 & 1.40e@xmath17705 & 1.25e@xmath17704 & 4.26e@xmath17705 & 6.82e@xmath17705 & 1.82e@xmath17707 & 1.80e@xmath17704 & 5.67e@xmath17705 & 1.76e@xmath17704 & 7.16e@xmath17705 & 6.08e@xmath17704 & 3.95e@xmath17704 + @xmath226fe & 4.47e@xmath17716 & 2.44e@xmath17714 & 4.24e@xmath17715 & 1.10e@xmath17713 & 1.54e@xmath17713 & 1.10e@xmath17712 & 1.01e@xmath17715 & 2.10e@xmath17712 & 1.95e@xmath17712 & 2.44e@xmath17711 & 2.44e@xmath17711 & 1.11e@xmath17712 & 3.86e@xmath17713 + @xmath25ni & 7.00e@xmath17702 & 7.00e@xmath17702 & 7.00e@xmath17702 & 8.30e@xmath17702 & 1.07e@xmath17701 & 9.70e@xmath17702 & 7.93e@xmath17704 & 1.60e@xmath17701 & 5.04e@xmath17702 & 2.58e@xmath17701 & 1.08e@xmath17701 & 3.61e@xmath17701 & 2.34e@xmath17701 + @xmath224ni & 1.01e@xmath17703 & 1.08e@xmath17703 & 7.98e@xmath17704 & 1.77e@xmath17703 & 1.54e@xmath17703 & 1.55e@xmath17703 & 9.27e@xmath17706 & 3.03e@xmath17703 & 9.53e@xmath17704 & 4.17e@xmath17703 & 1.74e@xmath17703 & 7.35e@xmath17703 & 4.77e@xmath17703 crrrrrr p & 3.28e@xmath17702 & 3.28e@xmath17702 & 3.27e@xmath17702 & 3.27e@xmath17702 & 3.28e@xmath17702 & 3.28e@xmath17702 + d & 8.97e@xmath17718 & 8.97e@xmath17718 & 8.97e@xmath17718 & 8.97e@xmath17718 & 1.22e@xmath17715 & 1.22e@xmath17715 + @xmath178he & 2.72e@xmath17707 & 2.72e@xmath17707 & 2.72e@xmath17707 & 2.72e@xmath17707 & 2.72e@xmath17707 & 2.72e@xmath17707 + @xmath179he & 2.31e@xmath17702 & 2.31e@xmath17702 & 2.30e@xmath17702 & 2.30e@xmath17702 & 2.33e@xmath17702 & 2.32e@xmath17702 + @xmath180li & 2.75e@xmath17720 & 2.75e@xmath17720 & 2.75e@xmath17720 & 2.75e@xmath17720 & 4.04e@xmath17718 & 4.04e@xmath17718 + @xmath181li & 1.14e@xmath17711 & 1.14e@xmath17711 & 1.12e@xmath17711 & 1.12e@xmath17711 & 1.14e@xmath17711 & 1.14e@xmath17711 + @xmath182be & 2.40e@xmath17720 & 2.40e@xmath17720 & 2.50e@xmath17720 & 2.50e@xmath17720 & 2.09e@xmath17719 & 2.09e@xmath17719 + @xmath183b & 8.94e@xmath17717 & 8.94e@xmath17717 & 4.80e@xmath17717 & 4.80e@xmath17717 & 7.70e@xmath17718 & 7.70e@xmath17718 + @xmath184b & 2.74e@xmath17715 & 1.95e@xmath17715 & 7.68e@xmath17716 & 7.64e@xmath17716 & 4.17e@xmath17716 & 4.14e@xmath17716 + @xmath88c & 8.27e@xmath17704 & 7.76e@xmath17704 & 8.35e@xmath17704 & 7.84e@xmath17704 & 8.56e@xmath17704 & 8.07e@xmath17704 + @xmath89c & 2.24e@xmath17710 & 2.15e@xmath17710 & 2.26e@xmath17710 & 2.18e@xmath17710 & 2.43e@xmath17710 & 2.36e@xmath17710 + @xmath185n & 3.41e@xmath17706 & 3.41e@xmath17706 & 3.38e@xmath17706 & 3.38e@xmath17706 & 3.41e@xmath17706 & 3.41e@xmath17706 + @xmath186n & 4.89e@xmath17710 & 4.28e@xmath17710 & 4.87e@xmath17710 & 4.27e@xmath17710 & 4.99e@xmath17710 & 4.40e@xmath17710 + @xmath136o & 7.32e@xmath17703 & 4.91e@xmath17703 & 7.40e@xmath17703 & 4.98e@xmath17703 & 7.62e@xmath17703 & 5.23e@xmath17703 + @xmath187o & 4.23e@xmath17709 & 4.22e@xmath17709 & 4.16e@xmath17709 & 4.15e@xmath17709 & 4.23e@xmath17709 & 4.22e@xmath17709 + @xmath188o & 6.25e@xmath17709 & 6.24e@xmath17709 & 6.26e@xmath17709 & 6.26e@xmath17709 & 6.24e@xmath17709 & 6.24e@xmath17709 + @xmath189f & 4.18e@xmath17711 & 4.08e@xmath17711 & 4.18e@xmath17711 & 4.09e@xmath17711 & 1.80e@xmath17711 & 1.70e@xmath17711 + @xmath190ne & 1.37e@xmath17703 & 1.16e@xmath17703 & 1.38e@xmath17703 & 1.17e@xmath17703 & 1.41e@xmath17703 & 1.21e@xmath17703 + @xmath191ne & 1.24e@xmath17707 & 1.07e@xmath17707 & 1.25e@xmath17707 & 1.08e@xmath17707 & 1.22e@xmath17707 & 1.08e@xmath17707 + @xmath192ne & 8.56e@xmath17708 & 7.03e@xmath17708 & 8.59e@xmath17708 & 7.05e@xmath17708 & 8.53e@xmath17708 & 7.11e@xmath17708 + @xmath193na & 4.57e@xmath17706 & 4.17e@xmath17706 & 4.58e@xmath17706 & 4.19e@xmath17706 & 4.55e@xmath17706 & 4.23e@xmath17706 + @xmath194 mg & 5.54e@xmath17704 & 3.91e@xmath17704 & 5.59e@xmath17704 & 3.95e@xmath17704 & 5.50e@xmath17704 & 3.87e@xmath17704 + @xmath195 mg & 9.14e@xmath17707 & 8.27e@xmath17707 & 9.16e@xmath17707 & 8.28e@xmath17707 & 8.23e@xmath17707 & 7.43e@xmath17707 + @xmath196 mg & 8.94e@xmath17707 & 8.47e@xmath17707 & 8.95e@xmath17707 & 8.48e@xmath17707 & 9.02e@xmath17707 & 8.54e@xmath17707 + @xmath196al & 1.60e@xmath17708 & 1.41e@xmath17708 & 1.63e@xmath17708 & 1.43e@xmath17708 & 1.83e@xmath17708 & 1.15e@xmath17708 + @xmath197al & 9.94e@xmath17706 & 7.54e@xmath17706 & 9.96e@xmath17706 & 7.56e@xmath17706 & 1.02e@xmath17705 & 7.49e@xmath17706 + @xmath198si & 6.09e@xmath17704 & 3.78e@xmath17704 & 6.48e@xmath17704 & 4.06e@xmath17704 & 5.55e@xmath17704 & 3.49e@xmath17704 + @xmath199si & 2.84e@xmath17706 & 1.81e@xmath17706 & 2.88e@xmath17706 & 1.84e@xmath17706 & 2.36e@xmath17706 & 1.50e@xmath17706 + @xmath200si & 2.57e@xmath17706 & 1.87e@xmath17706 & 2.58e@xmath17706 & 1.88e@xmath17706 & 2.73e@xmath17706 & 1.85e@xmath17706 + @xmath201p & 9.63e@xmath17707 & 6.98e@xmath17707 & 9.74e@xmath17707 & 7.06e@xmath17707 & 9.85e@xmath17707 & 6.83e@xmath17707 + @xmath202s & 2.25e@xmath17704 & 1.38e@xmath17704 & 2.38e@xmath17704 & 1.46e@xmath17704 & 2.14e@xmath17704 & 1.35e@xmath17704 + @xmath203s & 6.90e@xmath17707 & 3.64e@xmath17707 & 7.33e@xmath17707 & 3.97e@xmath17707 & 6.16e@xmath17707 & 3.54e@xmath17707 + @xmath204s & 1.27e@xmath17706 & 6.41e@xmath17707 & 1.39e@xmath17706 & 7.29e@xmath17707 & 1.40e@xmath17706 & 7.13e@xmath17707 + @xmath205s & 3.25e@xmath17711 & 1.91e@xmath17711 & 3.25e@xmath17711 & 1.91e@xmath17711 & 3.25e@xmath17711 & 1.97e@xmath17711 + @xmath206cl & 1.67e@xmath17707 & 1.07e@xmath17707 & 1.91e@xmath17707 & 1.25e@xmath17707 & 2.10e@xmath17707 & 1.31e@xmath17707 + @xmath207cl & 5.80e@xmath17708 & 2.72e@xmath17708 & 6.86e@xmath17708 & 3.46e@xmath17708 & 7.43e@xmath17708 & 3.87e@xmath17708 + @xmath205ar & 3.34e@xmath17705 & 2.10e@xmath17705 & 3.42e@xmath17705 & 2.12e@xmath17705 & 3.46e@xmath17705 & 2.19e@xmath17705 + @xmath208ar & 4.35e@xmath17707 & 2.02e@xmath17707 & 6.24e@xmath17707 & 3.29e@xmath17707 & 6.09e@xmath17707 & 2.58e@xmath17707 + @xmath209ar & 2.06e@xmath17713 & 1.33e@xmath17713 & 2.10e@xmath17713 & 1.37e@xmath17713 & 2.00e@xmath17713 & 1.33e@xmath17713 + @xmath210k & 7.01e@xmath17708 & 3.03e@xmath17708 & 1.08e@xmath17707 & 5.71e@xmath17708 & 1.02e@xmath17707 & 4.33e@xmath17708 + @xmath209k & 8.63e@xmath17712 & 4.66e@xmath17712 & 9.77e@xmath17712 & 5.44e@xmath17712 & 9.60e@xmath17712 & 5.28e@xmath17712 + @xmath211k & 1.15e@xmath17708 & 4.72e@xmath17709 & 1.45e@xmath17708 & 6.74e@xmath17709 & 1.25e@xmath17708 & 5.65e@xmath17709 + @xmath209ca & 2.71e@xmath17705 & 1.77e@xmath17705 & 2.66e@xmath17705 & 1.69e@xmath17705 & 2.92e@xmath17705 & 1.85e@xmath17705 + @xmath212ca & 9.76e@xmath17709 & 4.13e@xmath17709 & 1.55e@xmath17708 & 7.88e@xmath17709 & 1.60e@xmath17708 & 5.63e@xmath17709 + @xmath213ca & 4.19e@xmath17710 & 3.30e@xmath17710 & 3.87e@xmath17710 & 3.05e@xmath17710 & 8.50e@xmath17710 & 5.80e@xmath17710 + @xmath42ca & 2.86e@xmath17707 & 1.94e@xmath17707 & 2.33e@xmath17707 & 1.55e@xmath17707 & 6.98e@xmath17707 & 4.00e@xmath17707 + @xmath214ca & 3.20e@xmath17714 & 2.80e@xmath17714 & 3.78e@xmath17714 & 3.38e@xmath17714 & 2.04e@xmath17714 & 1.77e@xmath17714 + @xmath43ca & 7.40e@xmath17715 & 7.39e@xmath17715 & 6.30e@xmath17715 & 6.29e@xmath17715 & 3.85e@xmath17715 & 3.85e@xmath17715 + @xmath215sc & 3.73e@xmath17710 & 1.98e@xmath17710 & 7.31e@xmath17710 & 5.15e@xmath17710 & 1.25e@xmath17708 & 6.35e@xmath17709 + @xmath214ti & 1.94e@xmath17708 & 1.49e@xmath17708 & 3.08e@xmath17708 & 2.43e@xmath17708 & 3.11e@xmath17708 & 2.21e@xmath17708 + @xmath216ti & 4.10e@xmath17708 & 3.00e@xmath17708 & 6.41e@xmath17708 & 4.86e@xmath17708 & 4.90e@xmath17708 & 3.88e@xmath17708 + @xmath43ti & 5.58e@xmath17707 & 4.06e@xmath17707 & 4.62e@xmath17707 & 3.34e@xmath17707 & 9.10e@xmath17707 & 5.76e@xmath17707 + @xmath217ti & 9.45e@xmath17709 & 7.63e@xmath17709 & 1.33e@xmath17708 & 1.05e@xmath17708 & 6.87e@xmath17708 & 3.94e@xmath17708 + @xmath218ti & 1.38e@xmath17714 & 1.25e@xmath17714 & 1.33e@xmath17714 & 1.19e@xmath17714 & 7.05e@xmath17715 & 5.30e@xmath17715 + @xmath218v & 7.70e@xmath17714 & 4.89e@xmath17714 & 1.11e@xmath17713 & 7.21e@xmath17714 & 1.35e@xmath17713 & 6.19e@xmath17714 + @xmath219v & 6.87e@xmath17708 & 5.36e@xmath17708 & 1.03e@xmath17707 & 8.07e@xmath17708 & 1.23e@xmath17707 & 9.00e@xmath17708 + @xmath218cr & 3.25e@xmath17708 & 2.47e@xmath17708 & 7.81e@xmath17708 & 5.93e@xmath17708 & 1.03e@xmath17707 & 6.57e@xmath17708 + @xmath220cr & 4.66e@xmath17706 & 3.72e@xmath17706 & 4.45e@xmath17706 & 3.53e@xmath17706 & 5.28e@xmath17706 & 3.97e@xmath17706 + @xmath221cr & 1.75e@xmath17707 & 1.44e@xmath17707 & 2.44e@xmath17707 & 1.96e@xmath17707 & 3.14e@xmath17707 & 2.33e@xmath17707 + @xmath222cr & 1.89e@xmath17712 & 9.63e@xmath17713 & 9.03e@xmath17712 & 6.09e@xmath17712 & 1.14e@xmath17711 & 3.10e@xmath17712 + @xmath223mn & 2.92e@xmath17707 & 2.63e@xmath17707 & 6.64e@xmath17707 & 5.29e@xmath17707 & 7.88e@xmath17707 & 5.88e@xmath17707 + @xmath222fe & 1.64e@xmath17706 & 1.34e@xmath17706 & 5.10e@xmath17706 & 3.82e@xmath17706 & 5.31e@xmath17706 & 3.74e@xmath17706 + @xmath25fe & 3.81e@xmath17704 & 2.89e@xmath17704 & 3.82e@xmath17704 & 2.92e@xmath17704 & 3.91e@xmath17704 & 2.95e@xmath17704 + @xmath224fe & 6.33e@xmath17706 & 4.69e@xmath17706 & 4.98e@xmath17706 & 3.59e@xmath17706 & 5.25e@xmath17706 & 3.73e@xmath17706 + @xmath128fe & 1.32e@xmath17712 & 4.56e@xmath17713 & 4.26e@xmath17712 & 2.48e@xmath17712 & 5.62e@xmath17712 & 1.24e@xmath17712 + @xmath225co & 1.03e@xmath17706 & 7.78e@xmath17707 & 1.58e@xmath17706 & 1.25e@xmath17706 & 2.77e@xmath17706 & 1.94e@xmath17706 + @xmath128ni & 2.33e@xmath17706 & 1.74e@xmath17706 & 2.81e@xmath17706 & 2.13e@xmath17706 & 5.35e@xmath17706 & 3.50e@xmath17706 + @xmath226ni & 1.16e@xmath17705 & 8.52e@xmath17706 & 1.21e@xmath17705 & 8.96e@xmath17706 & 1.38e@xmath17705 & 9.82e@xmath17706 + @xmath227ni & 1.86e@xmath17707 & 1.42e@xmath17707 & 1.34e@xmath17707 & 1.01e@xmath17707 & 1.60e@xmath17707 & 1.14e@xmath17707 + @xmath228ni & 1.16e@xmath17707 & 8.64e@xmath17708 & 1.27e@xmath17707 & 9.99e@xmath17708 & 1.61e@xmath17707 & 1.12e@xmath17707 + @xmath44ni & 1.30e@xmath17714 & 1.21e@xmath17714 & 9.99e@xmath17715 & 9.24e@xmath17715 & 7.71e@xmath17715 & 6.32e@xmath17715 + @xmath229cu & 3.28e@xmath17708 & 2.40e@xmath17708 & 3.93e@xmath17708 & 2.97e@xmath17708 & 6.47e@xmath17708 & 4.41e@xmath17708 + @xmath230cu & 2.38e@xmath17709 & 1.61e@xmath17709 & 1.76e@xmath17709 & 1.16e@xmath17709 & 3.46e@xmath17709 & 2.03e@xmath17709 + @xmath44zn & 1.04e@xmath17706 & 7.43e@xmath17707 & 9.53e@xmath17707 & 6.75e@xmath17707 & 1.83e@xmath17706 & 1.15e@xmath17706 + @xmath231zn & 1.13e@xmath17708 & 7.72e@xmath17709 & 7.09e@xmath17709 & 4.81e@xmath17709 & 2.32e@xmath17708 & 1.35e@xmath17708 + @xmath232zn & 2.96e@xmath17710 & 2.13e@xmath17710 & 2.41e@xmath17710 & 1.73e@xmath17710 & 2.26e@xmath17709 & 1.28e@xmath17709 + @xmath233zn & 4.57e@xmath17710 & 2.87e@xmath17710 & 5.14e@xmath17710 & 3.23e@xmath17710 & 2.75e@xmath17710 & 1.94e@xmath17710 + @xmath234zn & 7.74e@xmath17715 & 7.23e@xmath17715 & 7.44e@xmath17715 & 7.20e@xmath17715 & 5.12e@xmath17715 & 4.64e@xmath17715 + @xmath235ga & 7.83e@xmath17711 & 5.08e@xmath17711 & 8.78e@xmath17711 & 5.67e@xmath17711 & 8.48e@xmath17711 & 5.51e@xmath17711 + @xmath236ga & 5.87e@xmath17714 & 5.70e@xmath17714 & 5.44e@xmath17714 & 5.36e@xmath17714 & 1.85e@xmath17714 & 1.74e@xmath17714 + @xmath234ge & 5.90e@xmath17711 & 4.33e@xmath17711 & 5.35e@xmath17711 & 3.90e@xmath17711 & 8.06e@xmath17711 & 5.34e@xmath17711 + @xmath237ge & 3.81e@xmath17714 & 3.40e@xmath17714 & 4.45e@xmath17714 & 4.05e@xmath17714 & 1.34e@xmath17714 & 1.01e@xmath17714 + @xmath238ge & 4.78e@xmath17714 & 4.56e@xmath17714 & 4.46e@xmath17714 & 4.18e@xmath17714 & 2.23e@xmath17714 & 2.01e@xmath17714 + @xmath239ge & 1.91e@xmath17714 & 1.84e@xmath17714 & 1.50e@xmath17714 & 1.45e@xmath17714 & 8.33e@xmath17715 & 7.80e@xmath17715 cccccccccccc 13 & 1 & 0.07 & 0.09 & @xmath1772.55 & @xmath1770.08 & 0.37 & 1.57 & & & 1 & fig . [ fig : ye]a + 15 & 1 & 0.07 & 0.07 & @xmath1772.55 & 0.15 & 0.27 & 1.48 & & & 1 & fig . [ fig : ye]b + 18 & 1 & 0.07 & 0.16 & @xmath1772.55 & 0.40 & 0.63 & 1.65 & & & 1 & fig . [ fig : ye]c + 20 & 10 & 0.08 & 0.17 & @xmath1773.34 & 0.50 & 0.58 & 1.88 & 1.52 & 2.02 & 0.28 & fig . [ fig : ye]d + 25 & 5 & 0.10 & 0.14 & @xmath1772.98 & 0.50 & 0.41 & 2.39 & 1.79 & 2.61 & 0.27 & fig . [ fig : ye]i + 25 & 10 & 0.10 & 0.16 & @xmath1773.25 & 0.50 & 0.48 & 2.90 & 1.79 & 2.99 & 0.07 & fig . [ fig : ye]e + 30a & 20 & 0.16 & 0.22 & @xmath1773.31 & 0.50 & 0.41 & 3.27 & 1.65 & 3.59 & 0.17 & fig . [ fig : ye]f + 30b & 20 & 0.05 & 0.04 & @xmath1773.82 & 0.55 & 0.20 & 6.73 & 1.65 & 7.01 & 0.05 & fig . [ fig : ye]f + 40a & 30 & 0.26 & 0.34 & @xmath1773.26 & 0.50 & 0.39 & 5.53 & 2.24 & 6.00 & 0.12 & fig . [ fig : ye]g + 40b & 30 & 0.11 & 0.09 & @xmath1773.63 & 0.46 & 0.20 & 10.70 & 2.24 & 11.16 & 0.05 & fig . [ fig : ye]g + 50a & 40 & 0.36 & 0.57 & @xmath1773.22 & 0.50 & 0.47 & 3.51 & 1.89 & 4.15 & 0.28 & fig . [ fig : ye]h + 50b & 40 & 0.24 & 0.20 & @xmath1773.40 & 0.35 & 0.20 & 10.99 & 1.89 & 13.05 & 0.18 & fig . [ fig : ye]h + crrrrrrrrrrrr p & 6.59e+00 & 7.58e+00 & 8.43e+00 & 8.77e+00 & 1.06e+01 & 1.05e+01 & 1.17e+01 & 1.17e+01 & 1.40e+01 & 1.40e+01 & 1.63e+01 & 1.63e+01 + d & 1.49e@xmath17716 & 1.69e@xmath17716 & 1.28e@xmath17716 & 8.66e@xmath17717 & 2.04e@xmath17716 & 2.06e@xmath17716 & 1.09e@xmath17714 & 1.09e@xmath17714 & 1.66e@xmath17714 & 1.66e@xmath17714 & 1.24e@xmath17715 & 1.24e@xmath17715 + @xmath178he & 4.12e@xmath17705 & 4.09e@xmath17705 & 3.33e@xmath17705 & 4.76e@xmath17705 & 2.11e@xmath17704 & 2.11e@xmath17704 & 2.06e@xmath17704 & 2.06e@xmath17704 & 2.56e@xmath17705 & 2.56e@xmath17705 & 2.86e@xmath17705 & 2.86e@xmath17705 + @xmath179he & 4.01e+00 & 4.40e+00 & 5.42e+00 & 5.96e+00 & 8.03e+00 & 7.82e+00 & 9.54e+00 & 9.51e+00 & 1.18e+01 & 1.18e+01 & 1.56e+01 & 1.55e+01 + @xmath180li & 3.65e@xmath17723 & 1.11e@xmath17722 & 4.37e@xmath17723 & 1.36e@xmath17722 & 2.28e@xmath17720 & 2.72e@xmath17720 & 3.50e@xmath17717 & 3.50e@xmath17717 & 5.39e@xmath17717 & 5.39e@xmath17717 & 2.02e@xmath17718 & 2.02e@xmath17718 + @xmath181li & 2.17e@xmath17710 & 2.94e@xmath17710 & 7.34e@xmath17711 & 2.79e@xmath17710 & 5.68e@xmath17709 & 5.12e@xmath17709 & 2.36e@xmath17708 & 2.36e@xmath17708 & 3.42e@xmath17711 & 3.42e@xmath17711 & 8.78e@xmath17712 & 8.78e@xmath17712 + @xmath182be & 1.77e@xmath17720 & 3.22e@xmath17722 & 1.05e@xmath17722 & 4.83e@xmath17720 & 5.00e@xmath17717 & 3.96e@xmath17717 & 3.09e@xmath17718 & 3.09e@xmath17718 & 9.03e@xmath17718 & 9.03e@xmath17718 & 6.04e@xmath17717 & 6.04e@xmath17717 + @xmath183b & 2.92e@xmath17721 & 8.30e@xmath17720 & 3.92e@xmath17721 & 2.88e@xmath17719 & 2.79e@xmath17715 & 8.28e@xmath17714 & 2.95e@xmath17714 & 2.95e@xmath17714 & 1.41e@xmath17714 & 1.41e@xmath17714 & 1.51e@xmath17717 & 5.27e@xmath17718 + @xmath184b & 2.94e@xmath17716 & 3.30e@xmath17716 & 7.14e@xmath17716 & 1.09e@xmath17715 & 5.91e@xmath17714 & 1.06e@xmath17712 & 6.84e@xmath17713 & 6.80e@xmath17713 & 3.25e@xmath17713 & 3.24e@xmath17713 & 2.98e@xmath17714 & 5.55e@xmath17715 + @xmath88c & 7.41e@xmath17702 & 1.72e@xmath17701 & 2.18e@xmath17701 & 1.90e@xmath17701 & 2.79e@xmath17701 & 2.86e@xmath17701 & 3.16e@xmath17701 & 2.86e@xmath17701 & 3.72e@xmath17701 & 3.51e@xmath17701 & 1.56e+00 & 1.17e+00 + @xmath89c & 8.39e@xmath17708 & 6.21e@xmath17708 & 2.63e@xmath17709 & 1.18e@xmath17708 & 2.90e@xmath17708 & 7.43e@xmath17708 & 6.32e@xmath17708 & 6.03e@xmath17708 & 8.19e@xmath17708 & 7.85e@xmath17708 & 2.38e@xmath17707 & 1.70e@xmath17707 + @xmath185n & 1.83e@xmath17703 & 1.86e@xmath17703 & 1.89e@xmath17704 & 5.42e@xmath17705 & 5.92e@xmath17704 & 5.29e@xmath17704 & 4.18e@xmath17705 & 4.17e@xmath17705 & 3.39e@xmath17706 & 3.33e@xmath17706 & 4.38e@xmath17704 & 4.37e@xmath17704 + @xmath186n & 6.38e@xmath17708 & 6.86e@xmath17708 & 2.40e@xmath17708 & 2.95e@xmath17708 & 1.64e@xmath17707 & 1.73e@xmath17707 & 2.20e@xmath17707 & 1.16e@xmath17707 & 6.54e@xmath17707 & 6.51e@xmath17707 & 3.30e@xmath17707 & 8.15e@xmath17708 + @xmath136o & 4.50e@xmath17701 & 7.73e@xmath17701 & 1.38e+00 & 2.03e+00 & 2.60e+00 & 2.55e+00 & 3.92e+00 & 1.37e+00 & 6.32e+00 & 2.40e+00 & 8.86e+00 & 4.04e+00 + @xmath187o & 1.69e@xmath17706 & 1.57e@xmath17706 & 2.79e@xmath17707 & 7.13e@xmath17708 & 1.49e@xmath17706 & 1.33e@xmath17706 & 3.81e@xmath17708 & 2.64e@xmath17708 & 1.23e@xmath17708 & 8.65e@xmath17709 & 8.60e@xmath17706 & 8.56e@xmath17706 + @xmath188o & 5.79e@xmath17708 & 4.89e@xmath17706 & 4.63e@xmath17706 & 2.33e@xmath17708 & 4.66e@xmath17707 & 4.15e@xmath17707 & 5.03e@xmath17707 & 5.02e@xmath17707 & 2.93e@xmath17707 & 2.92e@xmath17707 & 6.00e@xmath17706 & 5.99e@xmath17706 + @xmath189f & 1.17e@xmath17710 & 1.97e@xmath17709 & 7.91e@xmath17709 & 2.12e@xmath17709 & 1.43e@xmath17709 & 1.77e@xmath17709 & 7.88e@xmath17709 & 6.41e@xmath17709 & 1.17e@xmath17707 & 1.16e@xmath17707 & 8.49e@xmath17709 & 4.07e@xmath17709 + @xmath190ne & 1.53e@xmath17702 & 3.27e@xmath17701 & 4.94e@xmath17701 & 7.49e@xmath17701 & 3.91e@xmath17701 & 3.06e@xmath17701 & 5.20e@xmath17701 & 2.60e@xmath17701 & 2.64e@xmath17701 & 2.19e@xmath17701 & 2.48e+00 & 1.30e+00 + @xmath191ne & 5.42e@xmath17707 & 3.76e@xmath17705 & 9.12e@xmath17705 & 3.58e@xmath17705 & 1.47e@xmath17705 & 1.31e@xmath17705 & 3.51e@xmath17705 & 2.26e@xmath17705 & 1.41e@xmath17705 & 1.07e@xmath17705 & 2.40e@xmath17704 & 1.15e@xmath17704 + @xmath192ne & 1.98e@xmath17707 & 1.61e@xmath17705 & 2.57e@xmath17705 & 5.51e@xmath17705 & 1.30e@xmath17705 & 9.24e@xmath17706 & 3.52e@xmath17705 & 1.98e@xmath17705 & 1.66e@xmath17705 & 1.45e@xmath17705 & 2.10e@xmath17704 & 1.08e@xmath17704 + @xmath193na & 1.44e@xmath17704 & 2.45e@xmath17703 & 2.08e@xmath17703 & 2.31e@xmath17703 & 6.74e@xmath17704 & 4.74e@xmath17704 & 7.36e@xmath17704 & 3.24e@xmath17704 & 3.28e@xmath17704 & 1.66e@xmath17704 & 4.27e@xmath17703 & 1.99e@xmath17703 + @xmath194 mg & 8.62e@xmath17702 & 6.82e@xmath17702 & 1.57e@xmath17701 & 1.65e@xmath17701 & 1.43e@xmath17701 & 1.64e@xmath17701 & 2.17e@xmath17701 & 4.20e@xmath17702 & 3.37e@xmath17701 & 9.00e@xmath17702 & 5.70e@xmath17701 & 1.97e@xmath17701 + @xmath195 mg & 1.56e@xmath17704 & 2.98e@xmath17704 & 5.83e@xmath17704 & 1.07e@xmath17704 & 4.52e@xmath17705 & 4.90e@xmath17705 & 1.45e@xmath17704 & 9.00e@xmath17705 & 5.95e@xmath17704 & 4.11e@xmath17704 & 3.10e@xmath17704 & 1.17e@xmath17704 + @xmath196 mg & 7.07e@xmath17705 & 3.98e@xmath17704 & 8.73e@xmath17704 & 2.09e@xmath17704 & 4.31e@xmath17705 & 4.17e@xmath17705 & 8.00e@xmath17705 & 2.62e@xmath17705 & 6.90e@xmath17705 & 1.23e@xmath17705 & 2.32e@xmath17704 & 9.73e@xmath17705 + @xmath196al & 1.01e@xmath17706 & 1.14e@xmath17706 & 3.34e@xmath17706 & 1.32e@xmath17706 & 1.22e@xmath17706 & 1.40e@xmath17706 & 3.01e@xmath17706 & 1.36e@xmath17706 & 3.80e@xmath17705 & 3.43e@xmath17705 & 5.37e@xmath17706 & 1.34e@xmath17706 + @xmath197al & 3.78e@xmath17703 & 1.37e@xmath17703 & 3.14e@xmath17703 & 1.50e@xmath17703 & 8.59e@xmath17704 & 9.58e@xmath17704 & 1.55e@xmath17703 & 1.59e@xmath17704 & 7.52e@xmath17703 & 1.34e@xmath17703 & 3.29e@xmath17703 & 8.12e@xmath17704 + @xmath198si & 8.04e@xmath17702 & 7.32e@xmath17702 & 1.16e@xmath17701 & 1.03e@xmath17701 & 2.62e@xmath17701 & 2.41e@xmath17701 & 2.47e@xmath17701 & 3.17e@xmath17702 & 7.20e@xmath17701 & 2.29e@xmath17701 & 4.77e@xmath17701 & 1.25e@xmath17701 + @xmath199si & 7.50e@xmath17704 & 2.39e@xmath17704 & 4.42e@xmath17704 & 2.95e@xmath17704 & 5.15e@xmath17704 & 5.77e@xmath17704 & 8.85e@xmath17704 & 4.90e@xmath17705 & 3.72e@xmath17703 & 1.29e@xmath17703 & 9.96e@xmath17704 & 1.91e@xmath17704 + @xmath200si & 1.42e@xmath17703 & 1.50e@xmath17704 & 3.45e@xmath17704 & 1.15e@xmath17704 & 5.88e@xmath17705 & 6.90e@xmath17705 & 1.49e@xmath17704 & 1.06e@xmath17705 & 2.82e@xmath17703 & 4.84e@xmath17704 & 2.20e@xmath17704 & 5.21e@xmath17705 + @xmath201p & 4.88e@xmath17704 & 5.65e@xmath17705 & 1.32e@xmath17704 & 7.79e@xmath17705 & 5.34e@xmath17705 & 6.38e@xmath17705 & 1.17e@xmath17704 & 7.55e@xmath17706 & 1.01e@xmath17703 & 2.24e@xmath17704 & 2.13e@xmath17704 & 4.61e@xmath17705 + @xmath202s & 2.37e@xmath17702 & 3.20e@xmath17702 & 4.07e@xmath17702 & 4.27e@xmath17702 & 1.18e@xmath17701 & 9.43e@xmath17702 & 8.49e@xmath17702 & 1.48e@xmath17702 & 2.59e@xmath17701 & 6.11e@xmath17702 & 2.00e@xmath17701 & 6.03e@xmath17702 + @xmath203s & 8.98e@xmath17705 & 7.55e@xmath17705 & 1.03e@xmath17704 & 1.44e@xmath17704 & 2.15e@xmath17704 & 2.47e@xmath17704 & 3.02e@xmath17704 & 1.61e@xmath17705 & 8.45e@xmath17704 & 1.29e@xmath17704 & 5.42e@xmath17704 & 1.02e@xmath17704 + @xmath204s & 2.81e@xmath17704 & 2.03e@xmath17704 & 2.85e@xmath17704 & 1.89e@xmath17704 & 5.43e@xmath17705 & 1.37e@xmath17704 & 2.75e@xmath17704 & 1.92e@xmath17705 & 2.09e@xmath17703 & 1.34e@xmath17704 & 5.94e@xmath17704 & 1.32e@xmath17704 + @xmath205s & 1.48e@xmath17708 & 1.43e@xmath17709 & 5.34e@xmath17709 & 8.33e@xmath17710 & 6.69e@xmath17711 & 5.77e@xmath17711 & 1.41e@xmath17709 & 7.84e@xmath17711 & 5.04e@xmath17708 & 2.66e@xmath17709 & 5.58e@xmath17711 & 1.04e@xmath17711 + @xmath206cl & 5.54e@xmath17705 & 1.56e@xmath17705 & 2.69e@xmath17705 & 4.28e@xmath17705 & 2.33e@xmath17705 & 3.70e@xmath17705 & 4.98e@xmath17705 & 7.51e@xmath17706 & 1.82e@xmath17704 & 2.58e@xmath17705 & 1.50e@xmath17704 & 5.16e@xmath17705 + @xmath207cl & 3.04e@xmath17706 & 5.83e@xmath17706 & 9.12e@xmath17706 & 1.89e@xmath17705 & 2.54e@xmath17705 & 3.10e@xmath17705 & 2.39e@xmath17705 & 1.35e@xmath17706 & 7.96e@xmath17705 & 4.48e@xmath17706 & 7.61e@xmath17705 & 1.46e@xmath17705 + @xmath205ar & 3.24e@xmath17703 & 5.28e@xmath17703 & 5.67e@xmath17703 & 6.79e@xmath17703 & 1.86e@xmath17702 & 1.38e@xmath17702 & 1.15e@xmath17702 & 2.59e@xmath17703 & 3.55e@xmath17702 & 5.86e@xmath17703 & 3.06e@xmath17702 & 1.05e@xmath17702 + @xmath208ar & 5.30e@xmath17705 & 6.32e@xmath17705 & 1.70e@xmath17704 & 1.28e@xmath17704 & 4.98e@xmath17705 & 1.84e@xmath17704 & 9.20e@xmath17705 & 6.00e@xmath17706 & 7.84e@xmath17704 & 4.22e@xmath17705 & 8.75e@xmath17704 & 1.67e@xmath17704 + @xmath209ar & 8.01e@xmath17711 & 1.78e@xmath17711 & 3.96e@xmath17711 & 3.79e@xmath17711 & 5.17e@xmath17712 & 6.41e@xmath17712 & 1.95e@xmath17711 & 3.51e@xmath17712 & 2.77e@xmath17710 & 2.93e@xmath17711 & 3.64e@xmath17712 & 6.74e@xmath17713 + @xmath210k & 5.74e@xmath17706 & 8.88e@xmath17706 & 1.98e@xmath17705 & 2.54e@xmath17705 & 2.27e@xmath17705 & 4.25e@xmath17705 & 2.38e@xmath17705 & 3.90e@xmath17706 & 1.33e@xmath17704 & 1.17e@xmath17705 & 1.71e@xmath17704 & 4.44e@xmath17705 + @xmath209k & 1.14e@xmath17709 & 9.10e@xmath17710 & 1.96e@xmath17709 & 4.40e@xmath17709 & 1.84e@xmath17709 & 1.51e@xmath17709 & 2.52e@xmath17709 & 1.34e@xmath17710 & 1.18e@xmath17708 & 6.21e@xmath17710 & 5.89e@xmath17709 & 1.09e@xmath17709 + @xmath211k & 3.54e@xmath17707 & 8.05e@xmath17707 & 1.72e@xmath17706 & 3.73e@xmath17706 & 6.75e@xmath17706 & 6.94e@xmath17706 & 3.66e@xmath17706 & 2.15e@xmath17707 & 2.13e@xmath17705 & 1.20e@xmath17706 & 1.43e@xmath17705 & 2.82e@xmath17706 + @xmath209ca & 2.92e@xmath17703 & 4.41e@xmath17703 & 4.40e@xmath17703 & 4.77e@xmath17703 & 1.49e@xmath17702 & 1.02e@xmath17702 & 8.22e@xmath17703 & 2.25e@xmath17703 & 2.86e@xmath17702 & 5.18e@xmath17703 & 2.22e@xmath17702 & 8.98e@xmath17703 + @xmath212ca & 9.77e@xmath17707 & 1.23e@xmath17706 & 3.62e@xmath17706 & 3.41e@xmath17706 & 1.29e@xmath17706 & 5.53e@xmath17706 & 1.62e@xmath17706 & 8.49e@xmath17708 & 1.97e@xmath17705 & 1.03e@xmath17706 & 2.75e@xmath17705 & 5.08e@xmath17706 + @xmath213ca & 6.75e@xmath17708 & 4.58e@xmath17708 & 3.65e@xmath17708 & 2.06e@xmath17707 & 5.54e@xmath17708 & 9.96e@xmath17708 & 1.68e@xmath17707 & 5.18e@xmath17708 & 1.50e@xmath17707 & 5.83e@xmath17708 & 4.79e@xmath17707 & 3.10e@xmath17707 + @xmath42ca & 9.55e@xmath17706 & 1.19e@xmath17705 & 8.75e@xmath17706 & 1.09e@xmath17704 & 2.86e@xmath17705 & 5.25e@xmath17705 & 1.49e@xmath17704 & 4.63e@xmath17705 & 1.53e@xmath17704 & 6.28e@xmath17705 & 5.37e@xmath17704 & 3.51e@xmath17704 + @xmath214ca & 1.07e@xmath17712 & 1.76e@xmath17712 & 9.27e@xmath17712 & 1.15e@xmath17711 & 2.61e@xmath17712 & 2.66e@xmath17711 & 8.42e@xmath17712 & 6.77e@xmath17712 & 4.61e@xmath17711 & 3.37e@xmath17711 & 1.04e@xmath17712 & 1.93e@xmath17713 + @xmath43ca & 1.14e@xmath17717 & 4.21e@xmath17714 & 4.17e@xmath17716 & 5.74e@xmath17716 & 8.89e@xmath17712 & 1.08e@xmath17711 & 5.76e@xmath17713 & 5.72e@xmath17713 & 6.20e@xmath17712 & 6.20e@xmath17712 & 7.67e@xmath17714 & 1.42e@xmath17714 + @xmath215sc & 2.09e@xmath17708 & 4.00e@xmath17708 & 5.52e@xmath17708 & 7.30e@xmath17707 & 2.51e@xmath17707 & 2.32e@xmath17707 & 6.85e@xmath17708 & 1.29e@xmath17708 & 6.27e@xmath17707 & 4.89e@xmath17708 & 5.89e@xmath17707 & 1.59e@xmath17707 + @xmath214ti & 8.17e@xmath17706 & 4.97e@xmath17706 & 4.96e@xmath17706 & 1.09e@xmath17705 & 3.38e@xmath17706 & 7.57e@xmath17706 & 8.92e@xmath17706 & 2.65e@xmath17706 & 1.21e@xmath17705 & 2.42e@xmath17706 & 3.38e@xmath17705 & 1.58e@xmath17705 + @xmath216ti & 1.22e@xmath17705 & 8.51e@xmath17706 & 7.79e@xmath17706 & 1.86e@xmath17705 & 1.27e@xmath17705 & 1.82e@xmath17705 & 3.21e@xmath17705 & 9.99e@xmath17706 & 3.18e@xmath17705 & 1.34e@xmath17705 & 8.35e@xmath17705 & 5.47e@xmath17705 + @xmath43ti & 5.66e@xmath17705 & 7.32e@xmath17705 & 7.18e@xmath17705 & 1.23e@xmath17704 & 1.06e@xmath17704 & 1.13e@xmath17704 & 2.36e@xmath17704 & 7.36e@xmath17705 & 3.13e@xmath17704 & 1.31e@xmath17704 & 6.41e@xmath17704 & 4.20e@xmath17704 + @xmath217ti & 2.26e@xmath17706 & 3.30e@xmath17706 & 2.99e@xmath17706 & 3.80e@xmath17706 & 4.13e@xmath17706 & 3.30e@xmath17706 & 5.62e@xmath17706 & 1.75e@xmath17706 & 6.87e@xmath17706 & 2.70e@xmath17706 & 8.37e@xmath17706 & 5.26e@xmath17706 + @xmath218ti & 1.18e@xmath17712 & 9.55e@xmath17713 & 1.72e@xmath17712 & 9.35e@xmath17713 & 1.92e@xmath17712 & 3.02e@xmath17712 & 1.61e@xmath17712 & 1.26e@xmath17712 & 3.10e@xmath17711 & 2.68e@xmath17711 & 1.10e@xmath17712 & 2.03e@xmath17713 + @xmath218v & 1.35e@xmath17711 & 1.05e@xmath17711 & 4.60e@xmath17711 & 2.62e@xmath17711 & 8.16e@xmath17712 & 2.88e@xmath17711 & 9.04e@xmath17712 & 7.27e@xmath17713 & 1.13e@xmath17710 & 2.06e@xmath17711 & 1.42e@xmath17710 & 2.63e@xmath17711 + @xmath219v & 2.00e@xmath17705 & 1.60e@xmath17705 & 1.41e@xmath17705 & 4.05e@xmath17705 & 1.63e@xmath17705 & 2.55e@xmath17705 & 4.90e@xmath17705 & 1.53e@xmath17705 & 3.91e@xmath17705 & 1.63e@xmath17705 & 1.11e@xmath17704 & 7.21e@xmath17705 + @xmath218cr & 1.07e@xmath17705 & 1.61e@xmath17705 & 2.64e@xmath17705 & 1.98e@xmath17705 & 3.29e@xmath17705 & 2.60e@xmath17705 & 1.96e@xmath17705 & 5.88e@xmath17706 & 4.13e@xmath17705 & 9.90e@xmath17706 & 8.80e@xmath17705 & 2.88e@xmath17705 + @xmath220cr & 8.84e@xmath17704 & 1.09e@xmath17703 & 1.13e@xmath17703 & 6.89e@xmath17704 & 1.31e@xmath17703 & 1.17e@xmath17703 & 1.57e@xmath17703 & 4.89e@xmath17704 & 2.75e@xmath17703 & 1.16e@xmath17703 & 3.17e@xmath17703 & 2.07e@xmath17703 + @xmath221cr & 4.96e@xmath17705 & 6.76e@xmath17705 & 6.43e@xmath17705 & 3.34e@xmath17705 & 7.78e@xmath17705 & 6.64e@xmath17705 & 9.04e@xmath17705 & 2.82e@xmath17705 & 1.31e@xmath17704 & 5.46e@xmath17705 & 1.43e@xmath17704 & 8.85e@xmath17705 + @xmath222cr & 2.35e@xmath17710 & 4.13e@xmath17710 & 3.25e@xmath17709 & 2.61e@xmath17709 & 4.15e@xmath17710 & 6.12e@xmath17709 & 9.16e@xmath17711 & 7.16e@xmath17712 & 3.52e@xmath17709 & 2.17e@xmath17710 & 2.76e@xmath17708 & 5.11e@xmath17709 + @xmath223mn & 1.33e@xmath17704 & 1.86e@xmath17704 & 1.74e@xmath17704 & 8.17e@xmath17705 & 2.22e@xmath17704 & 1.86e@xmath17704 & 2.47e@xmath17704 & 7.70e@xmath17705 & 3.69e@xmath17704 & 1.50e@xmath17704 & 3.88e@xmath17704 & 2.25e@xmath17704 + @xmath222fe & 7.29e@xmath17704 & 1.24e@xmath17703 & 1.40e@xmath17703 & 7.18e@xmath17704 & 2.79e@xmath17703 & 1.73e@xmath17703 & 1.74e@xmath17703 & 5.41e@xmath17704 & 3.30e@xmath17703 & 1.09e@xmath17703 & 4.66e@xmath17703 & 1.93e@xmath17703 + @xmath25fe & 7.00e@xmath17702 & 7.00e@xmath17702 & 7.00e@xmath17702 & 8.27e@xmath17702 & 1.04e@xmath17701 & 1.03e@xmath17701 & 1.60e@xmath17701 & 4.97e@xmath17702 & 2.56e@xmath17701 & 1.08e@xmath17701 & 3.59e@xmath17701 & 2.35e@xmath17701 + @xmath224fe & 7.33e@xmath17704 & 6.88e@xmath17704 & 6.60e@xmath17704 & 1.13e@xmath17703 & 1.26e@xmath17703 & 1.40e@xmath17703 & 2.43e@xmath17703 & 7.57e@xmath17704 & 3.80e@xmath17703 & 1.60e@xmath17703 & 6.10e@xmath17703 & 3.99e@xmath17703 + @xmath128fe & 5.67e@xmath17711 & 1.76e@xmath17710 & 8.07e@xmath17710 & 1.28e@xmath17709 & 7.15e@xmath17710 & 2.77e@xmath17709 & 9.22e@xmath17711 & 6.69e@xmath17712 & 3.30e@xmath17709 & 2.25e@xmath17710 & 1.26e@xmath17708 & 2.32e@xmath17709 + @xmath225co & 2.77e@xmath17704 & 3.02e@xmath17704 & 2.30e@xmath17704 & 6.42e@xmath17704 & 2.73e@xmath17704 & 3.57e@xmath17704 & 7.19e@xmath17704 & 2.24e@xmath17704 & 6.15e@xmath17704 & 2.59e@xmath17704 & 1.77e@xmath17703 & 1.16e@xmath17703 + @xmath128ni & 4.06e@xmath17704 & 4.57e@xmath17704 & 3.85e@xmath17704 & 9.68e@xmath17704 & 6.60e@xmath17704 & 8.66e@xmath17704 & 1.32e@xmath17703 & 4.12e@xmath17704 & 1.46e@xmath17703 & 5.67e@xmath17704 & 3.31e@xmath17703 & 2.08e@xmath17703 + @xmath226ni & 2.18e@xmath17703 & 1.68e@xmath17703 & 1.65e@xmath17703 & 2.97e@xmath17703 & 2.94e@xmath17703 & 3.29e@xmath17703 & 5.60e@xmath17703 & 1.75e@xmath17703 & 8.50e@xmath17703 & 3.58e@xmath17703 & 1.30e@xmath17702 & 8.49e@xmath17703 + @xmath227ni & 2.56e@xmath17705 & 1.72e@xmath17705 & 1.46e@xmath17705 & 4.06e@xmath17705 & 2.50e@xmath17705 & 3.68e@xmath17705 & 6.21e@xmath17705 & 1.94e@xmath17705 & 8.14e@xmath17705 & 3.43e@xmath17705 & 1.57e@xmath17704 & 1.03e@xmath17704 + @xmath228ni & 1.94e@xmath17705 & 1.51e@xmath17705 & 1.24e@xmath17705 & 3.99e@xmath17705 & 1.97e@xmath17705 & 6.66e@xmath17705 & 5.26e@xmath17705 & 1.64e@xmath17705 & 6.26e@xmath17705 & 2.64e@xmath17705 & 1.38e@xmath17704 & 9.04e@xmath17705 + @xmath44ni & 2.14e@xmath17715 & 1.01e@xmath17713 & 1.07e@xmath17714 & 8.67e@xmath17713 & 3.73e@xmath17712 & 7.27e@xmath17712 & 3.59e@xmath17712 & 2.59e@xmath17712 & 1.78e@xmath17711 & 1.78e@xmath17711 & 5.33e@xmath17712 & 9.92e@xmath17713 + @xmath229cu & 6.24e@xmath17706 & 5.46e@xmath17706 & 4.49e@xmath17706 & 1.53e@xmath17705 & 6.84e@xmath17706 & 9.48e@xmath17706 & 1.90e@xmath17705 & 5.92e@xmath17706 & 2.10e@xmath17705 & 8.84e@xmath17706 & 5.10e@xmath17705 & 3.34e@xmath17705 + @xmath230cu & 1.40e@xmath17707 & 1.23e@xmath17707 & 1.17e@xmath17707 & 4.78e@xmath17707 & 3.64e@xmath17707 & 4.84e@xmath17707 & 1.03e@xmath17706 & 3.21e@xmath17707 & 1.62e@xmath17706 & 6.81e@xmath17707 & 3.10e@xmath17706 & 2.03e@xmath17706 + @xmath44zn & 1.10e@xmath17704 & 9.81e@xmath17705 & 8.47e@xmath17705 & 3.34e@xmath17704 & 1.86e@xmath17704 & 2.49e@xmath17704 & 5.29e@xmath17704 & 1.65e@xmath17704 & 6.57e@xmath17704 & 2.77e@xmath17704 & 1.43e@xmath17703 & 9.37e@xmath17704 + @xmath231zn & 4.49e@xmath17707 & 4.05e@xmath17707 & 3.65e@xmath17707 & 2.53e@xmath17706 & 1.06e@xmath17706 & 2.47e@xmath17706 & 4.19e@xmath17706 & 1.31e@xmath17706 & 5.33e@xmath17706 & 2.25e@xmath17706 & 1.29e@xmath17705 & 8.47e@xmath17706 + @xmath232zn & 1.49e@xmath17708 & 1.59e@xmath17708 & 1.27e@xmath17708 & 1.56e@xmath17707 & 2.36e@xmath17708 & 4.92e@xmath17708 & 1.53e@xmath17707 & 4.78e@xmath17708 & 1.03e@xmath17707 & 4.36e@xmath17708 & 4.35e@xmath17707 & 2.85e@xmath17707 + @xmath233zn & 2.54e@xmath17708 & 3.22e@xmath17708 & 3.43e@xmath17708 & 1.15e@xmath17707 & 1.20e@xmath17707 & 1.32e@xmath17707 & 3.10e@xmath17707 & 9.66e@xmath17708 & 5.03e@xmath17707 & 2.12e@xmath17707 & 1.13e@xmath17706 & 7.40e@xmath17707 + @xmath234zn & 5.95e@xmath17716 & 3.56e@xmath17714 & 6.44e@xmath17715 & 1.08e@xmath17713 & 2.03e@xmath17713 & 2.15e@xmath17712 & 7.54e@xmath17713 & 4.92e@xmath17713 & 2.23e@xmath17711 & 2.23e@xmath17711 & 1.96e@xmath17712 & 3.68e@xmath17713 + @xmath235ga & 7.27e@xmath17709 & 6.01e@xmath17709 & 5.58e@xmath17709 & 2.02e@xmath17708 & 1.95e@xmath17708 & 2.44e@xmath17708 & 4.77e@xmath17708 & 1.49e@xmath17708 & 9.52e@xmath17708 & 4.02e@xmath17708 & 1.40e@xmath17707 & 9.15e@xmath17708 + @xmath236ga & 7.45e@xmath17715 & 1.01e@xmath17713 & 1.65e@xmath17714 & 1.49e@xmath17712 & 1.58e@xmath17712 & 3.17e@xmath17711 & 6.77e@xmath17712 & 5.92e@xmath17712 & 1.39e@xmath17710 & 1.39e@xmath17710 & 5.49e@xmath17712 & 1.08e@xmath17712 + @xmath234ge & 7.35e@xmath17709 & 4.76e@xmath17709 & 3.96e@xmath17709 & 2.03e@xmath17708 & 8.99e@xmath17709 & 1.64e@xmath17708 & 2.66e@xmath17708 & 8.27e@xmath17709 & 3.59e@xmath17708 & 1.52e@xmath17708 & 7.31e@xmath17708 & 4.78e@xmath17708 + @xmath237ge & 1.71e@xmath17714 & 7.98e@xmath17713 & 5.75e@xmath17713 & 5.96e@xmath17712 & 2.82e@xmath17712 & 3.08e@xmath17711 & 6.76e@xmath17712 & 3.15e@xmath17712 & 8.47e@xmath17711 & 8.45e@xmath17711 & 3.54e@xmath17711 & 6.59e@xmath17712 + @xmath238ge & 1.34e@xmath17714 & 1.19e@xmath17713 & 1.10e@xmath17713 & 3.68e@xmath17712 & 5.53e@xmath17712 & 2.78e@xmath17711 & 1.17e@xmath17711 & 8.68e@xmath17712 & 9.33e@xmath17711 & 9.32e@xmath17711 & 2.28e@xmath17711 & 4.28e@xmath17712 + @xmath239ge & 2.49e@xmath17715 & 1.21e@xmath17713 & 4.90e@xmath17714 & 2.02e@xmath17713 & 1.16e@xmath17712 & 8.92e@xmath17712 & 2.57e@xmath17712 & 2.05e@xmath17712 & 3.65e@xmath17711 & 3.64e@xmath17711 & 3.88e@xmath17712 & 7.32e@xmath17713 crrrrrrrrrrrr @xmath192na & 1.51e@xmath17707 & 7.70e@xmath17706 & 1.68e@xmath17705 & 4.65e@xmath17705 & 9.97e@xmath17706 & 7.01e@xmath17706 & 2.70e@xmath17705 & 1.34e@xmath17705 & 1.37e@xmath17705 & 1.19e@xmath17705 & 1.77e@xmath17704 & 9.14e@xmath17705 + @xmath196al & 9.84e@xmath17707 & 1.12e@xmath17706 & 3.33e@xmath17706 & 9.81e@xmath17707 & 1.21e@xmath17706 & 1.37e@xmath17706 & 2.92e@xmath17706 & 1.33e@xmath17706 & 2.13e@xmath17705 & 1.76e@xmath17705 & 5.24e@xmath17706 & 1.26e@xmath17706 + @xmath211ca & 3.54e@xmath17707 & 8.05e@xmath17707 & 1.72e@xmath17706 & 3.73e@xmath17706 & 6.75e@xmath17706 & 6.94e@xmath17706 & 3.66e@xmath17706 & 2.15e@xmath17707 & 2.13e@xmath17705 & 1.20e@xmath17706 & 1.43e@xmath17705 & 2.82e@xmath17706 + @xmath42ti & 9.55e@xmath17706 & 1.19e@xmath17705 & 8.75e@xmath17706 & 1.09e@xmath17704 & 2.86e@xmath17705 & 5.25e@xmath17705 & 1.49e@xmath17704 & 4.63e@xmath17705 & 1.53e@xmath17704 & 6.28e@xmath17705 & 5.37e@xmath17704 & 3.51e@xmath17704 + @xmath226fe & 4.74e@xmath17716 & 7.17e@xmath17714 & 3.78e@xmath17715 & 1.09e@xmath17713 & 1.53e@xmath17713 & 2.37e@xmath17712 & 1.68e@xmath17712 & 1.11e@xmath17712 & 2.40e@xmath17711 & 2.40e@xmath17711 & 7.78e@xmath17713 & 1.45e@xmath17713 + @xmath25ni & 7.00e@xmath17702 & 7.00e@xmath17702 & 7.00e@xmath17702 & 8.27e@xmath17702 & 1.04e@xmath17701 & 1.03e@xmath17701 & 1.60e@xmath17701 & 4.97e@xmath17702 & 2.56e@xmath17701 & 1.08e@xmath17701 & 3.59e@xmath17701 & 2.35e@xmath17701 + @xmath224ni & 7.33e@xmath17704 & 6.88e@xmath17704 & 6.60e@xmath17704 & 1.13e@xmath17703 & 1.26e@xmath17703 & 1.40e@xmath17703 & 2.43e@xmath17703 & 7.56e@xmath17704 & 3.80e@xmath17703 & 1.60e@xmath17703 & 6.09e@xmath17703 & 3.99e@xmath17703 cccccccccccccccc 20 & 10 & 0.08 & 0.16 & @xmath1773.33 & 0.50 & 0.56 & 1.83 & 1.52 & 2.01 & 0.37 & 1/2 & fig . [ fig : yelow]a + 25 & 5 & 0.11 & 0.14 & @xmath1772.95 & 0.50 & 0.35 & 2.08 & 1.79 & 2.38 & 0.51 & 1/4 & fig . [ fig : yelow]f + 25 & 10 & 0.10 & 0.14 & @xmath1773.25 & 0.50 & 0.41 & 2.67 & 1.79 & 2.92 & 0.22 & 1/3 & fig . [ fig : yelow]b + 30a & 20 & 0.16 & 0.21 & @xmath1773.31 & 0.50 & 0.38 & 2.97 & 1.65 & 3.42 & 0.25 & 1/3 & fig . [ fig : yelow]c + 30b & 20 & 0.05 & 0.04 & @xmath1773.86 & 0.64 & 0.20 & 6.50 & 1.65 & 6.87 & 0.07 & 1/3 & fig . [ fig : yelow]c + 40a & 30 & 0.28 & 0.38 & @xmath1773.22 & 0.50 & 0.40 & 4.80 & 2.24 & 5.53 & 0.22 & 1/4 & fig . [ fig : yelow]d + 40b & 30 & 0.12 & 0.10 & @xmath1773.59 & 0.49 & 0.20 & 10.09 & 2.24 & 10.92 & 0.10 & 1/4 & fig . [ fig : yelow]d + 50a & 40 & 0.37 & 0.51 & @xmath1773.20 & 0.50 & 0.42 & 3.18 & 1.89 & 3.99 & 0.39 & 1/2 & fig . [ fig : yelow]e + 50b & 40 & 0.26 & 0.22 & @xmath1773.36 & 0.38 & 0.20 & 9.80 & 1.89 & 12.74 & 0.27 & 1/2 & fig . [ fig : yelow]e + crrrrrrrrr p & 8.77e+00 & 1.06e+01 & 1.06e+01 & 1.17e+01 & 1.17e+01 & 1.40e+01 & 1.40e+01 & 1.63e+01 & 1.63e+01 + d & 8.66e@xmath17717 & 2.10e@xmath17716 & 2.17e@xmath17716 & 4.19e@xmath17714 & 4.19e@xmath17714 & 4.43e@xmath17712 & 4.43e@xmath17712 & 1.23e@xmath17715 & 1.23e@xmath17715 + @xmath178he & 4.76e@xmath17705 & 2.11e@xmath17704 & 2.11e@xmath17704 & 2.06e@xmath17704 & 2.06e@xmath17704 & 2.56e@xmath17705 & 2.56e@xmath17705 & 2.86e@xmath17705 & 2.86e@xmath17705 + @xmath179he & 5.98e+00 & 8.07e+00 & 8.06e+00 & 9.64e+00 & 9.55e+00 & 1.21e+01 & 1.20e+01 & 1.57e+01 & 1.57e+01 + @xmath180li & 1.22e@xmath17722 & 3.06e@xmath17720 & 7.78e@xmath17720 & 1.37e@xmath17716 & 1.37e@xmath17716 & 1.47e@xmath17714 & 1.47e@xmath17714 & 2.00e@xmath17718 & 2.00e@xmath17718 + @xmath181li & 2.79e@xmath17710 & 5.68e@xmath17709 & 5.68e@xmath17709 & 2.36e@xmath17708 & 2.36e@xmath17708 & 3.50e@xmath17711 & 3.50e@xmath17711 & 8.78e@xmath17712 & 8.78e@xmath17712 + @xmath182be & 1.16e@xmath17720 & 3.12e@xmath17717 & 2.90e@xmath17717 & 6.90e@xmath17718 & 6.90e@xmath17718 & 6.87e@xmath17716 & 6.87e@xmath17716 & 9.43e@xmath17717 & 9.43e@xmath17717 + @xmath183b & 2.59e@xmath17719 & 1.95e@xmath17715 & 3.81e@xmath17715 & 5.17e@xmath17716 & 5.17e@xmath17716 & 2.20e@xmath17714 & 2.20e@xmath17714 & 9.04e@xmath17718 & 4.50e@xmath17718 + @xmath184b & 9.41e@xmath17716 & 7.26e@xmath17715 & 5.42e@xmath17713 & 3.94e@xmath17714 & 3.44e@xmath17714 & 6.67e@xmath17713 & 6.65e@xmath17713 & 1.13e@xmath17714 & 3.13e@xmath17715 + @xmath88c & 1.95e@xmath17701 & 2.91e@xmath17701 & 2.77e@xmath17701 & 3.25e@xmath17701 & 2.90e@xmath17701 & 4.26e@xmath17701 & 4.15e@xmath17701 & 1.61e+00 & 1.24e+00 + @xmath89c & 9.95e@xmath17709 & 3.83e@xmath17708 & 7.78e@xmath17708 & 4.32e@xmath17708 & 4.14e@xmath17708 & 1.05e@xmath17707 & 1.04e@xmath17707 & 4.37e@xmath17707 & 3.75e@xmath17707 + @xmath185n & 5.43e@xmath17705 & 5.91e@xmath17704 & 5.93e@xmath17704 & 4.07e@xmath17705 & 4.05e@xmath17705 & 8.90e@xmath17706 & 8.70e@xmath17706 & 4.37e@xmath17704 & 4.37e@xmath17704 + @xmath186n & 3.27e@xmath17708 & 1.77e@xmath17707 & 1.78e@xmath17707 & 2.13e@xmath17707 & 1.05e@xmath17707 & 7.01e@xmath17707 & 6.98e@xmath17707 & 3.05e@xmath17707 & 9.60e@xmath17708 + @xmath136o & 2.05e+00 & 2.79e+00 & 2.50e+00 & 4.03e+00 & 1.54e+00 & 6.96e+00 & 2.85e+00 & 9.07e+00 & 4.80e+00 + @xmath187o & 7.06e@xmath17708 & 1.49e@xmath17706 & 1.49e@xmath17706 & 4.74e@xmath17708 & 3.59e@xmath17708 & 5.24e@xmath17709 & 3.71e@xmath17709 & 8.60e@xmath17706 & 8.57e@xmath17706 + @xmath188o & 2.34e@xmath17708 & 5.27e@xmath17707 & 3.90e@xmath17707 & 2.71e@xmath17707 & 2.70e@xmath17707 & 3.01e@xmath17707 & 3.00e@xmath17707 & 6.80e@xmath17706 & 6.79e@xmath17706 + @xmath189f & 2.01e@xmath17709 & 1.49e@xmath17709 & 1.42e@xmath17709 & 7.54e@xmath17709 & 5.99e@xmath17709 & 2.97e@xmath17708 & 2.94e@xmath17708 & 9.35e@xmath17709 & 4.56e@xmath17709 + @xmath190ne & 7.61e@xmath17701 & 4.24e@xmath17701 & 3.05e@xmath17701 & 5.58e@xmath17701 & 2.86e@xmath17701 & 3.06e@xmath17701 & 2.70e@xmath17701 & 2.48e+00 & 1.43e+00 + @xmath191ne & 3.44e@xmath17705 & 9.09e@xmath17706 & 7.92e@xmath17706 & 3.66e@xmath17705 & 2.67e@xmath17705 & 1.31e@xmath17705 & 1.13e@xmath17705 & 2.35e@xmath17704 & 1.31e@xmath17704 + @xmath192ne & 5.61e@xmath17705 & 1.45e@xmath17705 & 9.07e@xmath17706 & 3.94e@xmath17705 & 2.30e@xmath17705 & 6.34e@xmath17706 & 5.16e@xmath17706 & 2.15e@xmath17704 & 1.25e@xmath17704 + @xmath193na & 2.33e@xmath17703 & 7.58e@xmath17704 & 4.70e@xmath17704 & 7.75e@xmath17704 & 3.92e@xmath17704 & 1.31e@xmath17704 & 7.59e@xmath17705 & 4.24e@xmath17703 & 2.32e@xmath17703 + @xmath194 mg & 1.62e@xmath17701 & 1.35e@xmath17701 & 1.38e@xmath17701 & 2.09e@xmath17701 & 3.80e@xmath17702 & 3.79e@xmath17701 & 1.01e@xmath17701 & 5.12e@xmath17701 & 2.18e@xmath17701 + @xmath195 mg & 1.05e@xmath17704 & 3.47e@xmath17705 & 3.41e@xmath17705 & 1.36e@xmath17704 & 8.47e@xmath17705 & 2.97e@xmath17704 & 1.20e@xmath17704 & 2.96e@xmath17704 & 1.41e@xmath17704 + @xmath196 mg & 2.09e@xmath17704 & 4.74e@xmath17705 & 4.12e@xmath17705 & 8.67e@xmath17705 & 3.03e@xmath17705 & 8.55e@xmath17705 & 2.49e@xmath17705 & 2.34e@xmath17704 & 1.16e@xmath17704 + @xmath196al & 5.29e@xmath17706 & 4.59e@xmath17706 & 4.35e@xmath17706 & 1.17e@xmath17705 & 3.51e@xmath17706 & 2.02e@xmath17705 & 8.02e@xmath17706 & 1.41e@xmath17705 & 8.44e@xmath17706 + @xmath197al & 1.48e@xmath17703 & 8.44e@xmath17704 & 8.72e@xmath17704 & 1.58e@xmath17703 & 1.91e@xmath17704 & 8.56e@xmath17703 & 1.12e@xmath17703 & 2.98e@xmath17703 & 1.01e@xmath17703 + @xmath198si & 9.31e@xmath17702 & 2.83e@xmath17701 & 1.96e@xmath17701 & 2.52e@xmath17701 & 3.31e@xmath17702 & 4.69e@xmath17701 & 9.11e@xmath17702 & 4.21e@xmath17701 & 1.48e@xmath17701 + @xmath199si & 2.69e@xmath17704 & 3.71e@xmath17704 & 4.48e@xmath17704 & 7.86e@xmath17704 & 5.88e@xmath17705 & 2.27e@xmath17703 & 2.37e@xmath17704 & 7.59e@xmath17704 & 2.16e@xmath17704 + @xmath200si & 1.20e@xmath17704 & 7.69e@xmath17705 & 6.95e@xmath17705 & 2.05e@xmath17704 & 1.90e@xmath17705 & 3.29e@xmath17703 & 3.60e@xmath17704 & 2.28e@xmath17704 & 8.25e@xmath17705 + @xmath201p & 8.11e@xmath17705 & 9.14e@xmath17705 & 6.23e@xmath17705 & 1.58e@xmath17704 & 1.53e@xmath17705 & 9.93e@xmath17704 & 1.16e@xmath17704 & 2.19e@xmath17704 & 7.74e@xmath17705 + @xmath202s & 3.91e@xmath17702 & 1.21e@xmath17701 & 7.70e@xmath17702 & 1.03e@xmath17701 & 1.77e@xmath17702 & 1.77e@xmath17701 & 4.24e@xmath17702 & 2.01e@xmath17701 & 7.70e@xmath17702 + @xmath203s & 1.44e@xmath17704 & 2.45e@xmath17704 & 1.76e@xmath17704 & 2.55e@xmath17704 & 1.92e@xmath17705 & 5.77e@xmath17704 & 5.79e@xmath17705 & 5.59e@xmath17704 & 1.58e@xmath17704 + @xmath204s & 1.61e@xmath17704 & 3.50e@xmath17704 & 5.51e@xmath17705 & 5.32e@xmath17704 & 4.36e@xmath17705 & 1.98e@xmath17703 & 2.00e@xmath17704 & 4.74e@xmath17704 & 1.57e@xmath17704 + @xmath205s & 7.55e@xmath17710 & 1.39e@xmath17710 & 6.81e@xmath17711 & 1.23e@xmath17709 & 9.11e@xmath17711 & 5.07e@xmath17708 & 4.88e@xmath17709 & 6.52e@xmath17711 & 1.78e@xmath17711 + @xmath206cl & 4.57e@xmath17705 & 6.31e@xmath17705 & 2.78e@xmath17705 & 9.85e@xmath17705 & 1.47e@xmath17705 & 1.93e@xmath17704 & 3.76e@xmath17705 & 1.61e@xmath17704 & 8.00e@xmath17705 + @xmath207cl & 2.21e@xmath17705 & 4.01e@xmath17705 & 2.24e@xmath17705 & 4.52e@xmath17705 & 6.46e@xmath17706 & 7.46e@xmath17705 & 1.59e@xmath17705 & 9.19e@xmath17705 & 2.76e@xmath17705 + @xmath205ar & 6.53e@xmath17703 & 1.82e@xmath17702 & 1.24e@xmath17702 & 1.82e@xmath17702 & 3.56e@xmath17703 & 2.95e@xmath17702 & 8.45e@xmath17703 & 3.33e@xmath17702 & 1.40e@xmath17702 + @xmath208ar & 7.42e@xmath17705 & 4.06e@xmath17704 & 2.64e@xmath17705 & 4.97e@xmath17704 & 3.75e@xmath17705 & 5.92e@xmath17704 & 6.04e@xmath17705 & 6.26e@xmath17704 & 1.74e@xmath17704 + @xmath209ar & 3.64e@xmath17711 & 1.44e@xmath17711 & 2.59e@xmath17712 & 2.12e@xmath17711 & 2.17e@xmath17712 & 2.45e@xmath17710 & 2.47e@xmath17711 & 5.33e@xmath17712 & 1.44e@xmath17712 + @xmath210k & 1.69e@xmath17705 & 6.74e@xmath17705 & 1.22e@xmath17705 & 9.09e@xmath17705 & 8.93e@xmath17706 & 9.57e@xmath17705 & 1.41e@xmath17705 & 1.14e@xmath17704 & 4.14e@xmath17705 + @xmath209k & 4.18e@xmath17709 & 7.13e@xmath17709 & 8.73e@xmath17710 & 5.14e@xmath17709 & 3.66e@xmath17710 & 9.05e@xmath17709 & 8.66e@xmath17710 & 5.80e@xmath17709 & 1.57e@xmath17709 + @xmath211k & 3.41e@xmath17706 & 1.06e@xmath17705 & 3.71e@xmath17706 & 7.99e@xmath17706 & 6.40e@xmath17707 & 1.30e@xmath17705 & 1.41e@xmath17706 & 1.62e@xmath17705 & 4.52e@xmath17706 + @xmath209ca & 4.73e@xmath17703 & 1.34e@xmath17702 & 1.01e@xmath17702 & 1.62e@xmath17702 & 3.41e@xmath17703 & 2.70e@xmath17702 & 8.62e@xmath17703 & 2.40e@xmath17702 & 1.18e@xmath17702 + @xmath212ca & 1.70e@xmath17706 & 8.77e@xmath17706 & 5.14e@xmath17707 & 1.63e@xmath17705 & 1.16e@xmath17706 & 1.47e@xmath17705 & 1.40e@xmath17706 & 2.05e@xmath17705 & 5.57e@xmath17706 + @xmath213ca & 4.60e@xmath17707 & 2.69e@xmath17707 & 2.02e@xmath17707 & 5.68e@xmath17707 & 1.57e@xmath17707 & 6.15e@xmath17707 & 2.58e@xmath17707 & 9.35e@xmath17707 & 6.54e@xmath17707 + @xmath42ca & 2.05e@xmath17704 & 1.68e@xmath17704 & 1.51e@xmath17704 & 4.93e@xmath17704 & 1.37e@xmath17704 & 8.70e@xmath17704 & 3.69e@xmath17704 & 1.26e@xmath17703 & 8.82e@xmath17704 + @xmath214ca & 1.12e@xmath17711 & 5.12e@xmath17712 & 8.54e@xmath17712 & 2.78e@xmath17712 & 9.64e@xmath17713 & 1.69e@xmath17711 & 9.43e@xmath17712 & 1.55e@xmath17712 & 4.20e@xmath17713 + @xmath43ca & 2.53e@xmath17715 & 1.03e@xmath17711 & 7.92e@xmath17712 & 4.12e@xmath17713 & 4.02e@xmath17713 & 1.71e@xmath17712 & 1.71e@xmath17712 & 3.57e@xmath17714 & 9.67e@xmath17715 + @xmath215sc & 3.20e@xmath17706 & 2.66e@xmath17706 & 3.03e@xmath17706 & 1.13e@xmath17705 & 3.11e@xmath17706 & 1.94e@xmath17705 & 8.13e@xmath17706 & 5.44e@xmath17706 & 3.61e@xmath17706 + @xmath214ti & 8.04e@xmath17706 & 8.24e@xmath17706 & 3.69e@xmath17706 & 1.71e@xmath17705 & 3.07e@xmath17706 & 1.49e@xmath17705 & 4.41e@xmath17706 & 2.69e@xmath17705 & 1.44e@xmath17705 + @xmath216ti & 1.42e@xmath17705 & 1.01e@xmath17705 & 8.39e@xmath17706 & 1.98e@xmath17705 & 5.51e@xmath17706 & 2.25e@xmath17705 & 9.55e@xmath17706 & 5.79e@xmath17705 & 4.07e@xmath17705 + @xmath43ti & 2.30e@xmath17704 & 2.53e@xmath17704 & 2.13e@xmath17704 & 5.35e@xmath17704 & 1.49e@xmath17704 & 1.03e@xmath17703 & 4.39e@xmath17704 & 1.29e@xmath17703 & 9.07e@xmath17704 + @xmath217ti & 1.71e@xmath17705 & 1.86e@xmath17705 & 1.71e@xmath17705 & 5.02e@xmath17705 & 1.39e@xmath17705 & 9.38e@xmath17705 & 3.98e@xmath17705 & 5.62e@xmath17705 & 3.93e@xmath17705 + @xmath218ti & 5.55e@xmath17713 & 1.75e@xmath17712 & 3.58e@xmath17712 & 1.93e@xmath17712 & 8.27e@xmath17713 & 7.21e@xmath17712 & 2.65e@xmath17712 & 1.69e@xmath17712 & 4.57e@xmath17713 + @xmath218v & 1.49e@xmath17711 & 9.90e@xmath17711 & 1.07e@xmath17711 & 1.17e@xmath17710 & 9.31e@xmath17712 & 1.05e@xmath17710 & 1.23e@xmath17711 & 1.44e@xmath17710 & 3.89e@xmath17711 + @xmath219v & 4.18e@xmath17705 & 3.70e@xmath17705 & 2.84e@xmath17705 & 6.22e@xmath17705 & 1.72e@xmath17705 & 8.21e@xmath17705 & 3.48e@xmath17705 & 1.31e@xmath17704 & 9.20e@xmath17705 + @xmath218cr & 2.08e@xmath17705 & 4.53e@xmath17705 & 2.01e@xmath17705 & 7.01e@xmath17705 & 1.30e@xmath17705 & 7.56e@xmath17705 & 2.71e@xmath17705 & 7.67e@xmath17705 & 4.10e@xmath17705 + @xmath220cr & 8.68e@xmath17704 & 1.57e@xmath17703 & 1.33e@xmath17703 & 1.87e@xmath17703 & 5.20e@xmath17704 & 4.58e@xmath17703 & 1.95e@xmath17703 & 3.99e@xmath17703 & 2.81e@xmath17703 + @xmath221cr & 4.44e@xmath17705 & 1.13e@xmath17704 & 9.41e@xmath17705 & 1.21e@xmath17704 & 3.25e@xmath17705 & 2.77e@xmath17704 & 1.17e@xmath17704 & 1.79e@xmath17704 & 1.24e@xmath17704 + @xmath222cr & 2.90e@xmath17710 & 9.47e@xmath17709 & 9.93e@xmath17711 & 1.93e@xmath17708 & 1.37e@xmath17709 & 3.51e@xmath17709 & 3.43e@xmath17710 & 1.59e@xmath17708 & 4.29e@xmath17709 + @xmath223mn & 9.23e@xmath17705 & 2.87e@xmath17704 & 2.26e@xmath17704 & 3.33e@xmath17704 & 8.60e@xmath17705 & 6.30e@xmath17704 & 2.65e@xmath17704 & 3.98e@xmath17704 & 2.68e@xmath17704 + @xmath222fe & 5.65e@xmath17704 & 2.42e@xmath17703 & 1.47e@xmath17703 & 2.94e@xmath17703 & 6.02e@xmath17704 & 3.70e@xmath17703 & 1.42e@xmath17703 & 2.83e@xmath17703 & 1.69e@xmath17703 + @xmath25fe & 8.41e@xmath17702 & 1.13e@xmath17701 & 1.01e@xmath17701 & 1.62e@xmath17701 & 4.51e@xmath17702 & 2.81e@xmath17701 & 1.20e@xmath17701 & 3.71e@xmath17701 & 2.61e@xmath17701 + @xmath224fe & 1.27e@xmath17703 & 1.46e@xmath17703 & 1.38e@xmath17703 & 2.67e@xmath17703 & 7.43e@xmath17704 & 4.27e@xmath17703 & 1.82e@xmath17703 & 6.23e@xmath17703 & 4.38e@xmath17703 + @xmath128fe & 3.22e@xmath17710 & 2.92e@xmath17709 & 1.73e@xmath17710 & 9.60e@xmath17709 & 6.86e@xmath17710 & 2.97e@xmath17709 & 2.99e@xmath17710 & 6.81e@xmath17709 & 1.84e@xmath17709 + @xmath225co & 9.29e@xmath17704 & 9.93e@xmath17704 & 8.51e@xmath17704 & 1.56e@xmath17703 & 4.36e@xmath17704 & 2.19e@xmath17703 & 9.33e@xmath17704 & 3.27e@xmath17703 & 2.30e@xmath17703 + @xmath128ni & 1.54e@xmath17703 & 1.79e@xmath17703 & 1.65e@xmath17703 & 3.39e@xmath17703 & 9.28e@xmath17704 & 5.04e@xmath17703 & 2.12e@xmath17703 & 6.20e@xmath17703 & 4.31e@xmath17703 + @xmath226ni & 3.51e@xmath17703 & 3.60e@xmath17703 & 3.30e@xmath17703 & 7.41e@xmath17703 & 2.07e@xmath17703 & 1.06e@xmath17702 & 4.53e@xmath17703 & 1.59e@xmath17702 & 1.12e@xmath17702 + @xmath227ni & 4.93e@xmath17705 & 4.33e@xmath17705 & 3.71e@xmath17705 & 8.34e@xmath17705 & 2.33e@xmath17705 & 1.22e@xmath17704 & 5.20e@xmath17705 & 1.92e@xmath17704 & 1.35e@xmath17704 + @xmath228ni & 5.51e@xmath17705 & 4.85e@xmath17705 & 4.03e@xmath17705 & 8.84e@xmath17705 & 2.47e@xmath17705 & 1.31e@xmath17704 & 5.59e@xmath17705 & 2.04e@xmath17704 & 1.44e@xmath17704 + @xmath44ni & 4.92e@xmath17713 & 3.38e@xmath17712 & 4.38e@xmath17712 & 4.65e@xmath17712 & 2.08e@xmath17712 & 1.23e@xmath17711 & 1.23e@xmath17711 & 7.15e@xmath17712 & 1.94e@xmath17712 + @xmath229cu & 2.26e@xmath17705 & 2.06e@xmath17705 & 1.70e@xmath17705 & 3.67e@xmath17705 & 1.02e@xmath17705 & 5.59e@xmath17705 & 2.38e@xmath17705 & 8.75e@xmath17705 & 6.15e@xmath17705 + @xmath230cu & 8.36e@xmath17707 & 8.25e@xmath17707 & 7.88e@xmath17707 & 2.37e@xmath17706 & 6.61e@xmath17707 & 4.23e@xmath17706 & 1.80e@xmath17706 & 5.59e@xmath17706 & 3.93e@xmath17706 + @xmath44zn & 5.76e@xmath17704 & 5.24e@xmath17704 & 4.72e@xmath17704 & 1.21e@xmath17703 & 3.39e@xmath17704 & 1.86e@xmath17703 & 7.93e@xmath17704 & 2.58e@xmath17703 & 1.81e@xmath17703 + @xmath231zn & 7.29e@xmath17706 & 6.25e@xmath17706 & 5.68e@xmath17706 & 1.66e@xmath17705 & 4.62e@xmath17706 & 2.83e@xmath17705 & 1.21e@xmath17705 & 3.56e@xmath17705 & 2.51e@xmath17705 + @xmath232zn & 7.52e@xmath17707 & 6.36e@xmath17707 & 5.66e@xmath17707 & 1.73e@xmath17706 & 4.83e@xmath17707 & 2.94e@xmath17706 & 1.25e@xmath17706 & 2.81e@xmath17706 & 1.98e@xmath17706 + @xmath233zn & 7.55e@xmath17708 & 6.89e@xmath17708 & 5.63e@xmath17708 & 1.33e@xmath17707 & 3.72e@xmath17708 & 1.74e@xmath17707 & 7.40e@xmath17708 & 7.00e@xmath17707 & 4.92e@xmath17707 + @xmath234zn & 1.84e@xmath17713 & 1.53e@xmath17713 & 4.52e@xmath17712 & 1.60e@xmath17712 & 1.13e@xmath17712 & 8.04e@xmath17712 & 8.04e@xmath17712 & 4.72e@xmath17712 & 1.28e@xmath17712 + @xmath235ga & 2.63e@xmath17708 & 1.92e@xmath17708 & 1.75e@xmath17708 & 5.19e@xmath17708 & 1.45e@xmath17708 & 8.18e@xmath17708 & 3.49e@xmath17708 & 1.32e@xmath17707 & 9.25e@xmath17708 + @xmath236ga & 7.46e@xmath17713 & 1.23e@xmath17712 & 2.84e@xmath17711 & 2.97e@xmath17712 & 1.72e@xmath17712 & 1.72e@xmath17711 & 1.71e@xmath17711 & 9.27e@xmath17712 & 2.63e@xmath17712 + @xmath234ge & 3.10e@xmath17708 & 2.19e@xmath17708 & 1.81e@xmath17708 & 4.93e@xmath17708 & 1.38e@xmath17708 & 7.30e@xmath17708 & 3.11e@xmath17708 & 1.12e@xmath17707 & 7.87e@xmath17708 + @xmath237ge & 4.43e@xmath17712 & 2.23e@xmath17712 & 1.08e@xmath17711 & 6.56e@xmath17712 & 2.34e@xmath17712 & 6.35e@xmath17712 & 5.98e@xmath17712 & 2.48e@xmath17711 & 6.79e@xmath17712 + @xmath238ge & 1.15e@xmath17712 & 1.52e@xmath17712 & 2.39e@xmath17711 & 6.92e@xmath17712 & 3.62e@xmath17712 & 3.00e@xmath17711 & 2.97e@xmath17711 & 1.47e@xmath17711 & 4.18e@xmath17712 + @xmath239ge & 1.27e@xmath17712 & 7.00e@xmath17713 & 5.45e@xmath17712 & 2.57e@xmath17712 & 1.84e@xmath17712 & 1.54e@xmath17711 & 1.54e@xmath17711 & 3.97e@xmath17712 & 1.11e@xmath17712 crrrrrrrrr @xmath192na & 4.74e@xmath17705 & 1.09e@xmath17705 & 6.81e@xmath17706 & 3.00e@xmath17705 & 1.54e@xmath17705 & 4.11e@xmath17706 & 3.33e@xmath17706 & 1.82e@xmath17704 & 1.06e@xmath17704 + @xmath196al & 8.17e@xmath17707 & 9.13e@xmath17707 & 1.04e@xmath17706 & 2.57e@xmath17706 & 9.78e@xmath17707 & 4.26e@xmath17706 & 1.22e@xmath17706 & 7.45e@xmath17706 & 3.74e@xmath17706 + @xmath211ca & 3.40e@xmath17706 & 1.06e@xmath17705 & 3.69e@xmath17706 & 7.94e@xmath17706 & 6.26e@xmath17707 & 1.30e@xmath17705 & 1.37e@xmath17706 & 1.62e@xmath17705 & 4.52e@xmath17706 + @xmath42ti & 2.05e@xmath17704 & 1.68e@xmath17704 & 1.51e@xmath17704 & 4.93e@xmath17704 & 1.37e@xmath17704 & 8.70e@xmath17704 & 3.69e@xmath17704 & 1.26e@xmath17703 & 8.82e@xmath17704 + @xmath226fe & 2.45e@xmath17713 & 2.25e@xmath17713 & 9.60e@xmath17713 & 5.11e@xmath17713 & 2.34e@xmath17713 & 2.74e@xmath17712 & 2.74e@xmath17712 & 1.69e@xmath17712 & 4.60e@xmath17713 + @xmath25ni & 8.41e@xmath17702 & 1.12e@xmath17701 & 1.01e@xmath17701 & 1.62e@xmath17701 & 4.51e@xmath17702 & 2.81e@xmath17701 & 1.20e@xmath17701 & 3.71e@xmath17701 & 2.61e@xmath17701 + @xmath224ni & 1.27e@xmath17703 & 1.46e@xmath17703 & 1.38e@xmath17703 & 2.66e@xmath17703 & 7.43e@xmath17704 & 4.27e@xmath17703 & 1.81e@xmath17703 & 6.22e@xmath17703 & 4.37e@xmath17703

we perform hydrodynamical and nucleosynthesis calculations of core - collapse supernovae ( sne ) and hypernovae ( hne ) of population ( pop ) iii stars . we provide new yields for the main - sequence mass of @xmath0 @xmath1 and the explosion energy of @xmath2 ergs to apply for chemical evolution studies . our hn yields based on the mixing - fallback model of explosions reproduce the observed abundance patterns of extremely metal - poor ( emp ) stars ( @xmath3 } < -3 $ ] ) , while those of very metal - poor ( vmp ) stars ( @xmath4 } < -2 $ ] ) are reproduced by the normal sn yields integrated over the salpeter initial mass function . moreover , the observed trends of abundance ratios [ x / fe ] against [ fe / h ] with small dispersions for the emp stars can be reproduced as a sequence resulting from the various combination of @xmath5 and @xmath6 . this is because we adopt the empirical relation that a larger amount of fe is ejected by more massive hne . our results imply that the observed trends with small dispersions do not necessarily mean the rapid homogeneous mixing in the early galactic halo at [ fe / h ] @xmath7 , but can be reproduced by the `` inhomogeneous '' chemical evolution model . in addition , we examine how the modifications of the distributions of the electron mole fraction @xmath8 and the density in the presupernova models improve the agreement with observations . in this connection , we discuss possible contributions of nucleosynthesis in the neutrino - driven wind and the accretion disk .

You are an expert at summarizing long articles. Proceed to summarize the following text: molecular modeling and simulation is a technology central to many areas of research in academia and industry . with the advance of computing power , the scope of application scenarios for molecular simulation is widening , both in terms of complexity of a given simulation and in terms of high throughput . nowadays , e.g. the predictive simulation of entire phase equilibrium diagrams has become feasible . however , in order to rely on simulation results , the methodology needs to be sound and the implementation must be thoroughly verified . in its first release @xcite , we have introduced the molecular simulation tool @xmath02 . results from @xmath02 have been verified and the implementation was found to be robust and efficient . as described in section [ hybrid ] , in version 2.0 of the simulation tool @xmath02 the existing molecular dynamics ( md ) mpi parallelization was hybridized with openmp , leading to an improved performance on multi - core processors . furthermore , the new release offers a wider scope of accessible properties . in particular , @xmath02 was extended to calculate massieu potential derivatives in a systematic manner , cf . section [ massieu ] . this augments the range of sampled properties significantly and , as was demonstrated in @xcite , it allows to straightforwardly develop competitive fundamental equations of state from a combination of experimental vle data and molecular simulation results . lastly , besides being now capable of simulating ionic substances , the time and memory demand for calculating transport properties was reduced significantly ( section [ algo ] ) . @xmath02 is freely available as an open source code for academic users at www.ms-2.de . the molecular simulation tool @xmath02 focuses on thermodynamic properties of homogeneous fluids . therefore , systems investigated with @xmath02 typically contain on the order of @xmath1 molecules . while for monte carlo simulations a perfect scaling behavior up to large numbers of cores can be trivially achieved , md domain decomposition the de facto standard for highly scalable md is not feasible for such system sizes , because the cut - off radius is in the same range as half the edge length of the simulation volume . this excludes domain decomposition and limits the scalability of the mpi parallelization . the present release of @xmath02 features an openmp parallelization , which was hybridized with mpi . at the point where mpi communication becomes a bottleneck , a single process still has enough load to distribute to multiple threads , improving scalability . three parts of @xmath02 were parallelized with openmp : the interaction partner search , the energy and the force calculations . all openmp parallel regions rely on loop parallelism , as the compute intensive parts of the algorithm all feature a loop over the molecules . in the force calculation , race conditions need to be considered , because every calculated force is written to both interacting molecules . introducing atomic updates or critical sections leads to massive overheads . instead , it is more efficient to assign forces from individual interactions to the elements of a list ( or an array ) which is subsequently summed up . the same holds true for torques . in figure [ hybrid ] the speed - up of hybrid mpi / openmp vs. pure mpi is plotted for 2048 cores , varying the number of threads per mpi process and the number of molecules in the simulation volume . as can be seen , using 2 to 4 threads per mpi process delivers a speed - up of around 20% for 2048 cores . the evaluation of the hybrid parallelization algorithm was performed on a cray xe6 supercomputer at the high performance computing center in stuttgart , which has an overall peak performance of one pflops . it consists of 3552 nodes , each equipped with two amd opteron 6276 ( interlagos ) processors . each processor has 16 cores , sharing eight fpus ( floating point units ) . nodes are equipped with 32 gb ram and are interconnected by a high - speed cray gemini network . additional runtime performance comparisons with the simulation tool _ gromacs _ @xcite are listed in table [ tab1 ] . cccccc cores & threads & @xmath2 & gromacs / s & @xmath02(rf ) / s & @xmath02(ew ) / s + 8 & 8 mpi & 500 & 164 & 416 & 785 + 8 & 8 mpi & 1000 & 299 & 874 & 1607 + 8 & 8 mpi & 2000 & 1284 & 4461 & 6777 + 16 & 16 mpi & 500 & 95 & 233 & 415 + 16 & 16 mpi & 1000 & 166 & 477 & 848 + 16 & 16 mpi & 2000 & 678 & 2298 & 3506 + 32 & 32 mpi & 500 & 62 & 152 & 245 + 32 & 32 mpi & 1000 & 106 & 296 & 487 + 32 & 32 mpi & 2000 & 361 & 1286 & 1898 + 64 & 64 mpi & 500 & 40 & 119 & 166 + 64 & 64 mpi & 1000 & 65 & 228 & 324 + 64 & 64 mpi & 2000 & 220 & 814 & 1261 + 128 & 128 mpi & 500 & 38 & 105 & 131 + 128 & 128 mpi & 1000 & 51 & 197 & 247 + 128 & 128 mpi & 2000 & 147 & 557 & 727 + 8 & 1 mpi , 8 omp / mpi & 500 & 167 & 483 & + 8 & 1 mpi , 8 omp / mpi & 1000 & 323 & 975 & + 8 & 1 mpi , 8 omp / mpi & 2000 & 1416 & 4831 & + 16 & 2 mpi , 8 omp / mpi & 500 & 105 & 253 & + 16 & 2 mpi , 8 omp / mpi & 1000 & 186 & 517 & + 16 & 2 mpi , 8 omp / mpi & 2000 & 763 & 2514 & + 32 & 4 mpi , 8 omp / mpi & 500 & 75 & 167 & + 32 & 4 mpi , 8 omp / mpi & 1000 & 121 & 316 & + 32 & 4 mpi , 8 omp / mpi & 2000 & 418 & 1362 & + 64 & 8 mpi , 8 omp / mpi & 500 & 60 & 119 & + 64 & 8 mpi , 8 omp / mpi & 1000 & 92 & 217 & + 64 & 8 mpi , 8 omp / mpi & 2000 & 261 & 785 & + 128 & 16 mpi , 8 omp / mpi & 500 & 49 & 101 & + 128 & 16 mpi , 8 omp / mpi & 1000 & 74 & 172 & + 128 & 16 mpi , 8 omp / mpi & 2000 & 170 & 496 & + + + + [ tab1 ] @xmath02 version 2.0 features evaluating free energy derivatives in a systematic manner , thus greatly extending the thermodynamic property types that can be sampled in single simulation runs . the approach is based on the fact that the fundamental equation of state contains the complete thermodynamic information about a system , which can be expressed in terms of various thermodynamic potentials @xcite , e.g. internal energy @xmath3 , enthalpy @xmath4 , helmholtz free energy @xmath5 or gibbs free energy @xmath6 , with number of particles @xmath2 , volume @xmath7 , pressure @xmath8 , temperature @xmath9 and entropy @xmath10 . these representations are equivalent in the sense that any other thermodynamic property is essentially a combination of derivatives of the chosen form with respect to its independent variables . the form @xmath11 , known as the massieu potential , is preferred in molecular simulations due to practical reasons @xcite . the statistical mechanical formalism of lustig allows for the simultaneous sampling of any @xmath12 in a single @xmath13 ensemble simulation for a given state point @xcite @xmath14 where @xmath15 is the gas constant , @xmath16 and @xmath17 . @xmath18 can be separated into an ideal part @xmath19 and a residual part @xmath12 @xcite . the calculation of the residual part is the target of molecular simulation and the derivatives @xmath20 , @xmath21 , @xmath22 , @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 were implemented in @xmath02 for @xmath13 ensemble simulations . the ideal part can be obtained by independent methods , e.g. from spectroscopic data or ab initio calculations . however , it can be shown that for any @xmath28 , where @xmath29 , the ideal part is either zero or depends exclusively on the density , thus it is known by default @xcite . note that the calculation of @xmath30 still requires additional concepts such as thermodynamic integration or particle insertion methods . from the first five derivatives @xmath31 , @xmath32 , @xmath33 , @xmath34 , @xmath35 every measurable thermodynamic property can be expressed ( see the supplementary material for a list of properties ) with the exception of phase equilibria . a detailed description of the implementation is in the supplementary material , here , only an overview is provided . + the calculation of the derivatives up to the order of @xmath36 requires the explicit mathematical expression of @xmath37 and @xmath38 with respect to the applied molecular interaction pair potential and has to be determined analytically beforehand @xcite . the general formula for @xmath39 can be found in ref . @xcite . for common molecular interaction pair potentials , like the lennard - jones potential @xcite , describing repulsive and dispersive interactions , or coulomb s law , describing electrostatic interactions between point charges , the analytical formulas for @xmath37 and @xmath38 can be obtained straightforwardly . + as molecular simulation is currently limited to operate with considerably fewer particles than real systems , the effect of the small system size thus has to be counter - balanced with a contribution to @xmath40 and @xmath39 called long range correction ( lrc ) @xcite . the mathematical form of the lrc depends on the molecular interaction potential and the cut - off method ( site - site or center - of - mass cut - off mode ) applied . for the lennard - jones potential , the lrc scheme was well described in the literature for both the site - site @xcite and the center of mass cut - off mode @xcite . the reaction field method @xcite was the default choice in the preceding version of @xmath02 for the lrc of electrostatic interactions modelled by considering charge distributions on molecules . the usual implementation of the reaction field method combines the explicit and the lrc part in a single pair potential @xcite from which @xmath39 ( including the lrc contribution ) is directly obtainable . however , practical applications show that the electrostatic lrc of @xmath37 and @xmath38 can be neglected in case of systems for which the reaction field method is an appropriate choice . e.g. , the contribution of the electrostatic lrc for a liquid system ( @xmath41 k and @xmath42 mol / l ) consisting of only 200 water and 50 methanol molecules with a very short cut - off radius of @xmath43 of the edge length of the simulation volume is still @xmath44 for both @xmath37 and @xmath38 . the supplementary material contains detailed elaborations on the lrc for the lennard - jones potential . [ [ transport - property - calculations ] ] transport property calculations + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in @xmath02 , transport properties are determined via equilibrium md simulations by means of the green - kubo formalism @xcite . this formalism offers a direct relationship between transport coefficients and the time integral of the autocorrelation function of the corresponding fluxes . an extended time step was defined for the calculation of the fluxes , the autocorrelation functions and their integrals . the extended time step is @xmath45 times longer than the specified md time step , where @xmath45 is a user defined variable . the autocorrelation functions are hence evaluated in every @xmath45-th md time step . as a consequence , the memory demand for the autocorrelation functions was reduced and the restart files , which contain the current state of the autocorrelation functions and time integrals , become accordingly smaller . in addition , the overall computing time of the md simulation was reduced significantly . + [ [ ewald - summation ] ] ewald summation + + + + + + + + + + + + + + + ewald summation @xcite was implemented for the calculation of electrostatic interactions between point charges . it extends the applicability of @xmath02 to thermodynamic properties of e.g. ions in solutions . in ewald summation , the electrostatic interactions according to coulomb s law are divided into two contributions : short - range and long - range . the short - range term includes all charge - charge interactions at distances smaller than the cut - off radius . the remaining contribution is calculated in fourier space and only the final value is transformed back into real space . this allows for an efficient calculation of the long - range interactions between the charges . the algorithm is well described in literature . currently , some of the new features , the calculation of massieu potential derivatives and hybrid mpi & openmp parallelization for md , are not available together with ewald summation . [ [ radial - distribution - function ] ] radial distribution function + + + + + + + + + + + + + + + + + + + + + + + + + + + + the radial distribution function ( rdf ) @xmath46 is a measure for the microscopic structure of matter . it is defined by the local number density around a given position within a molecule @xmath47 in relation to the overall number density @xmath48 @xmath49 therein , @xmath50 is the differential number of molecules in a spherical shell volume element @xmath51 , which has the width @xmath52 and is located at the distance @xmath53 from the regarded position . @xmath46 can be evaluated for every molecule of a given species . in the present release of @xmath02 , the rdf can be calculated during md simulation runs for pure components and mixtures on the fly . the rdf is sampled between all lj sites . in order to evaluate rdfs for arbitrary positions , say point charge sites , superimposed dummy lj sites with the parameters @xmath54 have to be introduced in the potential model file by the user . [ [ electric - conductivity ] ] electric conductivity + + + + + + + + + + + + + + + + + + + + + the evaluation of the electric conductivity @xmath55 was implemented in @xmath02 version 2.0 , being a measure for the flow of ions in solution . the green - kubo formalism @xcite offers a direct relationship between @xmath55 and the time - autocorrelation function of the electric current flux @xmath56 @xcite @xmath57 where @xmath58 is boltzmann s constant . the electric current flux is defined by the charge @xmath59 of ion @xmath60 and its velocity vector @xmath61 according to @xmath62 where @xmath63 is the number of molecules of component @xmath64 in solution . note that only the ions in the solution have to be considered , not the electro - neutral molecules . for better statistics , @xmath55 is sampled over all independent spatial elements of @xmath56 . [ [ thermal - conductivity - of - mixtures ] ] thermal conductivity of mixtures + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in the previous version of @xmath65 the determination of the thermal conductivity by means of the green - kubo formalism was implemented for pure substances only . in the present release , the calculation of the thermal conductivity was extended to multi - component mixtures . the thermal conductivity @xmath66 is given by the autocorrelation function of the elements of the microscopic heat flow @xmath67 @xmath68 in mixtures , energy transport and diffusion occur in a coupled manner , thus , the heat flow for a mixture of @xmath45 components is given by @xcite @xmath69\cdot \mathbf{v}_i^k \\ & -&\frac{1}{2}\sum_{i=1}^{n } \sum_{j=1}^{n } \sum_{k=1}^{n_i } \sum_{l \neq k}^{n_j } \bm{r}_{ij}^{kl}\cdot \big ( \mathbf{v}_i^k \cdot \frac{\partial u\left(r_{ij}^{kl}\right)}{\partial \bm{r}_{ij}^{kl}}+\mathbf{w}_i^k \mathbf{\gamma}_{ij}^{kl}\big)- \sum_{i=1}^{n } h_{i}\sum_{k=1}^{n_i } \mathbf{v}_{i}^{k}\mbox { , } \label{heatflow } \end{aligned}\ ] ] where @xmath70 is the angular velocity vector of molecule @xmath60 of component @xmath71 and @xmath72 its matrix of angular momentum of inertia . @xmath73 is the intermolecular potential energy and @xmath74 is the torque due to the interaction of molecules @xmath60 and @xmath75 . the indices @xmath71 and @xmath64 denote the components of the mixture . @xmath76 is the partial molar enthalpy . it has to be specified as an input in the @xmath65 parameter file and can be calculated from @xmath77 simulations . [ [ residence - time ] ] residence time + + + + + + + + + + + + + + the residence time @xmath78 defines the average time span that a molecule of component @xmath64 remains within a given distance @xmath79 around a specific molecule @xmath71 . it is given by the autocorrelation function @xmath80 where @xmath81 is the time , @xmath82 the solvation number around molecule @xmath71 at @xmath83 and @xmath84 is the heaviside function , which yields unity , if the two molecules are within the given distance , and zero otherwise . following the proposal of impey et al . @xcite , the residence time explicitly allows for short time periods during which the distance between the two molecules exceeds @xmath79 . also , the solvation number @xmath85 can be evaluated on the fly @xmath86 where @xmath87 is the number density of component @xmath64 and @xmath88 is the distance up to which the solvation number is calculated . the authors gratefully acknowledge financial support by the bmbf `` 01ih13005a skasim : skalierbare hpc - software fr molekulare simulationen in der chemischen industrie '' and computational support by the high performance computing center stuttgart ( hlrs ) under the grant mmhbf2 . the present research was conducted under the auspices of the boltzmann - zuse society for computational molecular engineering ( bzs ) . 19 natexlab#1#1[1]`#1 ` [ 2]#2 [ 1]#1 [ 1]http://dx.doi.org/#1 [ ] [ 1]pmid:#1 [ ] [ 2]#2 , , , , , , , , , : ( ) . , , , , : ( ) . 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a new version release ( 2.0 ) of the molecular simulation tool _ ms_2 [ s. deublein et al . , comput . phys . commun . 182 ( 2011 ) 2350 ] is presented . version 2.0 of _ ms_2 features a hybrid parallelization based on mpi and openmp for molecular dynamics simulation to achieve higher scalability . furthermore , the formalism by lustig [ r. lustig , mol . phys . 110 ( 2012 ) 3041 ] is implemented , allowing for a systematic sampling of massieu potential derivatives in a single simulation run . moreover , the green - kubo formalism is extended for the sampling of the electric conductivity and the residence time . to remove the restriction of the preceding version to electro - neutral molecules , ewald summation is implemented to consider ionic long range interactions . finally , the sampling of the radial distribution function is added .

You are an expert at summarizing long articles. Proceed to summarize the following text: dense nano - particle systems have been shown to exhibit collective behavior , as evidenced by aging , and non - equilibrium effects similar to spin glasses.@xcite whether that collective behavior is associated with a `` true '' spin glass phase transition is still controversial . in this paper , we show a dynamic scaling analysis to a spin glass transition for a highly concentrated fe - c nano - particle sample with a narrow particle size distribution . we also discuss why the scaling analysis may not indicate a phase transition for samples with less interactions or wider size distributions . the sample consisted of ferromagnetic nanoparticles of amorphous @xmath5 ( x@xmath60.22 ) , with an average particle size of @xmath7 nm , prepared by the method described by van wonterghem et al . in ref . the sample was studied in the frozen state and contained 17 vol% of particles . the ac - susceptibility measurements were performed in a non - commercial squid magnetometer for frequencies in the range @xmath8 17 mhz - 170 hz . fig . 1 shows @xmath9 and @xmath10 at different frequencies . a sample that exhibit a spin glass transition will show critical slowing down , and hence the characteristic relaxation time @xmath11 diverges at the transition temperature according to @xmath12 where @xmath13 is the transition temperature , @xmath14 is related to the relaxation time of the individual particle magnetic moments , and @xmath15 is a critical exponent . we extracted the freezing temperature @xmath16 , associated with a relaxation time ( @xmath17 ) , from the out - of - phase component of the ac - susceptibility as @xmath18 with @xmath19 . we also tried other criteria to make sure that the choice of criterion is not significantly influencing the results of the critical slowing down analysis . the critical slowing down analysis gives @xmath1 , @xmath20 s , and @xmath21 k ( see fig . we also performed a full scaling of @xmath3 according to , @xmath22 and found data collapse to a single function @xmath23 , for different frequencies , using @xmath2 ( see fig . 3 ) . the value of @xmath15 compares quite well with values found for spin glasses with long range interactions ( rkky ) , while the value of @xmath24 is slightly larger than typical spin glass values , but is consistent with the value of @xmath25 found by jonsson et al . @xcite for an interacting nanoparticle sample by a static scaling analysis . the dynamic scaling analysis will only reveal a phase transition if the single particle contribution to @xmath3 is vanishingly small for @xmath4 , i.e. all slow dynamics is due to collective behavior . two criteria have to be fulfilled for this to be possible ; i ) the interparticle interactions need to be strong and ii ) the particle size distribution needs to be narrow . if we compare the sample used for the scaling analysis with the same sample but much more dilute ( 0.05 vol.% ) we can see that the out - of - phase component is almost zero for @xmath4 ( see inset of fig . we conclude that the concentrated sample is appropriate to use for scaling analysis . 9 j. l. dormann , et al . , j. magn . magn . mater . * 187 * , l139 ( 1998 ) . h. mamiya , i. nakatani , and t. furubayashi , phys . lett . * 82 * , 4332 ( 1999 ) . p. jnsson , m. f. hansen , and p. nordblad , phys . b * 61 * , 1261 ( 2000 ) j. van wonterghem , s , mrup , s.w . charles , and s wells , j. colloid interface sci . * 121 * , 558 ( 1988 ) .

a highly concentrated ( 17 vol.% ) fe - c nano - particle system , with a narrow size distribution @xmath0 nm , has been investigated using magnetic ac susceptibility measurements covering a wide range of frequencies ( 17 mhz - 170 hz ) . a dynamic scaling analysis gives evidence for a phase transition to a low temperature spin - glass - like phase . the critical exponents associated with the transition are @xmath1 and @xmath2 . the reason why the scaling analysis works for this sample , while it may not work for other samples exhibiting collective behavior as evidenced by aging phenomena , is that the single particle contribution to @xmath3 is vanishingly small for @xmath4 and hence all slow dynamics is due to collective behavior . this criterion can only be fulfilled for a highly concentrated nano - particle sample with a narrow size distribution .

You are an expert at summarizing long articles. Proceed to summarize the following text: the study of the internal dynamics of stellar systems plays an essential role in astronomy . from the observed positions and velocities of the stars in galaxies and globular clusters it is possible to infer their total ( dark+luminous ) mass distribution , which , in particular , provides information on the presence and properties of dark halos and massive black holes . in turn , this structural knowledge constrains theories for the formation and evolution of these systems . the dynamical state of a stellar system is determined by its phase space distribution function , @xmath3 , which counts the stars as a function of position @xmath4 and velocity @xmath5 . typically , however , only three of the six phase - space coordinates are available observationally : the projected sky position @xmath6 , and the velocity @xmath7 along the line of sight ( los ) . proper motion observations can provide the additional velocities @xmath8 , but such data are generally not available ( with the notable exception of some galactic globular clusters ) . to make progress with the limited information available , the dynamical theorist is often forced to make simplifying assumptions about geometry ( e.g. , that the system is spherical ) or about the velocity distribution ( e.g. , that it is isotropic ) . such assumptions can have strong effects on the inferred mass distribution ( @xcite ) . to obtain the most accurate results it is therefore important to make models that are as general as possible . of particular importance for collisionless , unrelaxed systems such as galaxies is to constrain the velocity anisotropy using available data , rather than to assume it a priori . in a collisionless system the distribution function satisfies the collisionless boltzmann equation . analytical methods to find solutions of this equation usually rely on the jeans theorem , which states that the distribution function must depend on the phase - space coordinates through integrals of motion ( quantities that are conserved along a stellar orbit ) . in a spherical system all integrals are known analytically , namely , the energy @xmath9 and the components of the angular momentum vector @xmath10 . analytical models for spherical systems are therefore fairly easily constructed . in an axisymmetric system things are more complicated ( e.g. , @xcite ) . only two integrals are known analytically , @xmath9 and the vertical component @xmath11 of the angular momentum vector denote the coordinates intrinsic to the axisymmetric stellar system , with the plane @xmath12 being the equatorial plane , and @xmath13 the symmetry axis . these relate via the inclination @xmath14 to the observable coordinates @xmath6 on the plane of the sky ( aligned , respectively , along the projected major and minor axes of the stellar system ) , and @xmath15 the line - of - sight direction , positive in the direction away from us . ] , but there is generally a third integral for which no analytical expression exists . therefore , it is not generally possible to construct an axisymmetric model analytically . the special class of so - called ` two - integral ' ( @xmath16 ) models ( e.g. , @xcite ) has its uses ( e.g. , @xcite ) , but these have an isotropic velocity distribution in their meridional plane , which need not be a good fit to real dynamical systems . the most practical way to model a general axisymmetric system is to do it numerically . while a few methods exist to do this ( e.g. , @xcite ) , the most common approach uses schwarzschild s ( 1979 ) method . one starts with a trial guess for the gravitational potential @xmath17 and then numerically calculates an orbit library that samples integral space in some complete and uniform way . the orbits are integrated for several hundred orbital periods , and the time - averaged intrinsic and projected properties ( density , los velocity , etc . ) are stored as the integration progresses . the construction of a model consists of finding a weighted superposition of the orbits that : ( 1 ) reproduces the observed stellar or surface brightness distribution on the sky ; and ( 2 ) reproduces all available kinematical data to within the observational error bars . additional constraints can be added to enforce that the distribution function in phase space be smooth and reasonably well behaved , e.g. , through regularization or by requiring maximum entropy . several axisymmetric schwarzschild codes have been developed in the last decade ( e.g. , @xcite ) . these codes deal with the situation in which information on the line - of - sight velocity distribution ( losvd ) is available for a set of positions on the projected plane of the sky . this is the case , e.g. , when the kinematical data are from long - slit or integral - field spectroscopic observations of unresolved galaxies . the optimization problem for such data can be reduced to a linear matrix equation for which one needs to find the least - squares solution with non - negative weights @xcite . one dimension of the matrix corresponds to the number of orbits in the library , while the other corresponds to the number of ( luminosity , kinematical and regularization ) constraints that must be reproduced . both dimensions are typically in the range @xmath18@xmath19 . nonetheless , efficient numerical algorithms exist to find the solution , which yield the orbital and the velocity distribution of the model , as well as the @xmath20 of the fit to the kinematical data . the procedure must then be iterated with different gravitational potentials , to determine the potential that provides the overall best @xmath20 . the existing codes have been used and tested extensively ( e.g. , @xcite ) . some questions remain , e.g. , about the importance of smoothing in phase space , the exact meaning of the confidence regions determined using @xmath21 contours , and , in some situations , valid concerns have been raised regarding whether the available data contain enough information so as to warrant the conclusions of the schwarzschildmodeling @xcite . nevertheless , on the whole schwarzschild codes have now been established as an accurate and versatile tool to study a wide range of dynamical problems . a disadvantage of the existing codes is that they can not be easily applied to the large class of problems in which the kinematical observations come in the form of discrete velocity measurements , rather than as losvds . this is encountered , e.g. , when modeling the dynamics of galaxies at large radii , where the low - surface brightness prevents integrated - light spectroscopy . the only available data are then often of a discrete nature , e.g. , via the los velocities of individual stars in galaxies of the local group ( e.g. , @xcite ) , or via planetary nebulae ( e.g. , @xcite ) and globular clusters ( e.g. , @xcite ) surrounding giant ellipticals . the kinematical data available for clusters of galaxies , consisting of redshifts for individual galaxies , are of a similarly discrete nature ( e.g. , @xcite ) . the typical datasets in all these cases consist of tens to hundreds of los velocities . galactic globular clusters constitute another class of object for which kinematical data is often available only as discrete measurements , rather than in the form of losvds . from ground - based observations , data sets of individual los velocities can be available for up to thousands of stars in these systems ( e.g. , @xcite ) , and for @xmath22 cen it has been possible to assemble large samples of proper motions as well @xcite . with the capabilities of _ hst _ , accurate proper motion data sets with up to @xmath23 stars are now becoming available for several more galactic globular clusters ( e.g. , @xcite ) . note that discrete datasets do not necessarily provide better or worse information than datasets obtained from integrated - light measurements . both types of data have their advantages and disadvantages . for discrete datasets , for example , interloper contamination can be a problem ( see also the end of section [ sec : logl ] below ) . by contrast , for integrated - light measurements , it is often difficult to constrain the wings of the losvd due to uncertainties associated with continuum subtraction . which type of data is most appropriate and most easily obtained depends on the specific object under study . this is therefore not a question that we address in this paper . instead , we focus on the issue of how to best analyze discrete data , if that happens to be what is available . analyses of discrete datasets have often been more simplified than the analyses that are now common for integrated - light data . for example , the observations are analyzed using the jeans equations ( e.g. , @xcite ) , often with the help of data binning to calculate rotation velocity and velocity dispersion profiles ( see , however , the `` spherical '' schwarzschild models of m87 of @xcite ) . the disadvantage of such an approach is that not all the information content of the data is used , including information on deviations of the velocity histograms from a gaussian . such deviations are important because they constrain the velocity dispersion anisotropy of the system ( e.g. , @xcite ) . this anisotropy is an important ingredient in some existing controversies , e.g. regarding the presence of dark halos around elliptical galaxies @xcite . loss of information can be avoided when large numbers of datapoints are available , as is often the case for globular clusters . it is then possible to create velocity histograms for binned areas on the projected plane of the sky , after which analysis can be done with existing schwarzschild codes ( e.g. , @xcite ) . while this is possible for large datasets , such an approach is not viable for the more typical , smaller datasets that are often available . the availability of schwarzschild codes that can fully exploit the information content of such smaller datasets would therefore be valuable to advance this subject . motivated by these considerations we set out to adapt our existing schwarzschild code @xcite to deal with discrete datasets . this does not constitute a trivial change , since it changes the constrained superposition procedure from a linear matrix problem to a more complicated maximum likelihood one . for each observed velocity of a particle in the system the question becomes : what is the probability that this velocity would have been observed if the model is correct ? the overall likelihood of the data , given a trial model , is the product of these probabilities for all observations . such likelihood problems have previously been solved for spherical systems @xcite and the special class of axisymmetric @xmath24 systems @xcite . however , for the axisymmetric schwarzschild modeling approach the problem corresponds to finding the minimum of a function in a space with a dimension of @xmath18@xmath19 . we show in this work , via the schwarzschildmodeling of simulated datasets , that this problem can indeed be solved successfully and efficiently . moreover , we follow @xcite and implement in our new code the ability to calculate and fit proper motions in addition to los velocities . applications of the code to real datasets will be presented in forthcoming papers . the structure of the paper is as follows . in section [ sec : logl ] we phrase the new problem of fitting a schwarzschildmodel to a dataset of discrete velocities ( of one , two , or three dimensions ) of individual kinematic tracers in terms of a likelihood formalism . section [ sec : code ] describes the implementation of the discrete fitting procedure into our existing schwarzschildcode . at the same time , we summarize here the major steps involved in the construction of the probability matrix that describes the likelihood of a given kinematic data point belonging to some particular orbit of the library . we then present in section [ sec : tools ] sets of simulated data that we use for the purpose of testing the performance of the discrete schwarzschildcode . we also describe the known input distribution functions from which these data were drawn . the application of the code to the simulated datasets is presented in section [ sec : tests ] . we present a thorough analysis of the accuracy with which our discrete schwarzschildcode recovers the known distribution function , mass - to - light ratio and inclination used to generate the simulated data . finally , in section [ sec : end ] we summarize our findings and present our conclusions . in the schwarzschildscheme the properties of every orbit @xmath25 in the orbit library are computed and stored . the modeling consists in finding the superposition of orbital weights @xmath26 , i.e. , the fraction of particles in the system residing in each orbit , that best reproduces some set of constraints . the weights are written as squares to ensure that they never become negative . linear constraints are of the form @xmath27 here @xmath28 is a constraint value that needs to be reproduced , @xmath29 is its uncertainty , and @xmath30 is its model prediction @xmath31 the matrix @xmath32 represents here , for orbit @xmath25 , the probability distribution corresponding to the constraint @xmath33 . the constraints are generally one of the following : ( a ) the integrated light ( surface brightness ) of a stellar population in some aperture number in the projected plane of the sky , necessary to reproduce an observational measurement of the surface brightness ; ( b ) the mean los velocity , velocity dispersion , or for data of sufficient quality , a higher - order gauss - hermite moment in some aperture number in the projected plane of the sky , necessary to reproduce an observational measurement of the stellar kinematics ; ( c ) the integrated mass in some meridional @xmath34 plane grid point , necessary to provide a consistent model ; ( d ) a combination of distribution function moments in some meridional @xmath34 plane grid point , if a model with a particular dynamical structure is desired ( e.g. , one may want a model with @xmath35 equal to zero in order to simulate a two - integral @xmath24 model ) ; ( e ) a combination of orbit weights , if regularization constraints are desired to enforce smoothness of the model in phase space ( e.g. , one can set the n - th order divided difference of adjacent orbit weights to zero , with an uncertainty @xmath36 that measures the desired amount of smoothing ) . it is natural to choose the best - fitting model to be the one that produces the maximum likelihood . to determine the likelihood we need to write down an expression for the probability of measuring @xmath33 among all its possible values . to do this , we recall that any model is not an attempt to reproduce a set of observations to infinite accuracy , but instead to do it within the uncertainty @xmath37 . for observational constraints , such as those in ( a ) and ( b ) above , @xmath37 is equal to the measurement uncertainty . for other constraints , such as those in ( c)-(e ) above , @xmath37 can be used as a forcing parameter that compels how accurately the likelihood needs to peak around a particular value of @xmath33 . if one assumes that these uncertainties have a normal ( gaussian ) distribution , then the probability we are interested in is given by @xmath38 . \ ] ] the combined probability for the simultaneous occurrence of all @xmath39 linear constraints is then given by the product of the single probabilities , @xmath40 . using equation ( [ eq.non.linear.term ] ) , the logarithm of this linear part of the likelihood is therefore @xmath41 the first sum on the right - hand side of this expression does not depend on the orbital weights @xmath26 and , therefore , does not affect the likelihood maximization . the second term has the exact form of the @xmath20 statistic . maximizing the likelihood is therefore equivalent to the minimization of this @xmath20 . this can be done by finding the solution of the set of equations ( [ eq.constraint ] ) and ( [ eq.gamma.def ] ) , which can be rewritten as an overdetermined matrix equation . this matrix equation can be solved with the use of standard non - negative least - squares ( nnls ) algorithms ( see @xcite for a detailed description ) . in the case of discrete data , however , the introduction of constraints of a `` non - linear '' type is inevitable in order to adequately exploit the entire information content available , avoiding restrictive simplifications and loss of information due to binning . this occurs because the individual probabilities do not necessarily have the simple , gaussian form of equation ( [ eq.non.linear.term ] ) . the procedure for finding the maximum likelihood then can not be cast as the solution of a linear matrix equation anymore . suppose we have a kinematic dataset consisting of discrete measurements which we are trying to model using the schwarzschildtechnique . let @xmath42 be the phase - space probability distribution of any given orbit , properly averaged azimuthally , and normalized such that @xmath43 . we use @xmath44 to denote a vector of up to six euclidean spatial and velocity coordinates . whenever @xmath44 is shorter than 6 elements , it is understood that the distribution has been marginalized over the missing dimensions . then the total probability of drawing a particle from a superposition of orbits representing the whole system is @xmath45 we now need to consider the total probability of the ensemble of @xmath46 particles with kinematic information that constitute our discrete dataset . before this , however , it is necessary to make the distinction , in the language of probabilities , between the possible modes of sampling of the tracers available in a system of particles . the two main possibilities depend on whether the particles are randomly or non - randomly drawn from their spatial distribution , and we may refer to these , respectively , as random positional sampling and incomplete positional sampling . additionally , particles may be drawn with or without velocity information , thus adding up to a total of four possibilities . the case with incomplete positional sampling and no velocity information , however , does not provide any useful constraint to the analysis and therefore we restrict the discussion to the remaining three cases . for particles drawn randomly from the spatial distribution with no velocity information , the probability @xmath47 is @xmath48 where * r * represents a 2 or 3 dimensional position . this type of dataset could result from imaging of the resolved populations of a stellar system , where the positional information could be used as actual constraints . this would force the model to fit the underlying spatial distribution of discrete tracers , instead of making use of a parametrization of the ( continuous ) brightness profile of the system . of course , a dataset without velocity information can not by itself constrain the dynamical state or the mass of the system . in the case of random positional sampling including velocity information , particles are randomly drawn from both the spatial and velocity distributions . in this case , @xmath47 has the form @xmath49 where @xmath50 is the same as above and @xmath51 represents a general 1 , 2 , or 3 dimensional velocity . this would be the case when being able to obtain the velocities of particles in a given field without introducing any spatial or velocity bias , such as the proper motions of all stars ( brighter than some magnitude limit ) in a sufficiently sparse stellar cluster , or when los velocities are obtained for a complete ( or possibly magnitude - limited ) set of globular clusters or planetary nebulae in a galaxy . in contrast , having _ incomplete _ positional sampling means that the particles are drawn from a velocity distribution , with _ a priori _ fixed positions . this can occur , for example , when because of the usually limited availability of telescope time and resources , los velocities are measured only for stars within some distance from the photometric major or minor axes of a galaxy , or when because of the finite size of fibers in a fiber - fed spectrograph , not all the potentially observable kinematic tracers in the field can be actually acquired . incomplete positional sampling also arises when , even though particles can be randomly drawn spatially , this is the case only for a limited area . this occurs , for example , when the observations have to avoid the innermost regions of a galaxy or globular cluster , where , because of crowding , stars can not be individually resolved . in these case , @xmath47 has the form @xmath52 where , rather than just @xmath26 , the effective weights when summing together the individual orbital distributions are @xmath53 . once the individual probabilities for all possible cases of spatial sampling that comprise the data have been properly assigned , we can proceed to the construction of the total probability of observing the entire dataset . let @xmath54 and @xmath55 be the number of observational data points obtained under the mode of random positional sampling without and with velocity information , respectively , and @xmath56 the number of data points obtained with incomplete positional sampling with kinematic information . then , the total probability is simply the product of the individual probabilities , with logarithm given by @xmath57 . using equations ( [ eq.prob.r ] ) to ( [ eq.prob.rfix ] ) , and adopting the abbreviated notation @xmath58 , @xmath59 , and @xmath60 ( all known for each orbit @xmath25 and particle @xmath14 from the orbit library calculation ; see 3 ) , the quantity to maximize becomes @xmath61 joining the results in equations ( [ eq : loglinear ] ) and ( [ eq.total.likelihood.1 ] ) , the complete log - likelihood for a general application of the schwarzschildmethod , which is the full expression to be maximized with respect to the orbital weights @xmath62 , is the sum of the log - likelihoods for linear and discrete constraints @xmath63 finding the maximum likelihood corresponds to finding the solution of @xmath64 = 0 , for all @xmath65 . denoting @xmath66 , the expression for the first derivative is @xmath67 one important question that remains is regarding the estimation of confidence regions around the parameters of the best - fitting model , i.e. , the ( statistical ) uncertainties around the likelihood maximum in the case of non - linear constraints . recalling that maximizing @xmath68 is equivalent to minimizing the quantity @xmath69 , it is easy to realize that , if the probabilities involved in equation ( [ eq.total.likelihood.1 ] ) were all of gaussian form , then @xmath70 would simply reduce to the well known @xmath20 statistic , as we have already seen for the case with linear constraints in equation ( [ eq : loglinear ] ) . when dealing with non - linear constraints , however , the likelihood does not reduce to a simple @xmath20 form . nevertheless , one still can use another well known theorem of statistics which , used before by @xcite and @xcite , states that the `` likelihood - ratio '' statistic @xmath71 does tend to a @xmath20 statistic in the limit of large @xmath46 , with the number of degrees of freedom equal to the number of free parameters that have not yet been varied and chosen so as to optimize the fit . therefore , the likelihood - ratio statistic reduces to the @xmath72 statistic for @xmath73 , even though the probabilities in equation ( [ eq.total.likelihood.1 ] ) are not all individually gaussian . since in the present work we explore datasets consisting of 100 kinematic measurements or more , the condition of large @xmath46 should be reasonably fulfilled . therefore , following the likelihood - ratio statistic , we assume @xmath74 , and compute the confidence regions around the best - fit parameters in the usual way ( e.g. , @xcite ) , i.e. , with the @xmath75 error for a single parameter corresponding to wherever @xmath76 , and so forth . other approaches to quantify the uncertainties exist as well , e.g. , using bayesian statistics , but these are generally more difficult to implement ( e.g. , @xcite ) . the equations described above assume that any possible `` interloper '' contaminants have already been removed , and that the targets with observed velocities that enter the likelihood equations all belong to the system under study . for realistic datasets , contamination by interlopers can certainly be a problem @xcite ; i.e. , targets that happen to lie close to the line - of - sight of the stellar system under study and are difficult to reject from the sample . however , efficient interloper rejection schemes do exist for various types of samples and these have been well - described in the literature @xcite . moreover , the use of empirically - calibrated selection criteria ( independent of the measured velocity ) can produce extraordinarily clean samples for kinematic analysis @xcite . either way , interloper rejection is best discussed in the context of specific data sets . we therefore do not discuss it further in the present paper . interloper rejection for discrete data sets can also be built in as part of the likelihood analysis @xcite , so a simple modification of the likelihood equations given above could deal with interlopers explicitly . however , we have not yet explored this in the present context . given equations ( [ eq.logl ] ) and ( [ eq.1st.deriv ] ) , fitting a schwarzschildmodel to the data requires the following two steps : ( a ) determination of all the individual probabilities @xmath77 and matrix elements @xmath32 , so that the only unknowns in equation ( [ eq.1st.deriv ] ) are the coefficients @xmath78 ; and ( b ) performing the maximization of the total likelihood , i.e. , finding the set of orbital weights @xmath78 that satisfies @xmath64 = 0 , for all @xmath65 , and therefore best fits the available constraints . the elements of the matrix @xmath32 , corresponding to the linear constraints discussed in [ sec : logl ] , are calculated in the same way as in the old ( continuous ) implementation of the code , and for them we refer to @xcite , @xcite and @xcite . in what follows we concentrate on the probabilities @xmath77 associated with the discrete treatment that is the subject of this work . the matrix elements @xmath77 in equation ( [ eq.1st.deriv ] ) , which keep track of the probability that orbit @xmath25 of the library would have produced the measurement @xmath14 of the dataset ( each @xmath25 corresponding to some combination of the three integrals of motion @xmath9 , @xmath79 , and @xmath1 ) , are stored as the orbit in question is being computed . that is , at every time step during the orbit integration , we check whether the position and velocity along the orbit is consistent with any of the observational datapoints . to accomplish this , it is necessary to implement some degree of _ smoothing _ , both in position and velocity space , since otherwise the probability of having a particle on an orbit at exactly the observed position and velocity would be infinitesimally small . smoothing in the spatial coordinates is accomplished through the definition of an _ aperture _ around the position of each particle in the dataset , with the size of the aperture controlling the amount of smoothing . the optimal aperture size will be somewhat dependent on the sampling characteristics of the data . in general , apertures should not be too small , or otherwise few time steps during orbit integration will fall on any one of them . this would lead to large shot noise in the computed probabilities @xmath77 , unless the orbits are integrated for very long times . nor should the apertures be too large , so that information on the orbital structure of the model is not erased by excessive spatial smoothing . the choice of aperture shape is arbitrary and a matter of numerical convenience . we adopt square apertures as in previous implementations of the code ( long - slit observations naturally produce data for rectangular apertures ) , and set their sizes to a user - supplied fraction of @xmath80 , the radius in the projected plane at the aperture s position . once the spatial apertures are defined , and every time the projected position along the orbit being integrated falls within an aperture , we need to keep track of whether the orbital velocity matches the observed velocity . in the old ( continuous ) implementation , the losvd was computed and stored for every orbit @xmath25 at each aperture @xmath14 , with the size of the bins in the histogram determining the amount of smoothing in velocity space . in our discrete treatment of the problem , @xmath77 would simply be the histogram value for the bin that contains the observed velocity . a direct , though information ally incomplete , generalization of this implementation to kinematical data in three - dimensions would be to keep track of two additional histograms at each aperture to account for @xmath81 and @xmath82 . this has been done by @xcite and @xcite , who calculated moments of the three model velocity distributions and fitted them to those obtained from binning los and proper - motion observations of stars in @xmath22 cen and m15 , respectively ( note that these studies still handle the data in a continuous fashion , by reducing the initially discrete datasets to binned velocity distributions at a number of apertures on the sky , an approach only possible thanks to the very large number of stars with measured velocities in these systems ) . while reproducing the three - dimensional mean velocities and dispersions of the stars in a stellar system is already an improvement over all previous implementations of the schwarzschildtechnique , doing so is nevertheless a simplification of the problem . the reason is that it implicitly assumes that the three velocity components are independent of each other , i.e. , it does not account for the fact that there is a velocity ellipsoid whose cross terms are , in the most general case , not identical to zero . the most complete treatment would be to store a cube with entries for all possible combinations of @xmath83 , and do this at each spatial aperture where there is kinematical data available . this implementation would be , however , expensive in terms of memory storage and , moreover , not absolutely necessary , simply because we are not interested in the entire probability cube . instead , we only need probabilities in the cases when the model velocities are close to the observed ones . thus , in the framework of velocity histograms or full velocity cubes , and because of the discrete nature of the data , the large majority of the bins or entries would be filled with weights that do not affect the likelihood in equation ( [ eq.logl ] ) . therefore , we adopt an approach in which , instead of storing velocity histograms or cubes , every time an orbit @xmath25 passes through an aperture @xmath14 with kinematical data , we add a gaussian contribution to @xmath77 . this contribution is centered on the observed ( any - dimensional ) velocity and has a dispersion that reflects the measurement errors , and if desired , any amount of extra velocity smoothing . thus , denoting the actually measured components of the particle s velocity in aperture @xmath14 as @xmath84 and their associated uncertainties as @xmath85 , with @xmath86 corresponding to @xmath87 , @xmath88 , and @xmath89 , the multiplicative contribution @xmath90 to the probability has the form @xmath91},\end{aligned}\ ] ] where @xmath92 is the component @xmath93 of the velocity of a test particle on orbit @xmath25 . the quantity @xmath94 is the numerical smoothing assigned to velocity component @xmath93 . whenever a particular component @xmath93 of the velocity is not available , we set @xmath95 . finally , in order to account for the fact that we represent a continuous orbit by a discrete sequence of time steps , we weigh this gaussian factor by multiplying it by the timestep @xmath96 . therefore , for every orbit @xmath25 , and every time the orbit integration falls within an aperture , the probability is increased according to @xmath97 when the integration of orbit @xmath25 is done , the @xmath77 elements for all datapoints ( apertures ) are written to a file for later use by the algorithm that performs the maximization of the likelihood . in most practical applications one can set @xmath98 , since the error bars @xmath85 on the data already provide sufficient natural smoothing for numerical efficiency . we do this throughout the rest of this paper . however , we note that there may be situations in which non - zero @xmath94 may be beneficial . for example , if the observational errors @xmath85 are much smaller than the velocity dispersions @xmath37 of the system . it then takes very long integrations to beat down the shot noise in the orbital distributions @xmath77 . addition of a numerical smoothing @xmath94 with @xmath99 can then speed up the calculations without affecting the accuracy of the results . the approach of equations ( [ eq.weights ] ) and ( [ eq.pij ] ) assumes that the errors @xmath85 for the different datapoints are uncorrelated . sometimes this is not true , as in the case of the proper motions of stars in the globular cluster @xmath22 cen , where relative rotation between the old photographic plates used in their derivation produce an artifact overall rotation of the cluster @xcite . if problems like these can not be removed before modeling , a more complicated treatment than the one described here will be necessary . the non - linear nature of the discrete problem addressed in this paper requires the use of a non - linear optimizer , and there is no guarantee of a unique optimum . after experimentation with various optimization algorithms , we settled on the toms 500 conjugate gradient optimizer of @xcite . this code uses the function value and gradient to optimize along successive vectors ( lines ) in the space of the orbital weights , choosing the optimization direction at every pass in a manner that attempts to minimize the number of such line minimizations needed ( see chapter 10 of @xcite for details on conjugate gradient methods ) . in our code , we rely on the fact that the majority of orbits do not contribute to any particular linear constraint , or to the likelihood of any particular observational datum . in the notation of equation ( [ eq.1st.deriv ] ) , the linear constraints @xmath100 , and also the @xmath101 , are sparse matrices . accordingly , the code to evaluate the gradient in equation ( [ eq.1st.deriv ] ) is written to store and evaluate only non - zero terms of @xmath100 and @xmath102 , reducing the computational burden by a factor of four or five . to evaluate convergence and estimate the proximity of our final likelihood maximum to the true ( possibly local ) maximum , we plot the magnitude of the improvement of the likelihood @xmath103 as a function of the number of function evaluations @xmath46 . see figure [ fig : mkfitin ] . we find that @xmath103 is well represented by an exponential relation @xmath104 , where @xmath105 . therefore , the future change in the likelihood if the optimizer were allowed to run forever would be @xmath106 , where @xmath107 is the current change in likelihood at step @xmath108 . in practice , we terminate the optimization at @xmath109 , so that we expect to be within an additive factor of @xmath110 of the true likelihood maximum . this typically occurs after a number @xmath111 of function evaluations . the final accuracy is merely linear in the exponential coefficient @xmath112 , so that this accuracy estimate should be reasonably robust . we ran a variety of tests in order to establish whether the algorithm has a tendency of finding local extrema as opposed to global ones . in particular , for some of the test cases to be discussed later in [ sec : tests ] , we started the iterative algorithm from different initial conditions , to verify that the solutions thus obtained were always in ( statistical ) agreement . also , as will be shown in [ sec : tests ] , we find that the algorithm recovers the properties of known input models with reasonable accuracy . while this does not prove that the schwarzschildcode can not end up in a local maximum , at least it shows that the code does not end up in ( potential ) local maxima that are far from the correct solution . in practice we usually start the maximization procedure from a homogeneous set of initial mass weights . we also investigated whether the convergence to a solution can be sped up by starting the iterative process from initial conditions that may already be reasonably close to the final solution . for example , we ran tests starting from a set of weights corresponding to a two - integral df of the form @xmath24 that already fit the light ( surface brightness ) profile followed by the input data . such a solution is easily obtained as the nnls solution of a matrix equation . we found that the same final answer was reached in essentially the same number of iterations . in order to test the performance of our discrete schwarzschildcode , we generate sets of simulated data drawn from a known phase - space distribution function ( df ) . unlike the case of using actual observations of a real stellar system , this approach offers the advantage of unambiguously knowing in advance the input properties underlying the data , which an optimally - working code should be able to `` recover '' . it also provides flexibility by allowing the possibility of adapting the input data at will in order to test different aspects of the code ( [ sec : tests ] ) . we discuss here the construction of various sets of pseudo - data and the properties of the underlying models . our simulated input data are obtained from a set of @xmath24 dfs derived by @xcite , with the methodology for drawing n - body initial conditions from these dfs described in @xcite . the models have a constant mass - to - light ratio @xmath113 , and have neither a central black hole or extended dark halo . they provide good fits to available photometric and kinematic observations of the galaxy m32 over the radial range from @xmath114 arcsec . however , this property has no bearing on the present analysis . we only use the fact that there is a known df , and not that this df resembles any realistic stellar system . a two - integral @xmath24 df provides a useful test case ( see also @xcite , @xcite ) , and does not mean that the model results would be less valid for more general dfs . also , the use of a constant @xmath113 is motivated only to simplify the test environment . central black holes ( e.g. , @xcite ) and extended dark halos ( e.g. , @xcite ) can be easily implemented in any schwarzschildcode . the luminous mass density is assumed to be axisymmetric and is parameterized according to @xmath115^{\beta}[1+(m / c)^2]^{\gamma},\ ] ] with @xmath116 . here , @xmath117 is the ( constant ) intrinsic axis ratio , related to the projected ( observed ) axis ratio @xmath118 via the inclination angle @xmath14 , @xmath119 . the parameters in equation ( [ eq : light ] ) are set to @xmath120 , @xmath121 , @xmath122 , @xmath123 , @xmath124 , @xmath125 , and @xmath126 , with the @xmath127-band luminosity density @xmath128@xmath129pc@xmath130 , and @xmath131 the mass - to - light ratio in the @xmath127-band and in solar units . the adopted distance is 0.7 mpc . the models share the property of appearing the same in projection on the sky , but correspond to different intrinsic axis ratios as determined by the inclination angle @xmath14 . the even part @xmath132 of the df @xmath24 is uniquely determined by the mass density @xmath133 ( e.g. , @xcite ) . to specify the odd part @xmath134 of the df we follow @xcite and write @xmath135,\ ] ] with @xmath136 being the angular momentum of a circular orbit of energy @xmath9 in the equatorial plane ( @xmath137 ) , and the auxiliary function @xmath138 defined by @xmath139 & \mbox{$(u < 0)$}. \end{array}\right.\ ] ] the choice of the parameters @xmath140 and @xmath141 determines the degree of streaming of the dataset . these free parameters can have values in the ranges @xmath142 and @xmath143 , with @xmath140 controlling the fraction of stars in the equatorial plane with clock - wise rotation , and @xmath141 controlling the behavior of the stellar streaming with orbital shape . the family of functions @xmath138 is shown in figure 1 of @xcite . combinations of @xmath144 that fit data for m32 are also discussed in that paper . here we explore a variety of input datasets with different amounts of mean streaming and test the recovery of these properties by our discrete schwarzschildcode . we generated 6 different datasets to test our discrete schwarzschildcode . by dataset we mean a number of particle @xmath6 positions on the sky with corresponding proper motions @xmath145 and los velocities @xmath146 . for two chosen inclinations on the sky , @xmath147 and @xmath148 , we produced three datasets resembling systems with varying degrees of rotation : a non - streaming system ( @xmath149 and @xmath150 ) , a maximally - streaming system ( @xmath151 and @xmath152 ) , and a third system with intermediate streaming ( @xmath151 and @xmath153 ) . we label our different datasets as 90ns , 90is , and 90ms to indicate the non - streaming , intermediate - streaming , and maximally - streaming cases of @xmath147 , respectively . similarly , for the @xmath148 case , we have the 55ns , 55is , and 55ms datasets . the mass - to - light ratio used to generate the datasets is @xmath154 for @xmath147 and @xmath155 for @xmath148 , in units of @xmath156/@xmath157 . although we examined the performance of our schwarzschildcode with tests that involve all of the six simulated datasets introduced above , we chose to use the 55is dataset to show most of our results . figure [ fig : data55is ] shows some projections of the phase - space coordinates for the 55is dataset . in order to quantitatively judge the performance of the three - integral schwarzschildcode , it is desirable to make a comparison between the properties of the input df ( i.e. , that from which the pseudo - data were obtained ) and those of the fitted df ( i.e. , that found as the solution to the fitting or minimization problem ) . it is important to note in this context that the direct output of our schwarzschildcode is not in the form of a proper df @xmath158 , but rather in the form of `` mass weights '' @xmath159 associated to each set of integrals of motion @xmath160 that uniquely define an orbit . the relation between the df and the orbital mass weights is through a volume element dependent on the three integrals and an integration over the 3-dimensional space associated to the particular orbit ( see @xcite for a detailed discussion ) . such a conversion can be done in schwarzschild codes ( e.g. , @xcite ) , but this is not necessary for the goals of the present paper . we therefore limit ourselves to the comparison between the input and the fitted orbital mass weight distributions , which from now on we denote by @xmath161 and @xmath162 , respectively . to validate the weights @xmath162 returned by the schwarzschildcode , we need to know the weights @xmath161 for the model df @xmath24 . this is not a simple problem in the absence of an analytic expression for @xmath1 . however , two related functions are more easily accessible . the first is @xmath163 , defined as the projection of @xmath161 over the @xmath164 plane ( i.e. , integrated over @xmath1 ) . having the means of drawing n - body initial conditions from the df @xcite , we know that the energy and z - component of the angular momentum of each particle are given by @xmath165 and @xmath166 , respectively . therefore , @xmath163 is easily obtained by binning a large n - body dataset @xmath167 in @xmath9 and @xmath79 . the second related function that is easily accessible is @xmath168 , the distribution of mass weights for an @xmath24 model of axial ratio @xmath117 and a power - law density profile with logarithmic slope @xmath70 in a spherical kepler potential . this function is calculated analytically in de bruijne et al . ( 1996 ; their equation ( 38 ) ) , and has the form @xmath169.\ ] ] here , @xmath70 is the logarithmic slope of the mass distribution and @xmath170 is a known function . the spherical kepler potential is of course only an accurate approximation to our model at asymptotically large radii . nonetheless , we can combine @xmath163 and @xmath168 to obtain a reasonable approximation for @xmath161 throughout the system , namely @xmath171 with @xmath172}{\int j_{\lambda}\left[l_z / l_{z,{\rm max}}(e),i_3\right]{\rm d } i_3}.\ ] ] for @xmath70 we take the slope of the mass distribution of equation ( [ eq : light ] ) at @xmath173 , the radius of the circular orbit of energy @xmath9 in the equatorial plane @xmath174 . the function @xmath175 in equation ( [ eq : zetain ] ) is correct ( i.e. , reduces to @xmath176 ) when projected on the @xmath164 plane , and has approximately the correct distribution over @xmath1 at fixed @xmath177 . in this way , we construct sets of orbital mass weights for each of our 6 simulated datasets described in [ sec : data ] . using all the kinematic ( pseudo ) datasets and their corresponding input dfs described in [ sec : tools ] we now proceed to examine how accurately the discrete schwarzschildcode can recover the properties of the galaxy models used to generate the input datasets . by _ recovery _ we mean to determine how close or how far is the obtained solution from the known df , known mass - to - light ratio @xmath113 , and known inclination @xmath14 of the galaxy model corresponding to the simulated dataset that was provided as input to the code . at the same time , we investigate the reliability of the uncertainties provided by the code on each of these properties . in the general case of modeling real observations of an actual stellar system , the true radial mass density profile is not known a priori and is typically described following some parameterization . since mass may not necessarily follow light , or may do so in some complicated way , different plausible mass models should be attempted , as well as allowing for possible variations of the mass - to - light ratio with position . for the purposes of the present tests , however , the underlying mass distribution is assumed to be perfectly known from equation ( [ eq : light ] ) , except for the value of @xmath113 . therefore , the assumed parameterization for the mass distribution is only a 1-parameter family , and includes the `` correct '' distribution @xmath178 . in applications to real data , higher - parameter families may be necessary , and there is no guarantee that any member of the family would provide a good approximation to the true underlying distribution . the results of our tests are examined via three different exercises , which can be performed on each of the 6 different input datasets , providing a good baseline to judge the performance of our discrete schwarzschildcode . first , we explore the recovery of the internal orbital structure of the input dataset ( i.e. , the input df , or more specifically , the input mass weights @xmath179 ) by feeding the code with the correct inclination and mass - to - light ratio @xmath113 ( [ sec : getdf ] ) . second , we fix the inclination to the correct value of the input dataset and study whether the code finds the minimum of the @xmath21 function at the correct value of @xmath113 ( [ sec : getml ] ) . and third , we explore grids of schwarzschildmodels with different ( @xmath14,@xmath113 ) combinations to study how well these two properties are recovered when they are both assumed unknown ( [ sec : grids ] ) . we run all the above exercises for various subsets of each of our 6 datasets in order to explore the results as a function of relevant observational variables , particularly the size of the input dataset and the type of kinematical constraints available ( i.e. , only los velocities , only proper motions , or the complete three - dimensional velocities ) . this adds even more elements for a thorough assessment of the code s performance . it also provides insights into the types of datasets that will be necessary to constrain @xmath14 or @xmath113 to some given uncertainty in realistic situations . our schwarzschildcode has the capability of computing and storing , during a single orbit integration , the orbital properties for a series of different values of @xmath113 . thus , during the construction of the orbit library , different values of @xmath113 are converted into a dimensionless factor @xmath180 that multiplies all our original velocities , thus with @xmath113 scaling simply as @xmath181 . we stress that this allows us to explore several values of @xmath113 while computing only one orbit library . in our tests , we explore models for velocity factors in the range @xmath182 . given that our galaxy models with different inclinations have slightly different mass - to - light ratios @xmath131 , the use of this dimensionless representation also facilitates the visualization of the results in [ sec : getml ] and [ sec : grids ] . the correct ( input ) value of @xmath113 is always at @xmath183 . at each of its different steps , the schwarzschildcode requires the user to specify several settings ( or dials ) that control a corresponding number of tasks and functions of the modeling procedure . here we list the settings that we use for our standard run . we concentrate on the settings that are new to the discrete implementation . all other settings that are needed to fit a schwarzschildmodel ( e.g. , the resolution and limits of the polar grids used to compute the gravitational potential @xmath17 ; the required numerical accuracies in the fitting of the mass in the meridional plane and/or the projected plane of the sky ; etc . ) are identical to previous implementations of the code , so for those we refer to @xcite and @xcite . at the heart of the schwarzschildmethod lies the generation of a comprehensive library of orbits that should be representative of all types of orbits possible in the gravitational potential under study . this is achieved by adequately sampling the ranges of values that the three integrals of motion @xmath160 can acquire , each set of values uniquely determining one possible orbit . in this work we build models using two libraries that only differ in their size . most of our runs consist of the generation of initial conditions and libraries with @xmath184 orbits , obtained by sampling the available integral space with 20 energies @xmath9 , 14 angular momenta @xmath79 ( 7 positive and 7 negative ) , and 7 third integrals @xmath1 . in order to study the dependency of the results on the size of the orbit library , we also compute schwarzschildmodels using a much larger orbit library , with @xmath185 combinations of @xmath160 . the energy @xmath9 is sampled via the corresponding radius @xmath186 of the circular orbit of that energy ( that with maximum angular momentum ) in the equatorial plane @xmath174 . this radius is logarithmically sampled from a minimum value that we choose to be much smaller than the spatial resolution of the data , to a maximum value set much beyond the point at which most of the mass of the input distribution is actually encompassed . since totally unconstrained by the data , therefore , the few first and last energy bins will mostly be of no interest ( i.e. , no mass gets assigned to them in the process of optimization ) . the vertical component of the angular momentum , @xmath79 , is linearly sampled using the variable @xmath187 , where @xmath188 and @xmath189 is the angular momentum of the circular orbit with energy @xmath9 . while orbits with both positive and negative @xmath79 are included in the library , the latter need not be individually integrated because they are simply obtained by reversing the velocity vector at each point along the orbit . the third integral @xmath1 is sampled via an angle @xmath190 , where @xmath191 determines the position at which the `` thin tube '' orbit at the given @xmath177 touches its zero - velocity curve ( defined by the equation @xmath192 , where @xmath193 is the effective gravitational potential ; see @xcite for a detailed presentation ) . finally , in order to help alleviate the discrete nature of the numerical orbit library , some extra radial smoothing of the orbits is performed by randomly generating a small variation to the energy and computing and storing the contribution to the probabilities from the `` new '' orbit with integrals @xmath194 . it is possible to implement similar smoothing in @xmath79 and @xmath1 as well ( e.g. , @xcite ) , but we leave this for a future version of our code . this energy smoothing is repeated , at each timestep , for 7 random @xmath195 values . azimuthal averaging is also performed by randomly drawing 7 @xmath196 values at each timestep . smoothing in phase - space is accomplished with the use of apertures ( [ sec : pij ] ) . the size of the ( squared - shaped ) spatial apertures are defined as a fraction of @xmath80 , the distance to the center of the stellar system in the projected plane , and we set this fraction to 10% . in velocity space , and for most practical applications , the measurement errors @xmath85 themselves will provide sufficient `` natural '' smoothing for numerical purposes . thus , we set the factors @xmath94 in equation [ eq.weights ] to zero for all our tests ( see also discussion in [ sec : pij ] ) . in practice , the optimal value of @xmath94 will depend on the characteristics of the data ( particularly the size of the velocity errors ) as well as the stellar system under study . when dealing with actual data , therefore , at least a few different values should be tried in order to explore their impact on the results . additionally , the extra smoothing provided by @xmath94 can also be useful to explore the validity of the quoted errors in any given application . the uncertainties @xmath85 in the los velocities and/or proper motions are in practice determined by the details of the observations and , since obtained by different techniques ( spectroscopy versus astrometry ) , are of different size in general . furthermore , the uncertainties in the velocities tangential to the plane of the sky are affected by the uncertainty in the distance to the stellar system under study . here , however , since we deal with simulated data , we assume kinematical data of nowadays typical good quality , and simply set all these errors to a moderate and arbitrary value of @xmath197@xmath198 . the large majority of our tests were done on the simulated datasets as described in [ sec : data ] _ without _ the addition of simulated observational errors ( i.e. , random gaussian deviates with dispersion @xmath85 ) . this simplification was made early on in our project , based on the fact that the velocity errors should not matter much as long they are much smaller than the average one - dimensional velocity dispersion of the system under study . however , we realized later that this does induce a slight bias in our estimated mass - to - light ratios . our typical simulated datasets have dispersions of 48.4@xmath198and 46.3@xmath198 , for the 55is and 90is cases , respectively . therefore , by not adding any random velocity errors , the one - dimensional velocity dispersion of the pseudo - data that we actually analyzed is too small by a factor @xmath199 . as a consequence of the virial theorem , it follows that we should expect to infer a mass - to - light ratio that is too small by a factor of @xmath200 , corresponding to about 2.2% and 2.4% for the 55is and 90is datasets , respectively . instead of rerunning all our calculations , which would have been computationally expensive , we therefore simply corrected for this bias _ so when studying the recovery of the mass - to - light ratio in [ sec : getml ] and [ sec : grids ] , instead of comparing the inferred values to the value @xmath131 of the input model , we compare to the slightly smaller @xmath201 . this quantity is @xmath202 for @xmath203 and @xmath204 for @xmath205 . the sizes of currently existing kinematic datasets of discrete nature range from a few hundred datapoints ( red giants in local group dwarf galaxies , planetary nebulae in the outskirts of giant ellipticals ) to a few thousands ( stars in galactic globular clusters , systems of globular clusters around giant ellipticals ) . for our standard tests we adopt datasets with 1000 kinematic observational points , although we also study the consequences of studying datasets with sizes ranging from 100 to 2000 datapoints . in these tests , the small-@xmath46 datasets are subsets of the largest dataset ( @xmath206 ) , which means that there will be some correlation between the results of experiments done as a function of the number of available observations . this approach , we note , is of no substantial difference than having all the datasets of different @xmath46 but within the same simulation to be completely disjoint . the progression with @xmath46 should still follow the expected @xmath207 statistical - convergence behavior ( see fig . [ fig : errors_ml ] and [ sec : getml ] ) . the generation of one of our smaller @xmath208 orbit libraries , simultaneously storing discrete probabilities for a set of 1000 observational points with both los velocities and proper motions , takes 2.5 hours on a 3.6 ghz , pentium 4 , 64-bit cpu with 2 gb memory . an additional 0.5 hours are needed to find the maximum likelihood fit to the data . in practice , these steps must be iterated over a grid of gravitational potential parameterizations . in order to determine whether the best - fitting solution obtained by the discrete schwarzschildcode actually resembles the properties of the input data , we start by making detailed comparisons between the input and fitted dfs . to do this , we feed the code with the correct inclination and mass - to - light ratio @xmath113 used to generate the input datasets , and compare the fitted mass weights @xmath209 to those corresponding to the input data , @xmath161 , approximated using equation ( [ eq : zetain ] ) . we use datasets with 1000 los velocities and proper motions , and present results for both the small and big orbit libraries detailed in [ sec : settings ] . the comparison is best achieved via the analysis of corresponding one- and two - dimensional projections of the cubes of mass weights @xmath162 and @xmath161 , obtained by integrating over two and one of the integrals of motion , respectively ( figs . [ fig:1dplots ] to [ fig:2dplot55is ] ) . also , we make comparisons of two - dimensional @xmath210 slices of both cubes at selected values of the energy ( fig . [ fig : ebins55is ] ) . for al of these projections we quantify the agreement between fits and input data by computing the rms and median absolute deviation of the quantity @xmath211 , i.e. , the difference between fit and input mass weights normalized by the input mass weights . these statistics are listed for schwarzschildmodels run on all our input datasets in table 1 . since the rms can be biased disproportionately by a small number of large outliers , in our discussion below we use preferentially the median absolute residual . figure [ fig:1dplots ] shows , for the 55is case , the integrated mass weights as a function of each of the three integrals of motion , for both the input dataset and the discrete schwarzschildfit . inside the region actually constrained by kinematic data ( containing 99.83% of the total mass ) , the mean absolute deviations between the fitted and input distributions of mass weights are 3% , 16% , and 18% , for the integrated distributions as a function of @xmath9 , @xmath11 , and @xmath1 , respectively . as listed in table 1 , similar numbers are obtained for the other 5 simulated datasets , with the agreement between both distributions as a function of energy always better than 5% . as a function of @xmath79 , the largest disagreement actually corresponds to the one shown in figure [ fig:1dplots ] , the 55is case . it goes down to 7% for our case of closest agreement , the case labeled 55ns . the net rotation inherent to the 55is dataset ( reflected in the middle panel by all the mass weights with positive @xmath79 being larger than those with negative @xmath79 ) is clearly reproduced by the schwarzschildfit . as a function of the third integral , the median absolute deviation varies from 16% for the 55ns case to up to 25% for the 90ns case . note that , since we are showing orbital mass weights instead of the actual df , the @xmath1 distributions are not expected to be constant over @xmath1 , even though the input df underlying all simulated datasets is of the form @xmath24 . next , integrating only over @xmath1 , we show in figures [ fig:2dplot55ns ] and [ fig:2dplot55is ] the agreement between the fitted and input sets of mass weights as a function of @xmath9 and @xmath79 , for the 55ns and 55is cases , respectively . the upper panels of these figures show the results of the schwarzschildfit ( @xmath209 ) and the lower panels the original input distributions ( @xmath175 ) . the left - hand panels show the results for a @xmath160 library of @xmath212 orbits , 8 times larger ( i.e. , finer ) than that of the right - hand panels , which correspond to our standard case of @xmath208 orbits . only the energy range constrained by the respective sets of data is shown . black corresponds to zero weight , and the white ( brightest ) color in each pair of panels ( fit and model , or upper and lower ) has been assigned to the maximum orbital weight among the two panels , so that the comparison between fits and models is made using the same color scale . both figures [ fig:2dplot55ns ] and [ fig:2dplot55is ] show that the main features of the input @xmath164 distributions of mass weights are well reproduced by the 3-integral schwarzschildfits . in particular , the mean streaming properties of both datasets are satisfactorily recovered . in figure [ fig:2dplot55ns ] , the two prominent phase - space blobs occupying symmetrical locations on the negative and positive sides of the @xmath79-axis correspond well with the non - rotating overall nature of the 55ns dataset . moreover , this is recovered by both the models with standard and large orbit libraries ( right- versus left - hand panels ) . similarly , in figure [ fig:2dplot55is ] , the single phase - space blob at positive @xmath79 with a pronounced elongation towards negative @xmath79 ( in light blue and blue ) , indicative of the rotating nature of the 55is case , is reproduced by the schwarzschildfit as well . the median absolute deviations between the fitted and input @xmath164 distributions , always restricted to the energy range constrained by the data , are 14% and 19% for the 55ns and 55is cases , respectively ( table 1 ) . in figure [ fig : ebins55is ] we show the 3-dimensional distributions of mass weights of our 55is case in the form of a series of @xmath210 planes at different energies . here again , the upper panels show the results of the discrete schwarzschildfit ( @xmath209 ) , the lower panels the distribution of mass weights corresponding to the input data ( @xmath179 ) , and the color scale is set up in the same way as in the @xmath164 figures . as the energy @xmath9 is sampled via the radius @xmath186 of the circular orbit ( its value in arcmin indicated at the top of each pair of panels ) , this series of planes shows the variation of the @xmath210 distribution with increasing distance from the center of the galaxy . the fraction of the total mass at each energy slice is given as a percentage at the bottom of each panel . the bottom panels of figure [ fig : ebins55is ] indicate that , in the inner regions ( inside 0.2 arcmin ) , most of the mass in the 55is dataset is concentrated in orbits with @xmath79 near zero . the corresponding upper panels show that the schwarzschildfit recovers this @xmath213 component , but it distributes more weight than the input model into orbits with positive @xmath79 . these inner regions , nevertheless , have a relatively low mass content in comparison with regions at larger radii . as the radius increases , the @xmath214 region of phase - space gets progressively depleted of stars in favor of orbits with high @xmath79 . this transition is reasonably well reproduced by the schwarzschildsolution , and the agreement between fit and input data becomes better at large radii , at which point most of the mass at each energy is concentrated in orbits of high @xmath79 . note also that a common characteristic of figures [ fig:2dplot55ns ] , [ fig:2dplot55is ] , and [ fig : ebins55is ] is that schwarzschildfits typically present mass weight distributions that appear broader ( more extended ) and less peaked than the corresponding distributions displayed by the pseudo - data . the effect is most obvious among the right - most panels of figure [ fig : ebins55is ] , where one can see that the @xmath210 mass - weight distributions of the input data ( lower panels ) have higher peaks and overall sharper features than the corresponding fitted distributions ( upper panels ) . this is an expected effect and is due to the combined smoothing of the fitted distribution introduced both by the ( necessary ) use of velocity apertures for the computation of likelihoods ( see [ sec : pij ] ) , and by the regularization constraints imposed in order to enforce smoothness in phase space . while the first smoothing is particular to our discrete implementation , the second is a well - known procedure common to most schwarzschildcodes . models without regularization tend to be unrealistically noisy @xcite and unreliable for parameter estimation @xcite . thus , although we choose to plot the input distribution of mass weights as they actually are , the most fair of comparisons would be one in which the schwarzschildfit is compared with a smoothed version of the original mass weight distribution describing the input data . we explored this by convolving the input distribution of mass weights with a ( circular ) gaussian kernel , and then computing the same statistics shown in table 1 ( but this time using the smoothed version of the input distribution ) for different widths of the gaussian kernel . we have verified that indeed it is possible to find a kernel width for which the agreement between fit and input data is best , improving both the rms and mean absolute deviations of table 1 by factors between 1.2 and 1.5 . finally , we also note that the comparison in figure [ fig : ebins55is ] might be affected by the accuracy of the approximation in equation ( [ eq : zetain ] ) , which means that the values in table 1 are actually upper limits to the true accuracy of the schwarzschildfits . from these tests we conclude that our discrete schwarzschildcode can successfully recover the original df inside the region constrained by the kinematic data , at least for the case in which the inclination and mass - to - light ratio are assumed known . for a large range of potential applications of a schwarzschildcode , such as investigating dark matter halos in galaxies , the most important property that one is interested in measuring with confidence is the mass - to - light ratio . in the present tests , this quantity is a scalar , @xmath113 , although in more general applications it could be a function of radius . in this section we study in detail the capacity of our code to infer the correct @xmath113 when the inclination of the system is assumed known . tests were performed for a number of input datasets in order to investigate the dependence of the results on key observational variables such as the number of kinematic measurements and the type of kinematic constraints available ( i.e. , only - los velocities , only proper motions , as well as both los velocities and proper motions ) . all models in this section were computed using our small orbit library , with @xmath208 combinations of @xmath160 . the results of these experiments are summarized in figures [ fig : mlparabn]-[fig : errors_ml ] . for the 90is and 55is cases and using full 3-dimensional velocity information , figure [ fig : mlparabn ] shows the @xmath21 parabolae obtained when applying the discrete schwarzschildcode with a number of @xmath113 values , distributed around the correct one ( @xmath215 ) , for datasets of varying sizes . the quantity @xmath216 along the ordinate denotes the velocity scaling ; @xmath217 corresponding to a schwarzschildmodel with the input value @xmath215 , defined in [ sec : settings ] . the zero point of the vertical axis ( both in figures [ fig : mlparabn ] and [ fig : mlparab2 ] ) is arbitrary , but the difference @xmath72 between points on the same curve has its usual statistical meaning , and indeed we compute the ( random ) uncertainties on the determination of @xmath113 directly from them . figure [ fig : mlparabn ] shows that the difference @xmath72 between points on the same curve becomes larger ( the parabolae become narrower ) as the number of available kinematic measurements increases . the determination of the best - fit @xmath113 also depends on the type of available kinematic measurements . this is illustrated in figure [ fig : mlparab2 ] , where we plot the @xmath21 parabolae obtained when considering only los velocities , only proper - motions , or the full 3-dimensional velocity information . all cases are for the 55is dataset with 1000 kinematic measurements . in this case , the @xmath72 parabolae become narrower as the number of available velocity components increases . furthermore , the statistical errors are generally smaller for larger datasets , as well as when more velocity components are available . this is shown in figure [ fig : errors_ml ] , where we plot the behavior of the best - fit @xmath113 and its uncertainties as a function of @xmath218 , where @xmath46 is the number of datapoints . the uncertainties @xmath219 displayed in the upper panel of figure [ fig : errors_ml ] represent the @xmath75 intervals around the minimum of the parabolae in figures [ fig : mlparabn ] and [ fig : mlparab2 ] , and are defined as half the distance between the points on the curve where @xmath220 with respect to the minimum . the statistical errors scale roughly as @xmath207 over an interval of 1.3 dex in @xmath221 . also , the errors in the best - fit @xmath113 associated to datasets with only proper - motions ( triangles ) are smaller than those associated to only - los datasets ( open circles ) for any value of @xmath46 . in other words , our discrete schwarzschildcode satisfies the fundamental statistical expectation that it should become easier for the method to distinguish between models with different @xmath113 when the amount of observational information is larger . in the case of datasets with the full 3-d velocity information , the 55is uncertainties do not quite seem to follow the @xmath207 behavior expected from statistics . we attribute this to our tests having reached a fundamental floor due to the discrete nature of the models , a limit that can not be overcome by increasing the number @xmath46 of available measurements . this can cause an apparent flattening with respect to the regular @xmath207 behavior at large @xmath46 . to test the robustness of the errors estimated as above , we performed the following exercise . selecting 10 different ( disjoint ) realizations of the n - body data ( for the 55is case with 1000 measurements of only line - of - sight velocities , the case most often found in practice ) , we repeated the exercise of figures [ fig : mlparabn ] and [ fig : mlparab2 ] and computed discrete schwarzschildmodels for a set of different @xmath113 values distributed around the correct input one . this was done using our small orbit library . we obtained an average best - fit @xmath113 of 2.46 ( less than @xmath75 away from the input value , @xmath222 ) , with an rms of 0.074 ( corresponding to about 3% ) . when computing the statistical uncertainties using the @xmath223 parabolae as described above , the average @xmath75 error in the best - fit @xmath113 of the set of experiments turns out to be 0.204 , equivalent to a fractional error of 8% . this is a factor of 2.5 larger than the scatter in the results from multiple independent realizations of the pseudo - data . this gap is smaller when additional information about the individual kinematics of the tracers is available . indeed , repeating the above exercise for the same datasets but now using two - dimensional proper - motions instead of only line - of - sight velocities , the average error in @xmath113 computed from our @xmath21 parabolae is 0.112 , a factor of 1.7 larger than the scatter of the best - fit values , which was 0.067 . therefore , we conclude that our error estimation using @xmath72 is conservative . despite the smaller statistical errors for the case with proper - motions alone , the bottom panel of figure [ fig : errors_ml ] indicates that the best - fit @xmath113 is closer to the real value , @xmath215 , for the case with only - los velocities . while the best - fit @xmath113 from datasets with only los velocities are well within @xmath75 of @xmath215 for any @xmath46 , this is not the case for the datasets with only proper - motions , with best - fit @xmath113 values that are @xmath224 away from @xmath215 . still , the formal best - fit @xmath113 for the case of full 3-d velocities ( thick squares ) is on average within @xmath225 of the real value , @xmath215 , corresponding to better than @xmath226% accuracy . one contribution to the small systematic bias in @xmath113 may come from the fundamental nature of inverse problems in general ( of which schwarzschildmodeling is an example ) , namely , that there may not necessarily be a unique solution : it may be possible to change the mass profile and the df without appreciably changing the model predictions . if such is the case and there are multiple solutions , we do not necessarily expect a flat @xmath223 profile ( i.e. , with a number of equally acceptable solutions containing the correct one ) , most likely because of numerical noise and discretization effects . while we can not rule this out , our results do show that this probably does not affect the recovered mass - to - light ratio at more than the @xmath227% level ( based on figure [ fig : errors_ml ] , built with models using our smaller orbit library ) . unless superb data are available , random uncertainties are likely larger than such systematic errors . currently , the only exception to this are some galactic globular clusters , for which thousands of proper motions are being measured . however , such systems are often closer to spherical than galaxies , and hence one expects any theoretical degeneracies to be smaller . alternatively , numerical noise in the orbit library may be the cause of this systematic bias in @xmath113 seen in the bottom panel of figure [ fig : errors_ml ] . numerical noise may be reduced in part by the use of larger orbit libraries . indeed , we show in [ sec : grids ] below that a substantially larger orbit library tends to produce more accurate results overall . the likelihood ratio statistic @xmath21 in figures [ fig : mlparabn ] and [ fig : mlparab2 ] allows us to find the best - fit model parameters and their confidence intervals . however , it does not shed light on the question whether the best - fit model is actually statistically consistent with the data . the likelihood @xmath68 of the best - fit model also can not be used for this purpose . there is no theorem of mathematics that states what the value of @xmath68 should be for a statistically acceptable model , given that the underlying velocity distributions from which the particles are drawn are not known a priori ( and are not generally gaussian ) . nonetheless , many other statistics can be defined to address this issue once the best - fit model has been found . for example , the velocity moments of the best - fit model can be calculated ( as a function of position on the sky ) , and statistics can be defined that assess whether these moments are consistent with the observed data . alternatively , one can draw random realizations of the data from the best - fit model and use a kolmogorov - smirnov test to assess whether the data and the realization are consistent with being drawn from the same underlying distribution . we have explored a subset of these approaches and these suggested that the best - fit models are indeed statistically consistent with the pseudo - data they were designed to fit . in general neither the mass - to - light ratio nor the inclination of a stellar system under study are known in advance , and thus one has to explore models with several combinations of both parameters in a search for those values that provide the best fit to the data . in this section we present and discuss the results of running the discrete schwarzschildcode on grids of @xmath228 values to study whether the correct input combination is recovered . as in [ sec : getml ] , we perform tests on datasets with different types of kinematic constraints ( los velocities and/or proper motions ) . the results of tests are presented in figures [ fig:55is_los_mu ] and [ fig : grid_55_90 ] . they show @xmath72 contours that result when computing discrete schwarzschildmodels on a grid of @xmath228 values , including the correct input combination , for a variety of input datasets of the 55is and 90is cases . the goodness - of - fit parameter @xmath72 shown in these plots is obtained by first rebinning with a much finer grid the @xmath228 space explored by the models actually calculated ( indicated by small dots ) , and then computing the values on this new grid via interpolation between the nearest calculated models . we then determine the minimum on the finer grid ( whose location is indicated by the star ) and subtract it from all grid points to obtain the @xmath72 parameter , for which contours are shown . as in the case of figures [ fig : mlparabn ] and [ fig : mlparab2 ] , the mass - to - light ratio is parameterized by the dimensionless velocity scaling @xmath229 , so that the input value corresponds to @xmath230 . we start by showing in figure [ fig:55is_los_mu ] the results of running grids of models for input datasets composed of only - los velocities and only proper motions , in both cases for the 55is case with 1000 observational datapoints , and using our small orbit library with @xmath208 combinations of @xmath160 . overall , and in agreement with the results of figure [ fig : mlparab2 ] discussed in [ sec : getml ] , the @xmath72 contours indicate that proper motions ( bottom panel ) better constrain the best - fit @xmath228 combination than a dataset with only - los velocities ( upper panel ) . the @xmath231 uncertainties ( thick contours ) obtained from the only - los dataset are twice as large than those from the proper motions alone ( 31% and 16% , respectively ) . the input mass - to - light ratio @xmath215 is adequately recovered by both datasets ( to within the @xmath75 confidence region ) . the best - fit inclination , however , is offset from the actual input value @xmath148 for both datasets , although somewhat closer to the correct value in the case of proper motions only . the @xmath231 uncertainties in the best - fit inclination are @xmath232 and @xmath233 for the only proper motions and only los cases , respectively . difficulties in constraining the inclination using schwarzschildmodeling of stellar kinematics have been encountered in the past . a good recent example is that of @xcite who , based on integrated stellar los velocity profiles and ionized gas observations of the e4 galaxy ngc 2974 , carried out a study analogous to the present one by constructing simulated observations of this galaxy , which they feed to their `` continuous '' ( as opposed to discrete ) schwarzschildcode in order to study the recovery of the input mass - to - light ratio and inclination . they find that even with artificially perfect input kinematics the inclination is very poorly constrained . the same conclusion is reached when attempting to fit the actually observed los velocity profiles with schwarzschildmodels , so stellar los velocity profiles provide weak constraints on the inclination of this system , a statement they are confident about because the actual inclination of ngc 2974 is known from observations of its extended disc of neutral and ionized gas in rapid rotation . while one could expect that the availability of proper motion measurements in addition to los velocities would enhance the ability of the models to obtain useful constraints on the inclination of a stellar system in general , the reality is that the current state - of - the - art of schwarzschildmodeling does not have a definitive answer on this issue yet . as recent studies of the kinematics of stars in globular clusters seem to indicate , the chances of success are highly dependable on the quality and quantity of available data on the system under study ( compare , for example , the results of @xcite and @xcite regarding the best - fit inclinations of @xmath22 cen and m15 , respectively ) . there are at least two factors that may contribute to the difficulty in recovering the inclination from stellar kinematics : degeneracies inherent to schwarzschildmodels , and numerical noise . first , there is no guarantee that inclinations other than the correct one must fit the data worse . indeed , in their modeling of high signal - to - noise integral - field data of ngc 2974 , @xcite already observe that the differences between schwarzschildmodels with different inclinations are smaller than the differences between the best - fitting model and the data , which they interpret as indication of a fundamental degeneracy in the recovery of the inclination with three - integral models . numerical noise , on the other hand , is a consequence of schwarzschildmodels being in the end only discrete representations of a smooth , continuous distribution of possible orbits , and it could be argued that this discreteness might have a more negative effect for high inclinations . for example , even a simple and smooth circular orbit presents cusps or discontinuities when viewed close to edge - on . the turning points of such an orbit may get smoothed out differently for different inclinations . the issue of degeneracy , nevertheless , can be avoided in those cases where the inclination is known to be uniquely determined by the data . this is the case , e.g. , in the situations where the following conditions are met : ( 1 ) the kinematical dataset consists of proper motion measurements and los velocities , ( 2 ) the stellar system is reasonably close to axisymmetric , and ( 3 ) there exists an independent measurement of the distance @xmath234 to the system . as first used in practice by @xcite , the inclination then follows directly from the following relationship between the mean los velocity ( in units of @xmath198 ) and the mean proper motion along the short axis ( in units of masyr@xmath235 ) , @xmath236 where @xmath234 is the distance in kpc , and the brackets denote an integration along the line - of - sight . this relation is true at each projected position @xmath6 in any axisymmetric system , and has been successfully applied to the galactic globular clusters @xmath22 cen and m15 @xcite . here , in order to explore the applicability of this simple relationship , we take advantage of our a priori knowledge of the correct inclination for our simulated datasets , and study the circumstances under which the use of equation ( [ eq : inclination ] ) provides an accurate result . unlike the case of integrated light measurements ( where @xmath237 is simply the average of the losvd at any given projected position on the sky ) , in the context of discrete datasets neither @xmath237 nor @xmath238 are quantities that can be rigorously obtained from the data at any given @xmath6 . both quantities may , nevertheless , be approximated by averaging a number of kinematic measurements that fall within one or more apertures of a given size around projected positions @xmath6 . following this , we applied equation ( [ eq : inclination ] ) to a series of subsets of our 6 simulated datasets with varying number of kinematic measurements , and verified that indeed the correct inclination is reproduced provided : ( a ) the system is rotating ( otherwise , while the relation is still valid , both averages are nearly zero and hence the inclination is not really constrained ) ; ( b ) most of the datapoints are not located close to the minor axis ( where rotation velocities are too small ) ; and ( c ) the averages are computed from a sufficiently large number of kinematical measurements ( so that the error in @xmath239 is not too large ) . these are conditions that are certainly fulfilled by datasets on some galactic globular clusters , currently the only class of stellar system for which there are 3-dimensional kinematic information available . therefore , in those cases , equation ( [ eq : inclination ] ) can be safely applied . the schwarzschildmodeling can then concentrate on recovering the more interesting properties such as the orbital structure and mass - to - light ratios , which we have shown are successfully recovered when the inclination is assumed known . to better understand the problem of numerical noise , we explored the dependence of the results on the size of the orbit library used to construct the schwarzschildmodels . we did this for cases with 1000 datapoints with complete three - dimensional velocities , so that because of equation ( [ eq : inclination ] ) we know that there is no theoretical degeneracy in inclination . figure [ fig : grid_55_90 ] shows the @xmath72 contours resulting from fits of schwarzschildmodels using our standard library of @xmath208 orbits ( upper panels ; same library size as in figure [ fig:55is_los_mu ] ) in comparison with fits that use a library 8 times larger , i.e. , one with @xmath212 orbits ( lower panels ) . we show results for the 55is ( left - hand panels ) and 90is ( right - hand panels ) cases . in all four panels of figure [ fig : grid_55_90 ] , the best - fit mass - to - light ratio is always within @xmath75 of the input value @xmath215 , with the exception of the 55is case with the bigger library ( lower left ) , where they agree at the @xmath225 level . the size of the confidence regions on the mass - to - light ratio does not change significantly when the orbit library is increased in size . therefore , we conclude that libraries of @xmath208 orbits are large enough to properly constrain the mass - to - light ratio ( provided that one uses regularization as we do here ; see @xcite ) . this provides further justification for our use of this library size in section [ sec : getml ] . the top left panel in figure [ fig : grid_55_90 ] is directly comparable to the two panels of figure [ fig:55is_los_mu ] , but now with three components of velocities observed , instead of just one or two , respectively . consistent with the results in figures [ fig : errors_ml ] and [ fig:55is_los_mu ] , we see that the addition of an extra component of velocity decreases the size of the confidence regions . more interestingly , a secondary minimum in @xmath223 appears close to the @xmath228 values for the correct input model . this suggests that indeed all three components of velocity may be necessary to uniquely constrain the inclination of an axisymmetric stellar system . the bottom left panel shows the effect of increasing the orbit library size . there is now only a single minimum , centered at an inclination that agrees with the input value at the @xmath240 level . the right panels in figure [ fig : grid_55_90 ] show the situation for the 90is case . with the small library ( top right ) , the best - fit inclination is at @xmath241 , substantially far from the input value . when the orbit library size is increased ( lower right ) , the best - fit shifts to @xmath242 . this is only @xmath243 from the correct input value , which may well be acceptable for many realistic applications . on the other hand , the best fit and the input value are inconsistent at the many sigma level , which is certainly reason for some concern . a possible cause for this is that the turning points of orbits in edge - on systems have very sharp edges in projection . therefore , larger grid sizes than we have used may be necessary to correctly represent them in all the necessary detail . however , we have not explored this further for two reasons . first , information on all three velocity components may be necessary to be able to uniquely constrain the inclination . if that is available , then use of equation ( [ eq : inclination ] ) will be more accurate and efficient than use of schwarzschild modeling . second , in practice one is generally much more interested in the mass distribution than in the inclination . figure [ fig : grid_55_90 ] shows that the mass - to - light ratio is correctly recovered , even when the inclination is systematically biased . in conclusion , our tests demonstrate that the recovery of the most important properties of the system ( its orbital structure and the mass - to - light ratio ) by our discrete schwarzschildmodels is robust . correct recovery of the inclination appears to be the most complicated aspect of the modeling . sufficient observational data must be available and a large enough orbit library must be used . our code can then adequately recover the inclination of sufficiently inclined systems . however , for edge - on systems there remains a systematic inclination bias of @xmath244 that we have been unable to resolve . this is the primary shortcoming of our new approach that was unearthed by the pseudo - data tests that we have presented . this may be a generic property of schwarzschild codes , since other authors have also reported difficulties in recovering inclinations . either way , this is not believed to be a significant limitation for most potential practical applications of our code . discrete kinematic datasets , composed of velocities of individual tracers ( e.g. , red giants , planetary nebulae , globular clusters , galaxies , etc . ) , are routinely being assembled for a variety of stellar systems of all scales ( [ sec.intro ] ) . these include not only los - velocity surveys . high - quality proper - motion databases already exist for galactic globular clusters , and future facilities hold the promise of providing the same for stars in the nearest galaxies . however , the most sophisticated tools typically being used in the modeling of these observations were actually developed for the analysis of kinematic data in the form of losvds , a rather different type of velocity information than the case of the velocities of kinematic tracers on a one - by - one basis . as a consequence , the information content of any particular dataset of a discrete nature is likely not being fully exploited . we thus have developed a specific tool for the modeling of discrete datasets , which we have presented in this paper along with detailed tests of its performance based on the modeling of simulated data . the new tool consists of a schwarzschildorbit - superposition code that , adapted from the implementation of @xcite , can handle any number of ( one- , two- , or three - dimensional ) velocities of individual kinematic tracers without relying on any binning of the data . under the only assumptions that the system is in steady - state equilibrium ( i.e. , the gravitational potential is not changing in time ) and may be well approximated as axisymmetric , the code finds the distribution function ( a function of the three integrals of motion @xmath9 , @xmath79 , and @xmath1 ) that best reproduces the observations ( the velocities of the tracers as well as the overall light distribution ) in a given potential . the fact that the distribution function is free to have any dependence on the three integrals of motion allows for a very general description of the orbital structure , thus avoiding common restrictive assumptions about the degree of ( an)isotropy of the orbits . unlike previous implementations of the schwarzschildtechnique , we cast the problem of finding the best superposition of orbits using a probabilistic approach , i.e. , by building a likelihood function representing the probability that the entire set of measurements would have been observed assuming a particular form for the gravitational potential ( [ sec : logl ] ) . in this case , and in contrast with the old continuous versions , the dependence of the likelihood function on the orbital weights is non - linear , and the optimization problem can not be reduced to a linear matrix equation . instead , it becomes a problem of the maximization of a likelihood with respect to the set of weights associated to all possible combinations of the integrals @xmath160 that comprise the orbit library ( [ sec : logl ] ) , and which accounts for the observed positions and ( any - dimensional ) velocities of all particles in the dataset , including their uncertainties ( [ sec : pij ] ) . after extensive testing , a conjugate gradient algorithm was found to converge satisfactorily to the correct solution and was adopted for the remaining tests of the code s overall performance ( [ sec.mkfitin ] ) . in order to assess the reliability of our discrete schwarzschildcode , we applied it to several sets of simulated data , i.e. , artificially generated kinematic observations obtained from a model of an axisymmetric galaxy of which the orbital structure , mass distribution , and inclination are known in advance . pseudo - datasets were generated from a two - integral phase - space distribution function with varying degrees of overall rotation , types of velocity information ( only - los , only proper motions , and both ) , total number of particles , and for two different inclinations on the plane of the sky ( [ sec : data ] ) . using the various simulated datasets , we studied the recovery of the input orbital structure or df , mass - to - light ratio , and inclination . for the purposes of these tests , we assumed complete knowledge of the radial profile of the underlying mass distribution and a mass - to - light ratio that remains constant as a function of radius . these restrictions are easily ( and must be ) lifted when modeling data on real systems , in which case one needs to explore a range of plausible underlying potentials and allow for variations of the mass - to - light ratio to properly account for the possibility of central black holes and dark halos . inside the region constrained by data , we find that the distribution function ( represented by the corresponding distributions of orbital mass weights ) and streaming characteristics of the input datasets are satisfactorily recovered by the schwarzschildfits when the correct inclination and mass - to - light ratio are known ( figs . [ fig:1dplots ] to [ fig : ebins55is ] ) . as measured by the mean absolute deviations between the integrated weight distributions , the agreement between the fitted and the input orbital weight distributions as a function of @xmath9 , @xmath79 , and @xmath1 is typically of the order of 3% , 10% , and 20% , respectively ( the numbers for our worst case being 5% , 16% , and 25% ) . when eliminating the dependence on @xmath1 , the agreement between the fitted and input @xmath164 distributions is of the order of 15% , with the net rotational behavior of the input datasets cleanly recovered ( figs . [ fig:2dplot55ns ] and [ fig:2dplot55is ] ) . thus , we conclude that the discrete schwarzschildcode can successfully recover the orbital structure of the system under study . assuming that the inclination of the system on the plane of the sky is known , we quantified the recovery of the input mass - to - light ratio as a function of the size of the input dataset ( fig . [ fig : mlparabn ] ) and of the type of kinematic information available ( fig . [ fig : mlparab2 ] ) . we studied both the best - fit value as well as the uncertainty in its determination ( fig . [ fig : errors_ml ] ) . the statistical expectation of better results when the amount of observational information is larger ( either regarding the number of datapoints or the number of velocity components ) is clearly reproduced by our discrete schwarzschildmodels . for the smallest datasets used in our testing ( @xmath245 ) , and regardless of whether using only - los velocities , only proper motions , or both , the best - fit mass - to - light ratio is within 5 - 10% of the input value , with formal @xmath75 uncertainties of the order of 15% . when increasing either the number of available measurements or the number of measured velocity components , the mass - to - light ratio is always recovered to better than @xmath246 accuracy , with the corresponding random ( @xmath75 ) uncertainties in the range of 5 - 10% . the discrete schwarzschildcode , therefore , recovers the mass - to - light ratio of the input datasets to satisfactory levels of accuracy . the recovery of both the mass - to - light ratio and inclination when neither of these quantities are known in advance ( as is usually the case with real observations ) was studied using a grid of discrete schwarzschildmodels , exploring also the dependence on the type of velocity components available ( fig . [ fig:55is_los_mu ] ) . we find that the mass - to - light ratio was again successfully recovered , but the best - fit inclination was not identified correctly using small orbit libraries . we found that this was remedied by better sampling the available @xmath160 integral space using a larger orbit library ( fig . [ fig : grid_55_90 ] ) . for our input datasets with @xmath148 , the best - fit inclination obtained by our models with a large orbit library is @xmath247 , while for input datasets with @xmath147 we obtain a best - fit model with @xmath242 . given the known difficulty of schwarzschildmodels in general for determining the inclination of stellar systems , and considering the low relative importance of this parameter compared to other properties such as the orbital structure and the mass - to - light ratio , we regard this small disagreement for the high inclination datasets as acceptable . in summary , we have shown that our new schwarzschildcode , designed to adequately handle modern datasets composed of discrete measurements of kinematic tracers , doing this without any loss of information due to data binning or restrictive assumptions on the distribution function , is able to constrain satisfactorily the orbital structure , mass - to - light ratio , and inclination of the system under study . applications to data for galactic globular clusters and nearby de galaxies will be presented in future papers . these are only two examples of a large range of dynamical problems in astronomy to which a discrete schwarzschildcode like ours can be applied , so we expect this new tool will contribute to the better understanding of stellar systems in general . we are happy to thank marla geha and raja guhathakurta for their continued interest in the present work and its extension to the study of actual galaxies using their unique data on dwarf ellipticals . we also thank glenn van de ven for very useful discussions , his interest in the progress of this project and , last but not least , for his invaluable help with idl routines . this paper also benefited by comments from davor krajnovic , aaron romanowsky , and david merritt . thanks also to george meylan for his help with the writing of the hst theory proposal specified below , and to the anonymous referee , whose comments and suggestions improved the presentation of the paper . this work was carried out as part of hst theory project # 9952 and was supported by nasa through a grant from stsci , which is operated by aura , inc . , under nasa contract nas 5 - 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we present a new schwarzschildorbit - superposition code that is designed to model discrete datasets composed of velocity measurements of individual kinematic tracers in a dynamical system . this constitutes an extension of previous implementations that can only address continuous data in the form of ( the moments of ) velocity distributions , thus avoiding potentially important losses of information due to data binning . furthermore , the code can handle any combination of available velocity components , i.e. , only line - of - sight velocities , only proper motions , or a combination of both . it can also handle a combination of discrete and continuous data . the code determines the combination of orbital mass weights ( representing the distribution function ) as a function of the three integrals of motion @xmath0 and @xmath1 that best reproduces , in a maximum - likelihood sense , the available kinematic and photometric observations in a given axisymmetric gravitational potential . the overall best fit is the one that maximizes the likelihood over a parameterized set of trial potentials . the fully numerical approach ensures considerable freedom on the form of the distribution function @xmath2 . this allows a very general modeling of the orbital structure , thus avoiding restrictive assumptions about the degree of ( an)isotropy of the orbits . we describe the implementation of the discrete code and present a series of tests of its performance based on the modeling of simulated ( i.e. , artificial ) datasets generated from a known distribution function . we explore pseudo - datasets with varying degrees of overall rotation and different inclinations on the plane of the sky , and study the results as a function of relevant observational variables such as the size of the dataset and the type of velocity information available . we find that the discrete schwarzschildcode recovers the original orbital structure , mass - to - light ratio , and inclination of the input datasets to satisfactory accuracy , as quantified by various statistics . the code will be valuable , e.g. , for modeling stellar motions in galactic globular clusters , and modeling the motions of individual stars , planetary nebulae , or globular clusters in nearby galaxies . this can shed new light on the total mass distributions of these systems , with central black holes and dark matter halos being of particular interest .

You are an expert at summarizing long articles. Proceed to summarize the following text: blazars form one of the most energetically extreme classes of active galactic nuclei ( agn ) . blazars can be observed in all wavelengths , ranging from radio all the way up to @xmath0-rays . their spectral energy distribution ( sed ) is characterized by two broad non - thermal components , one from radio through optical , uv , or even x - rays , and a high - energy component from x - rays to @xmath0-rays . in addition to spanning across all observable frequencies , blazars are also highly variable across the electromagnetic spectrum , with timescales ranging down to just a few minutes at the highest energies . there are two fundamentally different approaches to model the seds and variability of blazars , generally referred to as leptonic and hadronic models ( see , e.g. , * ? ? ? * for a review of blazar models ) . in the case of leptonic models , where leptons are the primary source of radiation , synchrotron , synchrotron self - compton ( ssc ) , and external - compton ( ec ) radiation mechanisms are employed to explain the blazar sed ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the focus of the present study is also on a leptonic model . in hadronic models , the low - energy sed component is still produced by synchrotron emission from relativistic electrons , while the high - energy component is dominated by the radiative output from ultrarelativistic protons , through photo - pion induced cascades and proton synchrotron emission ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? one aspect common to all blazar models is a relativistic jet oriented at a small angle with respect to our line of sight , resulting in relativistic doppler boosting and the shortening of observed variability time scales . given computational limitations , the complex physical processes in relativistic jets can , realistically , only be evaluated with certain simplifying approximations . in order to facilitate analytical as well as numerical calculations , the two most common approximations employed in blazar jet models are to assume that the magnetic ( * b * ) field is randomly oriented and tangled , and that the lepton momentum distribution is isotropic in the comoving frame of the high - energy emission region . these two assumptions greatly simplify the evaluation of the synchrotron and compton emission by eliminating various integrals over the interaction and scattering angles . however , there is increasing evidence @xcite for a fairly well defined helical * b*-field structure within agn jets . these observations also suggest a spine - sheath geometry for agn jets . the differential velocity profiles within the jet is expected to create anisotropies in the particle distributions . it is therefore important to explore jet models where we can not only simulate an ordered * b*-field , but also study the resulting radiation behaviour with anisotropic lepton distributions . the standard approach to diagnosing the magnetic field properties is via synchrotron polarization . if the underlying distribution of emitting electrons is a power - law with power - law index @xmath1 , the maximum degree of synchrotron polarization is given by : @xmath2 where @xmath3 and @xmath4 are the synchrotron power per unit frequency in directions perpendicular and parallel to the projection of the magnetic field on the plane of the sky . using equation [ eq : synchpol ] we can see that for a power - law index of @xmath5 , the degree of polarization can be as high as 75% . it is therefore possible to estimate the magnetic field orientation based on polarization measurements , but an estimate of the field strength usually requires the consideration of flux and spectral properties of the synchrotron emission . furthermore , polarization measurements are notoriously difficult ( and even barely feasible at frequencies higher than optical ) , and may often not give realistic results due to faraday rotation and depolarization along the line of sight . in this work , we are interested in taking a complementary approach to estimating the magnetic field orientation where the difference in observed flux levels of the spectrum can give an estimate of how the magnetic orientation may be changing . ratio shows how the b - field estimate can vary depending on the electron power - law index and whether one assumes an isotropic b - field or a specific pitch angle.,scaledwidth=45.0% ] the principle behind this approach can be demonstrated when one compares the * b*-field estimates based on a power - law distribution of electrons with an arbitrary power - law index , @xmath1 , and pitch angle , @xmath6 . the comparison of synchrotron emission coefficients for a power - law distribution of electrons with and without pitch - angle ( @xmath6 ) dependence gives us a measure of how the estimated magnetic field strength can differ . the emission coefficients can be found in @xcite and are given by ( in the units of @xmath7 ) : @xmath8 and @xmath9 where @xmath10 is the electron distribution power - law normalization . the above two expressions can be solved for the magnetic field to obtain : @xmath11 this gives an estimate of how , for a given luminosity , the * b*-field estimates can differ depending upon whether we assume an isotropic pitch angle approximation or a given pitch angle ( which , in the case of relativistic electrons , is equal to the angle between the magnetic field and the line of sight ) . the above relation is only applicable in the optically thin regime . we can see in figure [ fig : bestimate ] that depending on the pitch angle assumption , and the electron distribution power - law index , the @xmath12 fraction can range from 0 ( there is negligible synchrotron emission along an ordered magnetic field ) to @xmath13 . because the compton emissivity is approximately isotropic for an isotropic distribution of electrons , the ratio @xmath14 will change with the pitch angle . it is therefore important to see how the overall synchrotron and synchrotron - self compton spectra differ with well ordered magnetic fields . one point worth noting is that in our set - up the lower limit on orientation is limited by the @xmath15 approximation ( see section [ sec : synch ] ) ; a magnetic field perfectly aligned with the observing direction will give zero output . however , in a more rigorous treatment , the lower limit on the minimum angle , @xmath6 , for the orientation will be determined by the relativistic beaming characteristic of synchrotron emission along an electron s direction of motion into a cone of opening angle @xmath16 , where @xmath0 is the electron lorentz factor . in the case of optical frequencies and magnetic fields of @xmath17 g , @xmath18 will be the order of @xmath19 , while in the x - ray regime @xmath20 . using the relation in equation [ eqn : bestimate ] it is possible to estimate the effects on the estimates between this value of @xmath18 and an isotropic magnetic field . for a power - law index of 3 , @xmath18 gives @xmath12 value of 0.0039 . the following section briefly outlines our model , including the synchrotron radiation and compton scattering treatments followed as well as the numerical techniques used to implement them . @xmath21{figure2.epsi } \\ \includegraphics[width=0.35\textwidth]{figure3.epsi}\\ \end{array}$ ] , the viewing angle with respect to the volume / jet is incorporated into doppler boosting calculations . , title="fig:",scaledwidth=40.0% ] + in our model the basic volume structure is a cubic cell . this allows the model to be modular and build an arbitrarily large volume with any desired anisotropies . each cell contains a magnetic field plus electron and photon distributions . the magnetic field can have an arbitrary orientation and strength in each cell . this means that the overall volume can be modelled to contain a completely uniform , partially anisotropic , or pseudo - random * b*-field . the purpose of the present work is to isolate the effects of the degree of order and orientation of the magnetic field on the emerging synchrotron emission . therefore , we choose the simplest conceivable approach concerning the electron distribution , and do not take electron cooling into account . this means , we only focus on static electron distributions which do not evolve due to energy losses . in future work we aim to include self - consistent cooling effects which would also allow us to probe how the pitch - angle dependence of the synchrotron cooling would give rise to different electron distributions in different cells , depending on the magnetic field set - up . the directional information in the electron and photon distributions and the is with respect to the cell . in the case of electrons the distribution is a function of energy and two angles with respect to the cell ( see figure [ fig : eleb ] ) . this gives us the ability to create anisotropies in the electron distributions as well by either having preferential direction for the electrons or by setting up the electron distributions differently in various cells . for both electrons and photons , the distributions energy grids ( lorentz factor @xmath0 for electrons and frequency @xmath22 for the photons ) are calculated using logarithmic binning . therefore each distribution is modelled using a 3 dimensional array with the dimensions of @xmath23 $ ] . figures [ fig : eleb ] ( right ) illustrates how various angles with respect to the cell are defined . angles @xmath24 and @xmath25 run from 0 to @xmath26 and 0 to 2@xmath26 respectively . in figure [ fig : volgeo ] we can see how the overall volume can be constructed from individual cells . for a given viewing angle , the emission from the visible outer layer of cells is combined to produce an overall spectrum from an effectively larger volume . the simulation currently transfers , from one cell to another , only the photons . in order to achieve this , we need to calculate which of the six cubic faces a given photon direction will intersect . to calculate this , we assume that all the photons are produced in the center of the cell , and then trace photon paths in any given direction towards the nearest boundary . even though our simulation considers a static situation , the transfer and radiative feedback between different cells requires an inherent time - dependence in the code . the time step for our radiation transfer approach is the light crossing time across a single cell , which is equal to the time it takes for the photons to travel from one cell to another . at the end of each time step and depending on the physical processes being modelled , the photon distribution is modified and passed to the appropriate neighbour . when being passed to a neighbour the entire photon distribution is passed . therefore at the end of a time - step each cell s ( intrinsic ) photon distribution is emptied into six neighbouring cells , unless it is a boundary cell . the six incoming ( transiting ) photon distributions are stored until the start of the following time - step when they are combined to form a single intrinsic photon distribution again . the physical processes are then carried out on this single photon distribution . synchrotron radiation is calculated first and the photons added to the intrinsic distribution . compton scattering is carried out after the synchrotron radiation . at this point we reach the end of a time - step and the process of transferring photon distributions to neighbouring cells begins again . the observed photon distribution originates from the boundary cells . the photon distributions emerging from visible faces of the boundary cells are combined to create a single observed photon distribution . this process of combining the photon distributions from the boundary cells in effect treats the whole multi - cell structure like a single cubic / cuboid structure . here we highlight the key points of the synchrotron radiation treatment that we follow . a more in - depth analysis and details can be found in @xcite . the synchrotron emissivity per electron , @xmath27 , is given by : @xmath28 where f(x ) is given by : @xmath29 @xmath30 , where @xmath31 is the critical frequency given by @xmath32c . @xmath6 , the pitch angle , is calculated using spherical trigonometry : @xmath33 the synchrotron emission coefficient is given by : @xmath34 the numerical bessel function integration in equation [ eqn : bessel ] can be time consuming . however , some fast routines to perform this integration are given by @xcite which we modified for our precision and computer language . in a full treatment of the synchrotron radiation the emitted photons are distributed within a solid angle ( @xmath35 ) about the pitch angle @xmath6 . however , for our purposes we assume the emitted photons travel in the same direction as the emitting electrons . a detailed calculation of synchrotron self - absorption can be found @xcite . the absorption coefficient when recast in terms of electron lorentz factors , @xmath0 , instead of @xmath36 , can be written as : @xmath37 the photons produced via synchrotron radiation are added to the intrinsic photon distribution of the cell . the photons received from neighbouring cells are added to the intrinsic photon distribution prior to calculating the synchrotron spectrum . therefore the photons passing through any cell are also synchrotron self - absorbed . the emission and absorption coefficients are used to calculate the total spectrum , @xmath38 where @xmath39 is the optical depth and _ l _ is the size of the emission zone / cell . to @xmath40 off electrons @xmath41,scaledwidth=40.0% ] in the limit @xmath42 , and in the electron rest frame , the incident photon travels in nearly the opposite direction to the electron . this is due to photon aberration : @xmath43 when @xmath44 , @xmath45 , we are in the head - on approximation regime , which we employ to greatly simplify compton cross section calculations . that is , we can assume that the scattered photon solid angle , @xmath46 , is well approximated by the electron solid angle @xmath47 . when the differential compton ( klein - nishina ) cross section is integrated over @xmath46 , we get @xcite : @xmath48 where h is a heaviside function , @xmath49 , and the compton kernel is given by : @xmath50 and @xmath51 . the compton cross section can then be used in the emission coefficient formula to obtain the comptonized spectrum . the head - on approximation simplifies the emission coefficient calculation by eliminating two integrals from the compton emissivity treatment without the approximation . the following relation can be used to obtain the number of interacting photons : @xmath52 the interacting photons are a combination of photons originating from synchrotron radiation and the photons received from neighbouring cells . at the start of a time - step , the photon distributions received from the neighbouring cells are combined , while preserving the direction information , to form the intrinsic photon distribution . synchrotron photons are also added to the intrinsic photon distribution . the total photon distribution is then used in the compton emissivity relation to obtain the compton spectrum in the head - on approximation , given by : @xmath53 after the intrinsic photon distribution has been compton scattered it is redistributed based on change in energy and direction . the redistributed photon distribution is then used to work out which neighbouring cells receive which proportion of the distribution . the overall spectrum is obtained by combining the synchrotron and compton spectra . as it stands in our model , synchrotron emission is the only source of photons which are then compton scattered by the same population of electrons . the resulting photon spectrum is given as a function of two cell angles @xmath24 and @xmath25 ( see figure [ fig : eleb ] ) . although not included in the present study , it is straight forward to include external compton effects by adding the external photon field to the photon distributions . once we have a spectrum , @xmath54 , the flux can be calculated using : @xmath55 for an emission zone with an area @xmath56 and luminosity distance @xmath57 . for a viewing angle @xmath58 the doppler factor @xmath59 is given by @xmath60^{-1 } \ . \label{eqn : doppler}\end{aligned}\ ] ] the ` @xmath61 ' corresponds to either an approaching or a receding component of the jet . in the case of blazars the observed emission is strongly dominated by the approaching jet , boosted with the doppler factor @xmath62 . also , any given frequency , @xmath22 , in the emission region rest frame will be shifted by a factor of : @xmath63 we follow the @xcite formulation to calculate the luminosity distance based on the redshift of an object . photons emitted at an angle @xmath64 in the cell rest frame will appear at an angle @xmath65 due to angle aberration , which can be expressed as follows : @xmath66 ( @xmath67 in jet frame ) . the spectra shown results of different b - field configurations , including when it is randomly oriented in different cells . the plot shows @xmath68 values ( @xmath69 , unless randomly oriented ) . the red dot - dot - dashed curve shows the corresponding calculation based on the angle averaged emissivity and spherical geometry , using the code of @xcite . see figure [ fig : eleb ] for details on various angles.,scaledwidth=45.0% ] , @xmath69 . see figure [ fig : eleb ] for details on various angles.,scaledwidth=45.0% ] in order to study the effects of the * b*-field orientation on the synchrotron and synchrotron self - compton spectra , we set up two scenarios . in the first set up the magnetic field is uni - directional in all the cells ( 27 in total ) and in the second scenario the magnetic field is randomly oriented in each of the cells . these set - ups are likely to be the two extreme scenarios for a jet . evidence points to the * b*-field being semi - ordered in agn jets ; for example , helical @xcite . in all the presented cases , the electron distribution is a power - law and distributed uniformly over the angles @xmath70 and @xmath71 . figure [ fig : b_angles_ssc ] shows synchrotron self - compton spectra with different configurations . various simulation parameters can be found in table [ tab : figparams ] . the spectra are for a fixed viewing angle . changing the orientation has a significant impact on the observed spectrum . there is a large difference in flux values depending on the orientation with respect to the observing direction . the minimum flux levels are observed at an orientation along the viewing angle while the maximum flux levels are observed when the is perpendicular to the line of sight . the figure also demonstrates the fact that only the synchrotron spectrum component is heavily affected by the orientation . the compton scattered component of the spectrum is almost independent of the orientation . the main reason for this is that the photon distribution anisotropies introduced by the orientation are lost when scattering off an isotropic distribution of electrons . the small variations that remain are due to the @xmath72 factor present in compton emissivity calculations . the line of sight and photon anisotropies therefore affect the extent to which the compton spectrum is boosted . additional anisotropies are introduced by the discretization of the photon and the electron angular distributions . figure [ fig : b_angles_ssc ] also shows a comparison with another ssc calculation @xcite which assumes a randomly oriented magnetic field . this simulation is set up with identical parameters to the ones outlined in table [ tab : figparams ] , except it uses a spherical volume instead of a cube ; the total volumes are identical , therefore the sphere has a radius of @xmath73 cm . we can see that the synchrotron components are in good agreement . however , the compton component in the @xcite calculation is much higher . the inverse compton to synchrotron ratio @xmath74 differs by a factor @xmath750.68 between the two calculations . this is most likely due to differing geometries . for a sphere , the average photon escape time is @xmath76 , whereas in our cubic set up , due to the way radiation transport between cells is treated ( see section [ sec : volume ] ) , a photon takes , on average , @xmath77 to escape the region ( @xmath78 is the width of an individual cell ) . since the flux ratio @xmath79 and the volume - averaged radiation energy density @xmath80 is proportional to the photon escape time scale , the longer photon escape time scale in the spherical geometry results in a larger ssc flux . figures [ fig : view_angles_ssc ] shows the effects of the viewing angle on the spectrum . it shows seds for a fixed azimuthal angle , but different viewing angles with a single configuration . as before , there are normalization differences between the seds when comparing uniform and randomly oriented , but the main factor in this case is the variation in the doppler boosting due different viewing angles . for the fit parameters.,scaledwidth=45.0% ] for the fit parameters.,scaledwidth=45.0% ] for the fit parameters.,scaledwidth=45.0% ] markarian 421 was the first extragalactic source to be detected in tev energies , hence making it an extensively studied source . we present spectral fits to _ xmm - newton _ and _ veritas _ data presented in @xcite . we refer the reader to the above paper for the details on data reduction . figures [ fig : mrk421_0402 ] , [ fig : mrk421_0802 ] , and [ fig : mrk421_1802 ] show fits to mrk 421 data using our code . our aim in the present paper is to explore the impact of the orientation on the fit parameters , especially the estimates . in figure [ fig : mrk421_0402 ] we can see that the synchrotron component is best fit for a particular orientation ( see table [ tab : mrkparams ] for fit parameters ) . the gamma - ray data , however , is fit well with all the orientations . this is due the fact that the self - compton spectrum is not affected much by the orientation ( see the discussion in the previous section ) . there is a significant difference in the synchrotron peak when comparing various orientations . the spectra in figure [ fig : mrk421_0402 ] show that at a good fit is achieved when the is oriented at @xmath81 with a strength of @xmath82 g. however , the fit shown is not unique . we can see in figures [ fig : mrk421_0802 ] and [ fig : mrk421_1802 ] that for identical parameters , except the strength , the best - fit orientation is very different . in one case a pseudo - random provides the best fit with @xmath83 g , while for @xmath84 g , a orientation of @xmath85 provides the best fit . therefore good fits can be achieved with different strengths at different orientations . the main point here is the fact that it is possible to over- or underestimate the strength when assuming it to be randomly oriented . in the cases presented here , the strength ranges from 0.18 g to 0.25 g for very similar fits to the data , but with different magnetic - field orientations . therefore it is possible to overestimate the magnetic field strength by at least @xmath86 if a particular orientation , whether uniform or tangled , is assumed . if the were pointed closely aligned with the line of sight , much higher values will be necessary to obtain similar fits ( see discussion in section [ subsec : bestimate ] ) . we also note that the bulk lorentz factor used in the fits are lower than the values obtained by some authors for fitting mrk 421 data ( e.g. see * ? ? ? the value of the bulk lorentz factor values used in our fits is likely to be on the lower end of the limits imposed by pair opacity arguments @xcite . however , the main point of this paper is not the determination of actual best - fit values for mrk 421 ( which would not be realistic due to our neglect of cooling effects anyway ) , but to demonstrate the orientation - dependent magnetic - field degeneracy in the course of blazar sed fitting . in this paper we have presented first results from a new relativistic jet radiation transfer code that we are currently developing . here we take the full angular dependence into account when modelling synchrotron and synchrotron self - compton processes . we are able to model the at arbitrary orientations and study its impact on the resulting spectra . we have seen that the orientation plays an important role on the normalization of the synchrotron spectrum . using fits to markarian 421 data , we have shown how the orientation can mislead into over / under - estimating its strength . any future work should therefore be mindful of the fact that the underlying assumption about the orientation will play a considerable role in the errors associated with the magnetic field strength estimates .

we report on the development of a numerical code to calculate the angle - dependent synchrotron + synchrotron self - compton radiation from relativistic jet sources with partially ordered magnetic fields and anisotropic particle distributions . using a multi - zone radiation transfer approach , we can simulate magnetic - field configurations ranging from perfectly ordered ( unidirectional ) to randomly oriented ( tangled ) . we demonstrate that synchrotron self - compton model fits to the spectral energy distributions ( seds ) of extragalactic jet sources may be possible with a wide range of magnetic - field values , depending on their orientation with respect to the jet axis and the observer . this is illustrated with the example of a spectral fit to the sed of mrk 421 from multiwavelength observations in 2006 , where acceptable fits are possible with magnetic - field values varying within a range of an order of magnitude for different degrees of b - field alignment and orientation .

You are an expert at summarizing long articles. Proceed to summarize the following text: gotthilf heinrich ludwig hagen is most renowned for his contributions to the study on laminar flow in pipes ; his measurements published in 1839 studied what is now well - known as the ( hagen-)poiseuille law @xcite . less well - known is hagen s work on granular systems . while janssen , with his 1895 paper , typically receives credit for the saturation effect in granular silos @xcite , it was hagen in his paper _ ber den druck und die bewegung des trocknen sandes _ @xcite who measured this effect earlier but also not for the first time , cf . @xcite and offered a first model that provided a qualitative understanding of the effect . hagen proposes a quadratic law ( with some cutoff ) for the pressure instead of the exponential form put forward by janssen more than 40 years later @xcite . in addition to the discussion of a static pile of sand , hagen examines in considerable detail the flow through an opening of the container , and discovers that the rate of discharge is proportional to the diameter of the opening raised to the power @xmath0 . this result is elegantly derived from dimensional analysis ( * ? ? ? * ch 10.2 ) ; but it fits the data best if instead of the real diameter some effective diameter is used the resulting law is known as the beverloo correlation @xcite . hagen finds an effective opening diameter that is smaller by twice the particle diameter , which is consistent with more recent measurements . hagen s work lays out the basis of the so - called hourglass theory where the flow of granular material is found to be independent of the filling height in the container , thus allowing the measure of time with an hourglass ( * ? ? ? * sec 10.4 ) . later work both confirms and extends hagen s early analysis @xcite . more recently , a lot of work has been devoted to understanding the fundamental differences between fluid and granular flows @xcite . while on the level of the individual grains the probability of arching at the opening is a matter of current investigations @xcite , hagen provides a successful route to a continuum description of the flow by the rescaling of the diameter of the opening . consider a container with a horizontal bottom that includes a circular opening of radius @xmath1 . in this opening is placed a disk which is easily movable but seals tightly ; on top of it is an extended filling of sand up to a height @xmath2 . as a result , there is a pressure exerted on the disk created by the weight of the cylinder of sand above it less the friction which is experienced by that cylinder from the sand surrounding it . the friction is proportional to the horizontal pressure , or the square of the height . let @xmath3 denote a friction dependent constant and @xmath4 be the weight of a unit volume of sand , then the pressure against the disk equals for a growing @xmath2 , this expression will increase in the beginning , reach a maximum , and subsequently decrease afterwards ; it will become not only zero but even negative . however , the sand cylinder is not rigidly connected throughout , and therefore the axial pressure , which its lower part exerts on the bottom disk , can not be compensated by the strong friction acting on the cylinder as a whole . hence , the pressure on the disk in fact remains unchanged for fillings higher than that height at which the pressure on the disk reaches its maximum value . for the case of pressures below the maximum , we seek to represent that pressure by the weight of a free - standing body of sand on the disk . the body is bounded by the surface of the filling . it is a conoid that is formed by the rotation of a parabola around its axis . here the parameter of the parabola is @xmath6 , while the height of the paraboloid is @xmath7 . the latter body joins the circumference of the opening . to compare these results with the real phenomenon , i created openings of radii of 0.3791 inch @xcite . ] and 0.7271 inch , respectively , in two brass plates used as the the bottom of the sand - filled container ; these openings were closed with suitable disks that were supported from underneath by hooks that were connected to one arm of a balance , while the other arm carried the counterweight . to reduce the counterweight to the pressure on the disk slowly and without concussion , sand was allowed to discharge through a small hole in the bottom of the plate . the constant error of this method could be found easily by measurement of the excess weight of the disk and the hook compared to the plate both by the discharge of sand and by direct weighing . while repeating measurements of the pressure due to the the sand against the disks multiple times , considerable deviations among the measurements were observed ; this was apparently due to different ways of settling . when the settling was as loose as possible , the weight of a cubic inch of the sand , a crude ferrous grit , was 2.9 loth @xcite . ] . however , the weight increased to 3 loth when there was modest agitation during filling and rose to 3.25 loth as soon as a fairly compact settling was generated by severe agitation or by pushing a wire into the container . in so doing , the friction increased by even more than the specific weight . hence , the pressure against the disk became remarkably smaller for the more compact settling . with the larger disk the pressure maximum was reached at a filling height of around 1 inch : for larger height the pressure decreased somewhat , since despite great care the sand settled in a slightly denser state . for the loosest fillings i found @xmath8 to 0.175 . in contrast , the deviations were less when the sand was dropped from a height of several inches in a narrow stream , and when the sand flow was cone - shaped : the value for @xmath3 was limited to between 0.21 and 0.22 . the influence of different settling states was also clearly noticeable as the sand discharged through the openings of various radii . when the settled filling was somewhat denser , there was less sand flowing within a second . as the discharge duration increased , the sand became agitated , particularly in the vicinity of the opening , resulting in an increased sensitivity , which in turn led to a diminished flow . incidentally , the height of the filling had no influence , as was already recognized by huber - burnand some time ago @xcite . to reduce the mentioned irregularities as much as possible , i limited the duration of each observation to 30 to 200 seconds and , additionally , tried to make the fillings rather uniform . to this end , the container was placed in a metal - sheet cylinder with a sieve - like bottom , filled with sand , and lifted slowly thereafter ; in this way the sand poured into the container in several hundred thin streams , each from a very small height . the radius reduction @xmath9 for the discharge opening is found to be close to the diameter of a grain of sand . from the constant @xmath10 can be determined the average distance from the opening at which the sand begins its free fall . assuming that the sand forms a compact mass until reaching the opening , the amount of sand discharged per second is @xmath11 where @xmath2 designates the mentioned height of fall and @xmath12 the effective opening . on the other hand , the observations yield @xmath13 equating both expressions and setting @xmath4 to 2.93 as obtained from the average loose packings , one finds @xmath14 if , in contrast , one assumes that for each unit of time a layer of sand of the same vertical height separates from the whole inner area of the paraboloid mentioned above in a free fall , then one can easily find the average velocity of this layer while passing the opening , and from the latter the average height of fall of the entire mass . this height is @xmath15 but from the data for loose packings it was found that @xmath16 and hence @xmath17 if instead one introduces the value @xmath18 which is valid for packings where sand is flowing sideways , which really happens during the discharge , then @xmath19 the result derived from the observations is in between the two when @xmath1 is interchanged with @xmath12 . this interchange is necessary because the sand only hits the edge of the opening when in motion ; in contrast , when at rest all the sand grains encountering the movable disk also load it . this confirms the assumption from above that the free fall of the sand starts on the surface of the paraboloid ; it also explains that the amount of sand flowing through the opening is proportional to the power @xmath20 of the effective radius of the opening . in the four inch wide and ten inch tall container , above the outlet , the entire surface of the sand packing subsided uniformly in the beginning . only gradually did a dip form vertically above the outlet . the dip grew continually , and above its sides sand fell down . concurrently , at the rim of the container a ring - shaped , almost horizontal surface remained . this surface also subsided , but without the granules of sand experiencing strong sideways motion . the flat ring gradually assumed a smaller width and disappeared completely when the funnel - shaped dip reached the outlet . from this it follows that the sand flows not only vertically towards the outlet but also along concentric inclined trajectories , and that the motion extends up to a slope , which a free surface of sand can exhibit . the motion in the inner part of the sand mass revealed itself very explicitly when i filled sand in a container having side walls made from a glass panel . since this glass panel touched the outlet , one could follow the motion of single grains of sand down to the opening . the strongest flow formed vertically above the outlet ; in fact the sand granules approached it with increasing yet moderate speed until , directly above , they were accelerated in a way that they could no longer be seen . nevertheless , the sand also flowed inwards from the side of the outlet , but this motion was interrupted frequently and only occurred periodically , presumably due to friction at the glass . underneath the opening , the stream of sand was not nearly as sharply bounded as a water - jet ; rather , it was surrounded by single granules that from time to time departed as far as several lines . because of this the streams flowing from the larger outlets showed a significant reduction in their diameters , extending about 2 inches deep . in addition , the measurement showed that even immediately under the disk the stream is already much weaker than at the outlet . the outlet was 0.335 inch in diameter while the stream was only 0.29 inch at a distance of @xmath21 lines , and contracted to 0.27 inch at greater depth . the reason for this effect is not the effective reduction of the opening mentioned above , since this would only explain the weakening of the stream by about 2% of an inch ; instead , the sand flowing sideways continues its motion towards the axis even after having passed the outlet , and the granules hitting the rim of the opening are also reflected towards the axis . the stream of sand hence experiences a contraction similar to the stream of a liquid ; and when one compares the diameter of the opening with the smallest diameter of the stream , the ratio appears as @xmath22 or for the ratio of cross sections @xmath23 this agrees closely with the known contraction ratios for liquid streams leaving openings in thin walls .

in a remarkable paper from 1852 , gotthilf heinrich ludwig hagen measured and explained two fundamental aspects of granular matter : the first effect is the saturation of pressure with depth in a static granular system confined by silo walls generally known as the _ janssen effect_. the second part of his paper describes the dynamics observed during the flow out of the container today often called the _ beverloo law _ and forms the foundation of the _ hourglass theory_. the following is a translation of the original german paper from 1852 .

You are an expert at summarizing long articles. Proceed to summarize the following text: resummation is often required in a perturbation theory to avoid infrared ( ir ) divergences to appear at finite energy scales . examples are 1pi diagram resummation ( schwinger - dyson equation ) to cure the on - shell singularity of the propagator , or daisy resummation @xcite to resum ir divergences of massless theories at finite temperature . we need leading ( or higher order ) logarithmic resummation near a second order phase transition point @xcite , and the htl resummation @xcite to be able to consistently define finite temperature observables in gauge theories . resummation means reorganization of the perturbative series in a way that the diagrams that are most important from the point of view of ir behavior , are taken into account first . however , ir importance does not necessarily means uv importance as well , and so it can happen that those ( counterterm ) diagrams that are needed to make the theory finite , are shifted to a later stage in the resummation process . then we observe that lower order results are divergent meaning that the resummed perturbation theory is _ inconsistent_. exceptions are the 1pi and leading log resummations , where , by chance , diagrams of ir and uv importance are the same , but other cases ( daisy , super - daisy , htl etc . resummations ) suffer from this difficulty . therefore , in these cases , the physically well - motivated resummation equations ( gap equations ) are meaningless in principle , and one usually makes some ad hoc assumption to obtain finite results . recently several papers were published that investigate this problem in various physical situations @xcite . in the present paper we try to go around the question of how to construct countertem diagrams for a given _ momentum independent _ resummation . we try to give two points of view on the problem : one is a direct method where we make explicit resummation for the counterterm diagrams , induced by the resummation of the `` normal '' diagrams ( section [ sec : massresum ] ) . the relevant counterterm diagrams can be constructed in each perturbative order , and finally we give an explicit form for the resummed lagrangian . to be more specific , we suggest , in @xmath0 theory , the following reorganization of the mass terms in the lagrangian in @xmath1 scheme to regularize the mass resummation : @xmath2 where @xmath3 is the multiplicative mass renormalization factor in @xmath1 scheme , and @xmath4 . this proposal is a generalization of the method of banerjee and mallik @xcite . at one hand this expression is equivalent with @xmath5 , so non - perturbatively we have the same physics . on the other hand we will prove that , if we take into account the under - braced terms first in one , and two loop levels , respectively , we obtain a mass resummed perturbation theory that consistently removes all the divergencies at any order . one of the most important corollary of these investigations is the following rule of thumb : we can renormalize the gap equations for mass resummation by _ first renormalizing the diagrams _ in an appropriate mass independent physical scheme ( eg . @xmath1 ) , and _ afterwards substituting the resummed parameters _ in the finite expressions . as it will be seen later on , although this is not the most generic renormalization method , still it is a possible and consistent way to deal with uv divergences . this strategy was already used in ref . @xcite , in the study of phase diagram of the quark - meson model . the other point of view treats resummed perturbation theory as a different renormalization scheme ( section [ sec : rg ] ) . we call this scheme , in which we impose environment ( @xmath6 or even time ) dependent renormalization conditions as _ resummation ( rs ) scheme_. this is not a physical choice , since changing the environment results in changing of the renormalization scheme . in order to have a meaningful result we have to relate the rs scheme and a bona fide renormalization scheme , eg . @xmath1 . this means constructing relations between the parameters of the renormalized lagrangians of the two schemes which formally appear as _ gap equations_. in several applications we need the inverse procedure : one has a physically motivated gap equation , and one would like to know the corresponding structure of counterterms . in this approach we consider the gap equations as relations between the @xmath1 and some rs scheme . expanding the gap equation in terms of the coupling constant we can read off the finite difference that has to be added to the @xmath1 counterterm to obtain the rs scheme counterterm . thus the rs scheme is defined , and so we can use perturbation theory to calculate any observables we need . if the choice of the gap equations was good enough , we will observe cancellations between the `` normal '' and counterterm diagrams . the diagrams that survive cancellation are not part of the resummation , and they can be used eventually to improve the resummation process . the paper ends with conclusions and an outlook ( section [ sec : conclusion ] ) . our basic model is the @xmath0 model where the lagrangian reads in terms of renormalized couplings and fields as @xmath7 as it is well known @xcite , in this model perturbation theory becomes unreliable at high temperatures , because of the presence of diagrams giving @xmath8 contribution . this ( ir ) problem can be cured by summing all the dominant tadpole contribution of the theory ( `` daisy diagrams '' ) . by this procedure one effectively replaces the mass @xmath9 by the resummed mass @xmath10 ( where , in this theory , @xmath11 to leading order ) , making the ir problem disappear ( apart from the phase transition point @xcite ) . since the daisy resummation does nothing else than replacing the mass with a different mass , it is equivalent with the `` thermal counterterm '' procedure @xcite . we add to and subtract from the lagrangian the same , temperature dependent term ; the relevant part of the lagrangian therefore reads @xmath12 as usual , the added term is treated as a part of the free lagrangian while the subtracted term , on the other hand , is taken into account one loop later as a ( thermal ) counterterm . by doing this we reinterpreted the perturbation theory , ie . we defined a resummation . the value of @xmath13 can be chosen to be the tadpole value , but its `` best value '' can be found out from the condition that at one loop order the self - energy is zero @xmath14 this is a gap equation @xcite which is to be solved to find @xmath13 . it turns out , however , that this naive definition is not consistent @xcite because of the mismatch in the counterterm diagrams . to see this we calculate the one loop self energy in the @xmath0 theory @xmath15 where @xmath16 is the bosonic tadpole function ; in dimensional regularization it reads : @xmath17 + \frac1{2\pi^2}\int\limits_m^\infty\ ! d\omega\,\sqrt{\omega^2-m^2}\,n(\omega),\ ] ] where @xmath18 is the bose - einstein distribution . we should write @xmath16 with argument @xmath19 , and that is what creates problem here . we , namely , already renormalized the theory at zero temperature , say , in @xmath1 scheme . that fixes the one - loop mass counterterm as @xmath20.\ ] ] so we have a divergency @xmath21 and a counterterm @xmath22 , that do not cancel each other . there is an unbalanced divergency , and so the rs scheme is inconsistent . since at the non - perturbative level ( infinite order ) the resummed theory is equivalent with the original one ( they have the same lagrangian ) , we expect that this divergence vanishes by higher loop effects . it has been shown @xcite that in the next order ( two loop in case of mass ) we indeed obtain contributions that cancel this divergence however , other new unbalanced divergences appear in this order . so finally the perturbation theory will not be consistent at any finite loop level . in order to make the theory consistent , we should find some method to shift up to one loop level those two - loop diagrams that are necessary to cancel the unbalanced divergences at one loop order : ie . we have to reorganize ( resum ) the . the idea is the same as in the case of the thermal mass . we find the correction to the counterterm , add it to and subtract it from the lagrangian , but classify them to belong to different loop orders . let us denote the necessary mass term by @xmath23 and call it _ compensating counterterm_. so , instead of ( [ eq : thermmass ] ) , we use the following mass terms in the lagrangian : @xmath24 in perturbation theory the first term is taken into account in the propagator , the second and third are taken into account first at one loop level , while the last term first contributes at two - loop level . @xmath23 has to be determined order by order in a way that it cancels the remaining divergence of the physical observable at a given order . although it is temperature dependent and also divergent in the present example , its value is irrelevant from the point of view of the consistency of the complete theory , since the lagrangian is the same as the original one , it does not depend on the value of @xmath25 . using the proposed decomposition of the lagrangian corresponding to ( [ eq : corrthermmass ] ) let us compute the resummed and renormalized self - energy of the @xmath0 model . at one loop level the self - energy reads with this new counterterm structure as @xmath26 @xmath27 is defined in ( [ msbardeltam2 ] ) ; it cancels a part of the divergence hidden in @xmath16 , the rest has to be canceled by @xmath28 . we can choose it in analogy with the @xmath1 form ( [ msbardeltam2 ] ) @xmath29.\ ] ] in principle we could add an arbitrary finite term to this form . this would modify the value of @xmath13 , and finally would lead to a different resummation process . in the language of section [ sec : rg ] it would correspond to a different renormalization scheme in the zero and finite temperature parts . for further details cf . section [ sec : rg ] . with this choice , using the form of @xmath16 from ( [ tb ] ) we find @xmath30 where @xmath31 is the @xmath1 renormalized tadpole diagram . we could obtain this result by first renormalizing the divergent diagrams in @xmath1 scheme , and afterwards by substituting the resummed mass into the finite expression . now let us turn to the determination of the self - energy at two loop level . the contribution from two - loop level diagrams is represented symbolically in the following way : @xmath32{setsun.eps } } + \frac{\lambda^2}4\ ; \raisebox{-0.25cm}{\includegraphics[height=1.6cm]{doublescoop.eps } } + \frac{\delta\lambda_1}2 \;\raisebox{-0.25cm}{\includegraphics[height=0.8cm]{tadpole.eps } } + \frac\lambda2 \;\raisebox{-0.25cm}{\includegraphics[height=0.85cm]{fish.eps } } -\delta m_{t1}^2 + \delta m_2 ^ 2 + \delta m_{t2}^2 - \delta z_2 p^2.\ ] ] the propagators have the squared mass @xmath33 , in this resummed theory . the value of the diagrams in dimensional regularization , writing out explicitly only their divergent part , are : @xmath34{tadpole.eps } } = \frac{m_t^2}{16\pi^2}\left(-\frac1{\varepsilon}+\gamma_e-1+\ln\frac{m_t^2 } { 4\pi\mu^2 } \right ) + t_b^{t\neq0},{\nonumber\\ } & & \raisebox{-0.25cm}{\includegraphics[height=0.85cm]{fish.eps } } = ( -\delta m_t^2 + \delta m_1 ^ 2 + \delta m_{t1}^2 ) \left [ \frac1{16\pi^2}\left(-\frac1{\varepsilon}+\gamma_e + \ln\frac{m_t^2 } { 4\pi\mu^2 } \right ) + i_\mathrm{fish}^{t\neq0}\right],{\nonumber\\}&&\raisebox{-0.3cm}{\includegraphics[height=0.8cm]{setsun.eps } } = \frac{3m_t^2}{(4\pi)^4}\left[\frac1{2{\varepsilon}^2}-\frac1{\varepsilon}\left(\gamma_e -\frac32 + \ln\frac{m_t^2}{4\pi\mu^2}\right)\right ] - \frac{p^2}{4(4\pi)^4 } \frac1{\varepsilon}- \frac3{16\pi^2}\,t_b^{t\neq0}\ , \frac1{\varepsilon}\;+\;i_\mathrm{setsun}^{finite}{\nonumber\\ } & & \raisebox{-0.4 cm } { \includegraphics[height=1.6cm]{doublescoop.eps } } = \frac{m_t^2}{(4\pi)^4 } \left[\frac1{{\varepsilon}^2 } -\frac1{\varepsilon}\left(2\gamma_e-1 + 2\ln\frac{m_t^2}{4\pi\mu^2}\right ) \right ] - \frac1{16\pi^2 } \frac1{\varepsilon}\left(t_b^{t\neq0 } + m_t^2 i_\mathrm{fish}^{t\neq0}\right ) \;+\;i_\mathrm{dsc}^{finite}.\end{aligned}\ ] ] since we do resummation in the @xmath1 scheme , the value of @xmath35 and @xmath36 counterterms are fixed , we can not modify them . @xmath37 , where @xmath38 the other counterterms read as @xmath39 using these values the divergent part is still not canceled , what remains is : @xmath40 to cancel this divergency we can choose @xmath41 , although , of course , an arbitrary finite part could be added to this expression . with this choice , however , we get further support to the practical observation of the one loop calculation : we first do renormalization in @xmath1 scheme , and substitute the resummation mass in the finite expressions . it should be clear , too , that we could add finite terms for @xmath28 and @xmath42 , and then this statement would not be true any more . thus this is just a comfortable possibility . one can conjecture that this feature persists at higher loop orders , since in @xmath1 scheme the overall divergency of a diagram always can be written as @xmath43 with some mass - independent @xmath44 factor . then , by choosing @xmath45 and @xmath46 , the overall divergency disappears . this should also be true for any other mass - independent schemes . we can , therefore , put forward a proposition : * proposition : * if in a renormalization scheme the zero temperature counterterm is @xmath47 and @xmath44 is mass - independent , then at finite temperature the theory with thermal counterterm @xmath13 and compensating counterterm @xmath48 yields finite result at each order of the perturbation theory . _ proof _ : let us consider a 2-point diagram @xmath49 in the original ( not resummed ) theory that has an overall divergence , and all the subdivergences were consistently subtracted . then , if the theory is renormalizable , the divergence of the diagram should be proportional to @xmath9 ; the proportionality constant can be obtained as @xmath50 . since the divergence is canceled by the counterterm @xmath47 , the divergent part should be equal to an appropriate part from the mass renormalization factor @xmath44 , which is mass independent @xmath51 in the resummed theory we have two effects . one is the substitution of the mass term by the resummed mass : @xmath52 . the divergent parts of the self energy diagrams , in our mass - independent scheme , are proportional to this new mass term . but we have also prepared the corresponding counterterm , since the third term of ( [ eq : corrthermmass ] ) reads @xmath53 . so this part of the divergences disappear in the same way as in the unresummed case . the other effect is the @xmath54 insertion coming from the second ( thermal ) counterterm of ( [ eq : corrthermmass ] ) . the sum of the single mass insertion diagrams is just @xmath55 , because at each propagator @xmath56 ( it is evident for feynman propagators , at finite temperature it can be seen using imaginary time propagators ) . its divergent part can be written , using ( [ dz ] ) , as @xmath57 . this is exactly canceled by the corresponding part of the compensating counterterm at this order : @xmath58 . so at each diagram the overall divergences are canceled by the proposed counterterms . using bphz argumentation , this is enough to make the complete theory finite . now we have a consistent method for how to reorganize the perturbation theory according to a given ( mass ) resummation in such a way as to have finite results in all orders in @xmath1 scheme . in order to generalize the result so as to be able to treat other schemes ( not just @xmath1 ) , or environment dependence of other couplings ( cf . vertex resummation ) , we change our viewpoint , and we rephrase the complete resummation procedure using the renormalization group ( rg ) argumentation : this will be the topic of the present section . first we demonstrate at one loop level that we can obtain the results of the previous section using a special renormalization scheme . we define the mass counterterm of the new scheme as @xmath59 where @xmath60 denotes the @xmath1 counterterm and @xmath61 is an arbitrary , finite expression . the other counterterms are the same as in @xmath1 scheme . then the one loop self energy reads @xmath62 this expression differs in a finite term from the corresponding @xmath1 expression . at one hand it assures finiteness , on the other hand this means that the new mass parameter must be different from the @xmath1 mass parameter in order to describe the same physics . in fact , applying the ideas of the renormalization theory @xcite to the present case , there must be a choice @xmath63 for which we have the same value _ for all observables _ in the two schemes at a given order . this function can be read off from the requirement that the bare parameters are the same in the two schemes @xcite : @xmath64 then the counterterm of the new scheme ( [ newscheme ] ) can be written as @xmath65 this is the result of the proposition of subsection [ sec : higherloop ] applied to the one loop case . for the definition of the new scheme we need to specify a value for @xmath61 . we may , for example , choose the mass shell scheme , where the full self - energy reads @xmath66 . from ( [ newschemeselfen ] ) and ( [ matchnewscheme ] ) we find for the value of @xmath61 @xmath67 where @xmath68 now means the tadpole diagram , renormalized in the @xmath69 scheme . this equation is the renormalized gap equation for the mass . this analysis suggests that the mass resummation is equivalent with the choice of a new renormalization scheme . using the relation of the masses of the two schemes , we can compute finite observables that can be interpreted as the result of the resummed @xmath1 scheme . the mass relation , together with the definition of the new scheme yields finite gap equation for the new ( resummed ) mass , which is the usual gap equation in case of the mass shell scheme . motivated by the previous subsection , we can start to build up the resummation strategy from the point of view of renormalization schemes . resummation itself means that some higher order diagrams are taken into account at lower levels with the consequence that these diagrams are missing from the higher order calculation . so , at least for the mass and coupling constant resummation , resummation is equivalent with a scheme that cancels these diagrams with a proper choice of the finite part of the counterterms . since we play only with the finite part of the counterterms , we will still have a consistent scheme . so this new scheme the _ resummation ( rs ) scheme _ differs from a general scheme ( eg . @xmath1 ) in that a certain set of diagrams is missing from the calculation . the rs scheme is a bona fide scheme while we keep the counterterms fixed : for example we can equally well use the ms and the @xmath1 schemes . however , at finite temperature , the finite parts of the counterterms will depend on temperature , and so the resummation scheme itself varies with temperature . in a more general case rs scheme can depend on the complete environment @xcite : beyond the temperature on chemical potential , background condensates , time , etc . since results in different schemes have different physical interpretation , rs scheme can not directly describe the effects of the environment variation . example is the mass shell scheme where @xmath70 , and we have no explicit temperature dependence . to be able to draw physical consequences , we have to project the results to a common reference scheme . technically what we should do is to relate the rs scheme to a physical renormalization scheme for concreteness we will use the @xmath1 scheme . since both of them are mathematically correct schemes , in case of renormalizable theories , this can be accomplished by changing the values of renormalized parameters of the lagrangian @xcite ( including wave function renormalization ) . in @xmath0 theory there must exist environment - dependent parameters @xmath71 such as for any @xmath72-point function with momenta @xmath73 we have @xmath74 in perturbation theory the equality holds only up to the computed order ; in fact the difference is the effect of resummation . the value of the `` running '' parameters can be computed using the fact that the bare quantities are independent of the scheme @xcite . in this way we can use the rs scheme for perturbation theory , enjoying its good ir convergence , and still have results in a physical renormalization scheme ( eg . @xmath1 ) , where also the parameters of the theory can be obtained from the usual renormalization conditions . sometimes we can explicitly define the new scheme ( eg . the mass shell scheme for mass resummation ) , but more often we just have a guess for the resummed parameters , and we hope that it performs some kind of resummation . that is we start with explicit functions of the form of ( [ eq : relation ] ) for the resummed mass and coupling constant ( and eventually , wave function renormalization ) . according to the line of thought above these relations can be interpreted as generators of a rs scheme : we expand them in power series in @xmath75 @xmath76 and we interpret @xmath77 as the @xmath72th order finite mass counterterm that has to be added to the @xmath1 counterterm ; similarly , @xmath78 is the finite coupling constant counterterm , @xmath79 is the finite wave function renormalization counterterm . having defined the counterterms , the rs scheme is appropriate to do perturbation theory , and the results can be related to the @xmath1 scheme by the same relations ( [ eq : relation ] ) . if our guess was good enough we will observe cancellation between `` normal '' and counterterm diagrams at higher orders . the main message , however , is that , independently of whether a choice represents a physically meaningful resummation or not , _ any choice _ leads to a finite , consistent perturbation theory . so as a possible strategy for resummation , we can do the following : we perform regularized perturbation theory in a generic scheme , keeping counterterms free , at a certain loop order . the divergent part of the counterterms are then fixed , as usual , to cancel the uv divergences of the diagrams , while the finite parts are fixed to cancel the ir dangerous parts . so we defined a scheme , which is , however , only a rs scheme , since the counterterms can depend on the environment . we determine its relation to the @xmath1 scheme of the form ( [ eq : relation ] ) by requiring that the bare parameters should be the same in the two schemes . this relation , by construction , corresponds to a resummation up to this perturbative order , and we should have some plausible guess to choose the higher order terms , hoping that it still performs some kind of resummation . different choices , of course , correspond to different resummations . this procedure can be repeated at each perturbative order , and so the resummation process can be improved . using the ideas of the previous subsection , in this section we perform both mass and coupling constant resummation . this latter is important to describe the `` softening '' of the theory close to a phase transition point @xcite , and also to consistently describe vertex resummation @xcite . at one loop level we can start from the effective potential @xmath80 where @xmath81 . the effective , background dependent mass and coupling constant then reads @xmath82 a possible way of doing coupling constant resummation that we make vanish all of the terms below @xmath83 . that means : @xmath84 for the mass resummation we demand that @xmath85 ; with the @xmath86 value defined above we obtain @xmath87 this counterterm depends on the background only at @xmath88 order . if @xmath89 depends on the environment , these two counterterms define an environment dependent rs scheme . if we denote by @xmath90 and @xmath91 the divergent parts according to the @xmath1 scheme , and by @xmath92 and @xmath93 the finite parts scheme @xmath94 and so @xmath95 , then we can write @xmath96 using the renormalization scheme independence of the bare parameters we can relate the resummed and the @xmath1 parameters at one loop level as @xmath97 these equations determine the relation between the parameters of the rs scheme and of the @xmath1 scheme ; but it can also be interpreted as the formulae for how to resum diagrams in the @xmath1 scheme . in fact , by the renormalization prescriptions of the rs scheme , @xmath98 is the complete static 4-point function and @xmath9 is the complete mass . the gap equations ( [ gapeq ] ) are derived at one loop level , and are not necessarily useful at higher orders , that is it does not necessarily resum any subset of diagrams . there are arguments , however , that help to guess the correct form . to achieve super - daisy resummation , for example , we should use the resummed mass in the one loop expressions : that suggest that we should always use @xmath9 in the right hand side of ( [ gapeq ] ) and @xmath98 if it is multiplied with @xmath99 . then all the possible mass resummations are done , so we should use @xmath75 for any other appearance of the coupling constant in the expression of the mass resummation . for the explicit coupling constants in the @xmath98 resummation we recall that the double scoop ( two bubbles ) diagram contributes @xmath100 , but if we iterate the expression for @xmath98 as a function of @xmath75 in ( [ gapeq ] ) , we would get @xmath101 . to cure this problem we should write @xmath102 instead of @xmath103 . so , finally , the proposed form for the resummation reads @xmath104 and @xmath81 , in agreement with 4pi suggestions @xcite . from ( [ newschemeselfen ] ) and ( [ eq : v ] ) we find @xmath105 , and so we can write @xmath106 it is instructive to examine these expressions for small masses . since the result of the finite temperature part has @xmath107 as the only dimensionless quantity , this is equivalent to the high temperature approximation @xcite @xmath108 where @xmath109 . then , using the notation @xmath110 , we have @xmath111 then the gap equations read @xmath112 the second term of the rhs of ( [ eq : second ] ) depends on @xmath113 at @xmath88 order , thus it is consistent to omit its background dependence . the last term is the consequence of the choice of the @xmath1 counterterms . so we have @xmath114 the solution therefore reads @xmath115 where @xmath116,\qquad w=\frac{\lambda_{{\ensuremath{\overline{\mathrm{ms}}}}}t}{16\pi\kappa},\qquad \kappa = 1 - \frac{\lambda_{{\ensuremath{\overline{\mathrm{ms}}}}}}{32\pi^2}\ln\frac{t^2}{\tilde\mu^2}.\ ] ] the solution has some remarkable properties . first of all it is a completely finite result , we do not need to bother with renormalization any more . secondly although coming from a high temperature expansion it nicely extrapolates between the 4d and 3d fixed points of the system . for high scales , namely , ( [ resumresult ] ) can be approximated as ( we set @xmath117 ) @xmath118 the formula for @xmath98 is the result of the non - perturbative renormalization method of @xcite , where it was computed by the explicit summation of bubble diagrams . we can realize here the running coupling and mass in the @xmath1 scheme , and write @xmath119 where @xmath120 . this corresponds to a 4d behavior in the @xmath1 scheme . at high temperatures the @xmath121 term takes over in the coupling constant , and the system follows a different trajectory . while the mass runs in the same way as before , @xmath122 this resummation describes correctly the second order nature of the phase transition ( as opposed to the daisy resummation ) . denoting @xmath123 , we find @xmath124 for @xmath125 . near the critical point the solution behaves as @xmath126 the critical exponent for the mass is @xmath127 in this case , the fixed point of the coupling constant is @xmath128 ( ie . it becomes gaussian ) , and the temperature dependence for the approach of this critical point is @xmath129 , where @xmath130 . this is exactly the behavior of the @xmath131 model at @xmath132 @xcite . indeed , in this simple approximation we resummed leading order diagrams in the large @xmath133 expansion , thus this agreement is not unexpected @xcite . in this paper we investigated the question of to what extent the uv and ir regularization of a system , ie . renormalization and resummation can be reconciled . we described two methods : one is a constructive method of how to resum higher order counterterm diagrams in order to cure the uv divergences appearing in mass resummation in @xmath1 scheme . similarly to the thermal mass counterterm idea we introduced a `` compensating counterterm '' that has to be added to and subtracted from the original lagrangian , the two terms contributing at different loop order . the other method is a rephrasing of this idea in the language of the renormalization group . here we defined a new scheme , the resummation ( rs ) scheme , where the infinite part of the counterterms cancels the uv divergences , the finite part cancels the ir divergences of the theory at a given order . the drawback of this scheme is that it depends explicitly on the environment , but , with help of the renormalization group , we can project the results of the resummation scheme to a reference scheme , eg . @xmath1 . from the point of view of the resummation , this projection formally appears as a set of gap equations . we demonstrated these ideas by performing coupling constant and mass resummation in the @xmath0 model . in practice usually the problem of resummation and renormalization comes up in two forms . the first arises when we work on a diagrammatic basis and we compute a gap equation . at first glance the divergences of the resummed diagrams and counterterms do not cancel each other , so we wonder whether there is a method to consistently renormalize the divergent gap equations . we give positive answer to this question , proposing to use the compensating counterterm , or renormalization group ideas described in the paper . as a rule of thumb we can use that prescription that , for mass resummation in mass independent scheme , we first renormalize the equation perturbatively , and we apply this finite expression to compute the resummed parameters . the other type of problem is when we modify our finite gap equation in a well motivated way , hoping that it catches important higher order effects , and we wonder whether the new form is compatible with renormalization . the answer is again positive : any form of the gap equation is compatible with renormalization . technically we should treat our gap equations as transformation equations from @xmath1 scheme to a specific rs scheme . expansion of the solution of gap equations in terms of the coupling constants yields the finite difference of the counterterms in the two schemes at a given order . with the so - 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resummation , ie . reorganization of perturbative series , can result in an inconsistent perturbation theory , unless the counterterms are reorganized in an appropriate way . in this paper two methods are presented for resummation of counterterms : one is a direct method where the necessary counterterms are constructed order by order ; the other is a general one , based on renormalization group arguments . we demonstrate at one hand that , in mass independent schemes , mass resummation can be performed by gap equations renormalized _ prior to _ the substitution of the resummed mass for its argument . on the other hand it is shown that any ( momentum - independent ) form of mass and coupling constant resummation is compatible with renormalization , and one can explicitly construct the corresponding counterterms .

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Proceed to summarize the following text: peculiar velocity surveys covering a fair fraction of the sky are now reaching to 6000 and beyond ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) and are being interpreted as evidence for substantial flows on these scales ( @xcite , @xcite , @xcite , @xcite , @xcite , @xcite ) . however , the amplitude , direction , and scale of these flows remain very much in contention , with resulting uncertainties in the theoretical interpretation and implications of these measurements ( @xcite , @xcite ) . indeed , recently published conflicting results suggest that the motion of the lg is either due , or is not due , to material within 6000 , and that _ iras _ galaxies either trace , or do not trace , the dark matter which gives rise to the observed peculiar velocities . the most recent potent reconstruction of the markiii velocities ( @xcite ) shows that the bulk velocity can be decomposed into two components arising from the mass fluctuation field within the sphere of radius @xmath3 about the lg and a component dominated by the mass distribution outside that volume . for convenience , we refer to this boundary at @xmath3 as the `` supergalactic shell '' since it includes the main local attractors in the supergalactic plane , the great attractor and perseus - pisces . this new analysis shows dominant infall patterns by the ga and pp but very little bulk flow within the supergalactic shell . the tidal component inside this volume is dominated by a flow of amplitude @xmath4 in the supergalactic direction @xmath5 , which is likely generated by the external mass distribution on very large scales ( see also @xcite , @xcite ) . this interpretation is also supported by an increasingly large number of tf / fp investigations ( based on the distribution and motion of abell clusters ) which report the detection of streaming motions of amplitudes greater than 700 beyond @xmath6 and away from the cmb dipole ( @xcite , @xcite , @xcite , @xcite ) . other investigations using nearly homogeneous samples of galaxies within and outside the supergalactic shell find motion consistent with the amplitude and direction of the cmb dipole @xcite . this suggests that the reflex motion of the local group could be explained by material contained within the supergalactic shell . this confusion stems , in large part , in our inability to perfectly match the many heterogeneous samples for flow studies into one self - consistent homogeneous catalogue . much of the problem lies in the fact that , with the exception of a few surveys beyond @xmath7 ( @xcite , @xcite , @xcite ) , none of the surveys within the supergalactic sphere sample the _ entire _ sky uniformly . in an attempt to overcome this problem , two of us ( jw & sc @xmath8 collaborators ) have recently combined the major distance - redshift surveys from both hemispheres ( published before 1994 ) into a catalog of 3100 galaxies ( @xcite ) , but showed that full homogenization at the @xmath9% level , the minimum required for a @xmath10 bulk flow detection at 6000 , can not be achieved . due to subjective reduction techniques and varying selection criteria , fundamental uncertainties remain when trying to match greatly disparate tf datasets ( @xcite ) . furthermore , a revised calibration of the markiii tf zero - points based on maximal agreement with the peculiar velocities predicted by the iras 1.2jy redshift survey suggests a possible source of systematic error for the data sets which cover the pp cone ( @xcite ) . this uncertainty has not seriously affected mass density reconstructions within the supergalactic shell ( @xcite ) but it could lead to spurious estimates of the bulk flows on larger scales . a newer calibration of the courteau / faber catalogue of northern spirals , not included in markiii , has been published ( @xcite , @xcite ) but a revision of the markiii catalogue is in progress ( @xcite ) . the need to tie all existing data bases for cosmic flow studies in an unambiguous fashion is clear . to that effect , we initiated a new survey in 1996 using noao facilities to measure tf distances for a complete , full - sky sample of sb@xmath0sc galaxies in the supergalactic shell for which we will obtain _ precise _ and _ uniform _ photometric and spectroscopic data . this will be the first well - defined full - sky survey to sample this scale , free of uncertainties from matching heterogeneous data sets . the sfi survey of giovanelli @xcite resembles ours in its scope and sky coverage , but it relies on a separate dataset ( @xcite ) for coverage of the southern sky and thus can not attain full - sky homogeneity . our survey , on the other hand , is designed from the outset to be homogeneous to the minimum level required for unambiguous bulk flow detection at the supergalactic shell . because of the overlap with existing surveys at comparable depth ( markiii + sfi ) , this new compilation will be of fundamental importance in tying the majority of existing data sets together in a uniform way , which will greatly increase their usefulness for global analyses of mass fluctuations in the universe . our sample is selected from the optical redshift survey ( @xcite ) , consisting of galaxies over the whole sky with m@xmath11 and @xmath12 from the ugc , eso , and esgc ( @xcite ) . it includes all non - interacting sb and sc galaxies with redshifts between 4500 and 7000 from the local group and inclinations between @xmath13 and @xmath14 , in regions where burstein - heiles extinction is less than 03 . this yields an all - sky catalog of 297 galaxies . following the approach of @xcite , we use the sample itself to calibrate the distance indicator relation ; this mitigates the need to tie the sample to external tf calibrators such as clusters ( although it precludes measurement of a monopole term in the velocity field ) . given a tf fractional distance error of 20% , the statistical uncertainty on a bulk flow from @xmath15 galaxies at common distance @xmath16 is @xmath17 . as the measured ( and much contested ) bulk motions on these scales are of the order of 300 , a detection of high statistical significance is well within reach . data taking and reduction techniques follow the basic guidelines of previous optical tf surveys ( @xcite , @xcite , @xcite , @xcite ) . our survey is now complete , which is essential to achieve our statistical requirements and ensure a rigorous analysis . the spectroscopy relies on measurement of h@xmath18 rotation velocities at 2.2 disk scale lengths for the tightest tf calibration and best match to analogous 21 cm line widths ( @xcite , @xcite ) . the photometry is based on the kron - cousins @xmath19 and @xmath20 systems which will allow direct matching with two largest tf field samples to date ( @xcite,@xcite ) . one of the key features of this study is not only its all - sky sample selection but the independent duplication of all data reductions ( by at least 2 , if not 3 , of us ) . these reductions and a first flow analysis based on the shellflow sample alone should be published soon ( @xcite ) . we also plan a more extensive analysis using the recalibrated markiii combined with other new catalogs not included in the original markiii . 99 corwin , h. g , & skiff , b. a. 1994 , extension to the southern galaxies catalogue , in preparation clutton - brock , m . , & peebles , p.j.e . 1981 , aj , 86 , 1115 courteau , s. 1992 , phd . thesis , uc santa cruz courteau , s. , faber , s.m . , dressler , a. , & willick , j.a . 1993 , apj , 412 , l51 courteau , s. 1996 , apjs , 103 , 363 courteau , s. 1997 , aj , 114 , 2402 courteau , s. ( + shellflow team ) 1999 ( in preparation ) da costa , l. n. , freudling , w. , wegner , g. , giovanelli , r. , haynes , m.p . , & salzer , j.j . 1996 , apj , 468 , l5 dekel , a. , eldar , a. , kolatt , t. , yahil , a. , willick , j. a. , faber , s. m. , courteau , s. , & burstein , d. 1999 , apj ( submitted ) giovanelli , r. , haynes , m.p . , freudling , w. , da costa , l. n. , salzer , j.j . , & wegner , g. 1998 , apjl , in print , astro - ph/9807274 han , m .- s . , & mould , j. r. 1992 , apj , 396 , 453 hudson , m. ( + smac team ) 1998 ( in preparation ) lauer , t. r. , & postman , m. 1994 , apj , 425 , 418 mathewson , d. s. , & ford , v. l. 1994 , apj , 434 , l39 mathewson , d. s. , ford , v. l , & buchhorn , m. 1992 , apjs , 81 , 413 [ m92 ] postman , m. 1995 , in _ dark matter _ , aip conf . series 336 , 371 postman , m. , & lauer , t. r. 1995 , apj , 440 , 28 riess , a. , press , w. , & kirshner , r. p. 1995 , apj , 445 , l91 santiago , b. x. , strauss , m. a. , lahav , o. , davis , m. , & huchra , j. p. 1995 , apj , 446 , 457 scaramella , r. , 1989 , nature , 338 , 562 schlegel , d. 1996 , phd . thesis , uc berkeley strauss , m.a . 1996 , in _ critical dialogues in cosmology _ , ed . neil turok ( singapore : world scientific ) strauss , m.a . , & willick , j.a . 1995 , physics reports , 261 , 271 willick , j. a. , courteau , s. , faber , s. m. , burstein , d. , dekel , a. , & strauss , m. a. 1997 , apjs , 109 , 333 willick , j. a. & strauss , m. s. 1998 , apj , in press ( astro - ph/9801307 ) willick , j.a . 1998 , apj ( submitted ) willick , j.a . 1999 ( in preparation )

we present a new optical tully - fisher ( tf ) investigation for a complete , full - sky sample of 297 sb@xmath0sc spirals with redshifts between 4500 and 7000 . the survey was specifically designed to provide _ uniform , well - calibrated _ data over both hemispheres . all previous tf surveys within the supergalactic shell ( @xmath1 ) have relied on matching separate data sets in the northern and southern hemispheres and thus can not attain full - sky homogeneity . analyses of the cosmological dipole and peculiar velocities based on these studies have produced contradictory claims for the amplitude of the bulk flow and whether it is generated by internal or external mass fluctuations . with shellflow , and further zero - point calibration of existing tf data sets , we expect a high - accuracy detection of the bulk flow amplitude and an unambiguous characterization of the tidal field at 6000 . # 1#1@xmath2

You are an expert at summarizing long articles. Proceed to summarize the following text: stone - wales defect @xcite @xmath1(55 - 77 ) is the simplest example of topological disorder in graphene and other @xmath2-hybridized carbon systems . it can be formed by rotating a c - c bond by @xmath3 with regard to the midpoint of the bond referred to as the sw transformation so that four hexagons are turned into two pentagons and two heptagons . this defect has received considerable amount of attention @xcite , because it has the lowest formation energy among all intrinsic defects in graphenic systems , and because it presumably plays an important role in plastic deformation of carbon nanotubes ( cnt ) under tension @xcite by relieving strain @xcite . it can also act as a source for dislocation dipoles @xcite . regardless of being the lowest energy defect in graphene @xcite and other @xmath2-hybridized carbon nanostructures , the @xmath1(55 - 77 ) needs about 5 ev to appear in graphene @xcite , and 35 ev in cnts with a diameter above 1 nm @xcite , which should lead to a negligible equilibrium concentration of such defects at room temperature . however , recent advances in hrtem have allowed the identification of such defects in graphene @xcite and cnts @xcite . moreover , sw transformations play an important role in the response of graphene to electron irradiation @xcite , leading to changes in the morphology of vacancy - type defects @xcite and to their migration . such changes are equally surprising , because the barrier for bond rotation is about 5 ev @xcite , which should exclude thermal activation as a cause for sw transformation at room temperature during experimentally relevant time scales . regarding irradiation effects , previous simulations @xcite showed that an energy of @xmath4 ev must be transferred to a c atom in graphene in the in - plane direction for a bond rotation to occur . also this can not explain the frequently observed sw transformations under the usual tem imaging conditions , since with typical acceleration voltages ( @xmath5 kv ) the transferred kinetic energy in the direction almost perpendicular to the electron beam will remain significantly below 10 ev . here , by combining aberration - corrected ( ac- ) hrtem with atomistic computer simulations , we show that topological defects associated with the sw transformation can be formed in @xmath2-hybridized carbon nanostructures by impacts of individual electrons at energies even _ below _ the threshold for a carbon atom displacement . we further study in detail the mechanisms of irradiation - driven bond rotations for pre - existing vacancy - type defect structures and how they transform and migrate due to electron impacts . at the same time we explain why electron irradiation at moderate energies ( @xmath6 kev ) tends to rather amorphize @xcite than perforate graphene . we also show via simulations that the @xmath1(55 - 77 ) can appear in curved graphitic structures due to `` incomplete '' recombination of irradiation - induced frenkel defects , reminiscent of the formation of wigner - type defects in silicon @xcite . graphene membranes used in our experiments were prepared by mechanical exfoliation of graphite on si / sio@xmath7 substrates and transfer of the resulting graphene sheets onto tem grids as described previously @xcite . for tem experiments we used an fei titan @xmath8 equipped with an image - side aberration corrector , operated at 80 kv . the spherical aberration was set to 15 @xmath9 m and images were recorded at scherzer defocus . the extraction voltage of the source was reduced to 2 kv and the condensor lens c2 was switched off in order to minimize the energy spread . under these conditions , dark contrast in the images can be directly interpreted in terms of the atomic structure . image sequences were recorded on a ccd camera with exposure times of 1 s and intervals of approximately 2 s. we carried out atomistic computer simulations based on the spin - polarized density functional theory ( dft ) implemented in the plane wave basis set vasp code @xcite . the projector augmented wave potentials @xcite were used to describe the core electrons , and the generalized gradient approximation of perdew , burke and ernzernhof @xcite for exchange and correlation . we included plane waves up to a kinetic energy of 300 ev . the @xmath10-point sampling of the brillouin zone was performed using the scheme of monkhorst - pack @xcite for the periodic dimensions . structure relaxation calculations were combined with molecular dynamics ( dft - md ) simulations with a lower kinetic energy threshold and fewer @xmath10-points . due to the high computational cost of the dft - md method , only a few simulations were carried out at this level . whenever statistics needed to be gathered , we calculated the forces using the non - orthogonal dft - based tight binding ( dftb ) method @xcite . the main results were checked against dft - md . in total , we carried out @xmath11 dynamical dftb - md simulations . the simulated structures consisted of 120200 atoms and were fully optimized . for the displacement threshold simulations , one of the atoms was assigned a kinetic energy @xmath12 with the initial velocity vector pointing to a preselected direction . the initial temperature of the system was set to 5 k , although we observed no differences when carrying out the simulations for initially stationary atoms . displacement threshold @xmath13 ( minimum kinetic energy required to eject the atom ) was found to be 22.50 ev , in a good agreement with earlier dftb results @xcite . it is also close to the dft value ( 22.03 ev ) @xcite . for the annihilation simulations , various system temperatures were studied ( 5001200 k ) both to fasten the migration of the adatoms and to understand the effect of an elevated temperature ( as will be mentioned below ) . we begin the presentation of our results with the description of experimental observations of @xmath1(55 - 77 ) in hrtem images . several long image sequences , typically containing hundreds of images from clean and initially defect - free graphene membranes , were recorded at 80 kv . occasionally , @xmath1(55 - 77 ) defects appear in individual exposures , as in the example shown in figure [ px : swexp]b ( figure [ px : swexp]c with structure overlay ) . remarkably , in most of the observed cases , isolated @xmath1(55 - 77 ) appeared in pristine graphene for one 1 s exposure , only to disappear in the following frame . hence , the lifetime of this defect under the 80 kv electron beam in terms of irradiation dose is of the order of @xmath14 , the dose used for a single exposure . ( 55 - 77 ) ( b ) . frame ( c ) shows the same image with a structure overlay . @xmath1(55 - 77 ) disappears in the following frame ( d ) . scale bar is 1 nm . ( see also video s1 in ref . @xcite . ) ] to understand the appearance and disappearance of @xmath1(55 - 77 ) , we carried out atomistic simulations for individual displacement events under the electron beam . after calculating @xmath13 , we extended this calculation to all in - plane ( @xmath15 ) and out - of - plane angles in the range @xmath16 $ ] , figure [ px::sect ] . displacements with @xmath17 would result in transferred kinetic energies of more than 2 ev below @xmath13 for an electron beam even at 120 kev . since the displacement threshold increases for increasing @xmath18 , it is unlikely that this restriction would lead us to miss any significant electron beam - induced structural changes , especially for electron energies similar to those used in this study ( 80 kev ) . the calculated displacement thresholds are shown in figure [ px::sect ] in a relative scale along with the space angles for which we observed the formation of @xmath1(55 - 77 ) . as a function of the displacement space angle ( @xmath18 , @xmath15 ) . crosses mark the angles for which we observed impact - induced sw transformations . ] it is evident from figure [ px::sect ] that the sw transformation is a very likely event at displacement angles slightly away from the graphene plane normal ( @xmath19 ) . the transferred kinetic energies ( @xmath12 ) required for this process are below the displacement threshold for the corresponding @xmath15 and @xmath18 ( @xmath20 ) since no actual removal of the recoil atom is required for the bond rotation to occur . typically , @xmath21 ev resulted in the @xmath1(55 - 77 ) formation , although for some space angles even @xmath22 ev was enough . the probability for @xmath1(55 - 77 ) formation is particularly high for certain space angles , which is related to different mechanisms of sw transformation , as described below . the above - presented result is in clear contrast with the earlier simulation results for graphite @xcite where no @xmath1(55 - 77 ) formation was observed for low @xmath18 . this discrepancy is caused by the neighboring graphene planes in the case of graphite : the displaced atom gets attached to the adjacent layer and does not therefore initiate a bond rotation . in figure [ px::swprocs ] we show the two processes which account for the majority of the sw transformations observed during our simulations . in the `` circle '' process ( figure [ px::swprocs]a ) , the displaced atom circles around its neighbor , whereas in the `` nudge '' process ( figure [ px::swprocs]b ) it nudges the neighbor to cause the bond rotation . note that the example cases are for the same @xmath15 and almost same @xmath12 , but for different @xmath18 . the resulting process for each displacement is an interplay of all three variables ( @xmath12 , @xmath18 , @xmath15 ) . similar mechanisms also exist for cnts . however , as two new parameters ( tube diameter and chirality ) should be introduced for quantitative analysis of sw transformation , we did not study this process in nanotubes at length due to unreasonably high computational cost . ): ( a ) the `` circle '' process , and ( b ) the `` nudge '' process . the black spheres correspond to the recoil ( displaced ) atoms , and the mono - vacancy structure is highlighted with numbered carbon rings in the second panels . the last panel shows schematically the route of the displaced atom . ( see also videos s8 and s9 in ref . @xcite . ) ] since we frequently observed formation of vacancy adatom pairs ( adatoms play the role of interstitials in graphene and cnts ) in our simulations of electron impacts onto graphene and earlier in cnts @xcite , we also explored another possible mechanism of @xmath1(55 - 77 ) formation , which is based on `` incomplete '' annihilation of a frenkel defect . this study was motivated by the peculiarities of the recombination of such a defect in bulk silicon . in that covalently - bonded material the recombination can give rise to either annihilation of the defect and restoration of the perfect crystal lattice , or to a wigner - type defect @xcite . such topological defects are imperfections in the crystal lattice with the locally `` correct '' number of atoms ( as opposed to vacancies and interstitials ) , with the atomic configuration separated from the perfect structure by a finite potential barrier . such defects are deemed to also exist in graphite @xcite , and @xmath1(55 - 77 ) can clearly be classified into this group . while carbon adatoms on graphene @xcite and cnts @xcite ( especially those inside nanotubes ) are mobile at room temperature , they can easily find vacancies in the system and annihilate . indeed , we occasionally observed disappearance of vacancies in hrtem image sequences . figure [ px::expannih ] shows an example of a mono - vacancy that disappears during observation . this proves that mobile carbon atoms are present under our experimental conditions , and may recombine with vacancy - type defects . however , we never noticed the creation of a @xmath1(55 - 77 ) after an observed mono - vacancy . ( 5 - 9 ) mono - vacancy , with an overlay in the inset , and ( b ) shows the same region in a later exposure , with no defect visible . scale bar is 1 nm . ( see also video s7 in ref . @xcite . ) ] to simulate the annihilation process , we created nearby frenkel defects ( separated by a few ngstrms ) in a graphene layer and small ( 6,6 ) zigzag and ( 10,0 ) armchair cnts ( radii @xmath23 and @xmath24 , respectively ) . we then heated the structures and collected statistics on the evolution of each system by running dynamical atomistic simulations at various temperatures ( 5001200 k ) . three possible outcomes emerged from the simulations : ( 1 ) perfect annihilation to the pristine structure ( similar to the experimental images in figure [ px::expannih ] ) , ( 2 ) formation of a @xmath1(55 - 77 ) ( figure [ px::swannih]a ) and , surprisingly , ( 3 ) sputtering of a c@xmath7 dimer with a remaining reconstructed di - vacancy @xmath25(5 - 8 - 5 ) ( figure [ px::swannih]b ) . ( 55 - 77 ) creation process as an outcome of a vacancy adatom annihilation in a ( 6,6 ) armchair nanotube . ( b ) carbon dimer sputtering process in a ( 10,0 ) zigzag nanotube . the black spheres stand for adatoms and the gray ones denote the dimer to be sputtered [ marked in the first panel of ( b ) ] . the nanotube structures are shown from inside the tube . the tube axis is in the horizontal direction . ( see also videos s10 and s11 in ref . @xcite . ) ] for graphene , we always observed perfect annihilation in accordance with the experiments . however , if the adatom in graphene was placed on top of one of those two bonds in the nine - membered carbon ring which are right next to the pentagon , a @xmath1(55 - 77 ) was spontaneously formed without an energy barrier . thus @xmath1(55 - 77 ) may also form in graphene due to recombination of frenkel defects , but the probability for this process must be much lower than in cnts . for ( 10,0 ) cnt , we obtained perfect annihilation in approximately 54% of cases , @xmath1(55 - 77 ) was formed in approximately 34% of cases and dimer sputtering occurred in approximately 12% of cases . for ( 6,6 ) cnt , the values were 53% , 42% and 4% , respectively . with increasing temperature the probability to sputter a dimer showed a slight tendency to increase . we also ran the calculations for a ( 8,8 ) cnt ( @xmath26 ) at 800 k in order to estimate the curvature dependency of the results . the values did not significantly differ from those for the ( 6,6 ) cnt , except for a somewhat increased tendency to perfect annihilation and decreased sputtering ( with probabilities of 56% and 2% , respectively ) . the formation energy of a spatially separated frenkel defect in graphene within the dftb model is approximately 11.1 ev . transformation to form sw defect from this initial setup leads to an energy gain of 5.4 ev . by sputtering a dimer , graphene would instead gain 0.4 ev . hence , all three observed outcomes are energetically reasonable also for graphene . for nanotubes , the formation energies of both @xmath1(55 - 77 ) and a di - vacancy are lowered due to the curvature and stronger c - c bonds at pentagons @xcite . the corresponding energy gains are also higher , which can explain why the probability for @xmath1(55 - 77 ) defect formation and dimer evolution is higher in curved carbon nanostructures . the actual energies depend on the local curvature . it is also plausible that the dimer evaporation process plays a role in shrinking fullerenes under electron irradiation @xcite . @xmath1(55 - 77 ) defect represents the elementary case of a topological change in the graphene structure , _ i.e. _ , a single bond rotation in the otherwise perfect structure . more abundant , however , are changes in the atomic configuration through bond rotations in the reconstructed vacancy defects , as recent experiments indicate @xcite . in presence of a ( multi-)vacancy , the atomic configuration of a defect can be transformed between different metastable structures via bond rotations . it was presumed @xcite that such sw transformations of vacancy - type defects would be stimulated by electron impacts , but the actual atomistic mechanism has not been hitherto unraveled . in order to get microscopic insight into irradiation - stimulated bond rotations near vacancy - type defects in graphene , we carried out a set of experiments and dedicated simulations aimed at assessing the probability of sw transformations in the defect structures . in our experiments , we initially generated `` defective '' graphene by brief 150 kv electron irradiation , and then recorded image sequences of di - vacancies using 80 kv ac - hrtem . as observed previously @xcite , the vacancies can transform between different configurations under the influence of the 80 kv electron beam . moreover , the di - vacancies migrate and transition between the different reconstructed configurations via sw transformations , typically until they cluster into larger defects ( see videos s5 and s6 in ref . @xcite ) . in figure [ px::mvacexp ] , we show an example of a di - vacancy defect that transforms between different reconstructed shapes [ @xmath25(5 - 8 - 5 ) , @xmath25(555 - 777 ) , @xmath25(5555 - 6 - 7777 ) ] under the electron beam . the changes in the atomic structure of these di - vacancy configurations can be described by sw transformations at the defect . moreover , multiple transformations allow migration of the di - vacancy . similar to @xmath1(55 - 77 ) defect formation discussed above , the activation energy for these transitions is far too high to allow a thermally activated process with an observable rate at room temperature . hence , on the basis of the observations , the activation energy for the transition must be provided by the electron beam . ( 5555 - 6 - 7777 ) transforming into @xmath25(555 - 777 ) ( b ) , and @xmath25(5 - 8 - 5 ) ( c ) . each of these transitions can be explained by a single bond rotation . in a later frame ( d ) , the defect is again a @xmath25(5555 - 6 - 7777 ) , but shifted by one lattice parameter . scale bar is 1 nm . ( see also videos s24 in ref . @xcite . ) ] in order to confirm that the sw transformations at these defects are caused by single electron impacts , we carried out atomistic simulations of such impacts onto atoms near di - vacancies . due to the computational cost related to many non - equivalent atoms present in the system and a large number of possible atomic configurations , we could not repeat the detailed analysis of the role of initial space angle of the displacement similar to pristine graphene , and therefore limited our simulations to the @xmath27 case for all non - equivalent atoms at reconstructed di - vacancy structures . however , because the defects break the symmetry of the lattice , a directional preference for the displacements arises ( atom displaced in the perpendicular direction will change its direction due to local strain ) . this effect is strong enough to facilitate bond rotations in reconstructed di - vacancy structures . in figure [ px::mvacsim ] we present examples of such processes for both @xmath25(5 - 8 - 5)@xmath28(555 - 777 ) and @xmath25(555 - 777)@xmath28(5555 - 6 - 7777 ) transformations . within these simulations , we never observed @xmath25(555 - 777)@xmath29(5 - 8 - 5 ) transformations . because the symmetry of the @xmath25(555 - 777 ) defect around the middle atom is the same as that of pristine graphene ( the middle atom is represented as a black sphere in the last panel of figure [ px::mvacsim]a ) , one would also expect that a @xmath30 displacement is required for this transformation , similar to the @xmath1(55 - 77 ) case , although in principle the surrounding atoms could also cause this transformation . we noticed that the most likely di - vacancy transformation , at least for @xmath31 , is the @xmath25(5555 - 6 - 7777)@xmath29(555 - 777 ) . the @xmath25(5 - 8 - 5)@xmath28(555 - 777 ) was the least likely one of those observed . we never observed a @xmath25(5 - 8 - 5)@xmath32(5 - 7 ) transformation @xcite during our simulations , which we also attribute to the limited simulated conditions ( @xmath31 ) . curiously , however , we did observe one transformation in which a @xmath25(5 - 8 - 5 ) di - vacancy directly migrated one step in the zigzag lattice direction . ) ; ( a ) @xmath25(5 - 8 - 5)@xmath28(555 - 777 ) and ( b ) @xmath25(555 - 777)@xmath28(5555 - 6 - 7777 ) . the black spheres indicate the recoil ( displaced ) atoms . structurally equivalent atoms to the displaced ones are marked with circles in first panels . in the last panels , the circled atoms are those which could cause a backward transformation upon displacement in addition to the recoil atom . ( see also videos s1214 in ref . @xcite . ) ] another interesting observation originating from these simulations is the fact that the displacement threshold for atoms in the central part of the reconstructed defects [ @xmath25(555 - 777 ) and @xmath25(5555 - 6 - 7777 ) ] are higher than that for pristine graphene ( by as much as 5% ) . this may explain why defect structures tend to grow into larger and larger amorphous patches instead of collapsing into holes under continuous electron irradiation at low voltages ( @xmath33 kev ) @xcite : even when atoms are removed from the defected area , the displacements occur at the edges of the existing defects rather than at the central part where the local atomic density is already lower . clearly , since the core structure of these defects consists of carbon hexagons , there must exist a limiting size above which the displacement threshold becomes similar to that of ideal graphene . to conclude , by combining ac - hrtem experiments and atomistic simulations , we have shown that the bond rotations which lead to creation of topological defects in carbon nanostructures are caused by single electron impacts or incomplete annihilation of frenkel defects . this explains the discrepancy between experimental observations of stone - wales defects and their relatively high formation energy and even higher energy barrier for bond rotation . the sw transformation in graphene can be initiated at least in two different ways upon electron impact ( involving a `` circling '' or `` nudging '' motion ) , and for almost any space angle , provided that enough energy is transferred from the electron to the target atom . our simulations indicate that @xmath1(55 - 77 ) can appear as a result of `` incomplete '' recombination of a frenkel defect reminiscent of the formation of wigner - type defects in silicon @xcite . however , this is much more likely in the case of local curvature , as in nanotubes . more surprisingly , we also observed sputtering of c@xmath7 dimer as a result of annihilation of a frenkel defect in carbon structures with high curvature . moreover , we showed that the displacement threshold of atoms in the central area of reconstructed defects is higher than that of pristine graphene , by as much as 5% , which explains why defected graphene under low - energy electron irradiation ( @xmath34 kev ) tends to turn graphene into a two - dimensional amorphous structure @xcite instead of a perforated membrane . for different di - vacancy structures , even displacements in direction perpendicular to the graphene layer can initiate sw transformation and thus local structural changes and defect migration . our results provide microscopic insight into the irradiation - induced changes in the atomic structure of carbon nanosystems under electron irradiation , and taking into account the interesting electronic properties of defects associated with sw transformations @xcite , may open new avenues for irradiation - mediated engineering @xcite of carbon nanostructures with next - generation electron microscopes . we acknowledge financial support by the german research foundation ( dfg ) , the german ministry of science , research and arts ( mwk ) of the state baden - wuerttemberg within the salve ( sub angstrom low voltage electron microscopy ) project and academy of finland through several projects . we also thank csc , espoo , finland , for generous grants on computer time .

observations of topological defects associated with stone - wales type transformations ( _ i.e. _ , bond rotations ) in high resolution transmission electron microscopy ( hrtem ) images of carbon nanostructures are at odds with the equilibrium thermodynamics of these systems . here , by combining aberration - corrected hrtem experiments and atomistic simulations , we show that such defects can be formed by single electron impacts , and remarkably , at electron energies below the threshold for atomic displacements . we further study the mechanisms of irradiation - driven bond rotations , and explain why electron irradiation at moderate electron energies ( @xmath0100 kev ) tends to amorphize rather than perforate graphene . we also show via simulations that stone - wales defects can appear in curved graphitic structures due to incomplete recombination of irradiation - induced frenkel defects , similar to formation of wigner - type defects in silicon .

You are an expert at summarizing long articles. Proceed to summarize the following text: the extensive air shower ( eas ) inverse approach to a problem of the primary energy spectra reconstruction in the region of @xmath0 pev energies has been an essential tool in the past decade @xcite . basically , it follows from the high accuracies of recent experiments @xcite and the availability of the eas simulation code @xcite , which was developed in the framework of contemporary interaction models in order to compute the kernel functions of a corresponding integral equation set @xcite . at the same time , the energy spectra of primary ( @xmath1 and @xmath2 ) nuclei obtained from the kascade experiment @xcite using the eas inverse approach disagree with the same data from the ongoing gamma experiment @xcite , where parameterization of the eas inverse problem is used . + below , a peculiarity of the eas inverse problem is investigated , and one of the possible reasons for the observed disagreements between the energy spectra in @xcite and @xcite is considered in the framework of the sibyll @xcite interaction model . the paper is organized as follows : in section 2 the eas inverse approach and the definition of the problem of uniqueness is described . it is shown , that the abundance of primary nuclear species leads to pseudo solutions for unfolded primary energy spectra . the existence and significance of the pseudo solutions are shown in section 4 . the pseudo solutions for primary energy spectra were obtained on the basis of simulation of kascade @xcite shower spectra . the eas simulation model is presented in section 3 . in section 5 the peculiarities of the pseudo solutions are discussed in comparison with the methodical errors of the kascade data . the eas inverse problem is ill - posed by definition and the unfolding of the corresponding integral equations does not ensure the uniqueness of the solutions . the regularized unfolding on the basis of _ a priori _ information on expected solutions ( smoothness , monotony and non - negativity ) in some cases can redefine the inverse problem @xcite and provide the appropriate solutions . however , the expected singularities ( e.g. knees ) in the primary energy spectra at @xmath3 ev may erroneously be smoothed by regularization algorithms and vice versa , be imitated by the unavoidable oscillations @xcite of the solutions . furthermore , the eas inverse problem implies evaluations of at least two or more unknown primary energy spectra from the integral equation set of fredholm kind @xcite . these peculiarities have not been studied in detail and the problem of the uniqueness of solutions can limit the number of evaluated spectra . + let @xmath4 be the energy spectrum of a primary nucleus @xmath5 over the atmosphere , @xmath6 be the probability density function describing the transformation of @xmath5 and @xmath7 parameters of the primary nucleus to a measurable vector @xmath8 . then the eas inverse problem , i.e. the reconstruction of the energy spectra of @xmath9 primary nuclei on the basis of the detected spectra @xmath10 of eas parameters , is defined by the integral equation @xmath11 evidently , if @xmath12 are the solutions of eq . ( 1 ) , the functions @xmath13 should also be the solutions of ( 1 ) , provided equation @xmath14 is satisfied for the given measurement errors @xmath15 and for at least one of the combinations of the primary nuclei @xmath16 the number of combinations ( 3 ) stems from a possibility of the existence of a set of functions @xmath17 for each of the primary nuclei ( @xmath5 ) , which can independently satisfy eq . + for example , suppose that @xmath18 . let us denote @xmath19 by @xmath20 and , for simplicity , set the right - hand side of eq . ( 2 ) to 0 . then , following expression ( 3 ) , we find @xmath21 independent combinations of eq . ( 2 ) : @xmath22 for @xmath23 and @xmath24 , @xmath25 , @xmath26 , @xmath27 and @xmath28 with different @xmath29 functions . the measurement errors @xmath30 on the right - hand side of these equations can both increase and decrease the domains of @xmath29 functions . + one may call the set of functions @xmath31 the pseudo functions with the corresponding pseudo solutions ( spectra ) @xmath13 . the oscillating @xmath32 functions at @xmath33 are responsible for the first @xmath9 equations @xmath34 , @xmath35 , due to the positive - definite probability density function @xmath36 . the pseudo solutions @xmath37 can be avoided by using iterative unfolding algorithms @xcite . + additional sources of the pseudo solutions originate from the mutually compensative effects at @xmath38 : @xmath39 inherent to eq . ( 2 ) for arbitrary groups of @xmath40 and @xmath41 primary nuclei . since there are no limitations on the types of the pseudo functions ( except for @xmath42 ) that would follow from expression ( 4 ) , and the number of possible combinations ( 3 ) rapidly increases with the number of evaluated primary spectra ( @xmath9 ) , the problem of the uniqueness of solutions may be insoluble for @xmath43 . moreover , the pseudo functions have to restrict the efficiency of unfolding energy spectra for @xmath44 , because the unification of @xmath45 primary nuclei spectra into @xmath46 nuclear species ( e.g. light and heavy ) inevitably increases the uncertainties of the kernel functions @xmath36 and thereby also increases the domains of the pseudo functions . + notice , that the pseudo solutions will always appear in the iterative unfolding algorithms if the initial iterative values are varied within large intervals . at the same time , it is practically impossible to derive the pseudo functions from the unfolding of equations ( 1,2 ) due to a strong ill - posedness of the inverse problem . however , for a given set of the measurement errors @xmath15 and the known kernel functions @xmath6 for @xmath35 primary nuclei , eq . ( 2 ) can be regularized by parametrization of the pseudo functions @xmath47 . the unknown parameters @xmath48 can be derived from a numerical solution of parametric eq . ( 2 ) , and thereby one may also evaluate the parametrized pseudo functions @xmath31 . + below ( section 3 ) , an eas simulation model for computing the kernel function @xmath36 and replicating the kascade @xcite eas spectral errors @xmath15 is considered . the primary energy spectra obtained in the kascade experiment were derived on the basis of the detected 2-dimensional eas size spectra @xmath49 and an iterative unfolding algorithm @xcite for @xmath50 primary nuclei @xcite . evidently , whether these solutions are unique or not depends on the significance of the arbitrary pseudo functions @xmath51 from eq . + we suppose that the convolution of the shower spectra @xmath52 at the observation level and corresponding measurement errors @xmath53 , @xmath54 @xcite are described by 2-dimensional log - normal distributions with parameters @xmath55 , @xmath56 , @xmath57 , @xmath58 and correlation coefficients @xmath59 between the shower size ( @xmath60 ) and the muon truncated size ( @xmath61 ) . we tested this hypothesis by the @xmath62 goodness - of - fit test using the corsika(nkg ) eas simulation code @xcite for the sibyll2.1 @xcite interaction model , 4 kinds of primary nuclei ( @xmath63 ) , 5 energies ( @xmath64 pev ) and simulation samples for each of @xmath7 and @xmath5 : 5000 , 3000 , 2000 , 1500 , 1000 respectively in @xmath65 zenith angular interval . the values of corresponding @xmath66 , ( @xmath67 , @xmath68 ) were distributed randomly in the interval @xmath69 for the measurement ranges of the kascade experiment ( @xmath70 and @xmath71 ) and the bin sizes @xmath72 . + notice , that the combined 2-dimensional log - normal distributions with parameters @xmath73 at @xmath74 , @xmath75 at @xmath76 , @xmath77 at @xmath78 and @xmath79 at @xmath80 , more precisely ( @xmath81 ) describe the shower spectra @xmath52 in the tail regions . + we performed an additional test of the log - normal fit of the @xmath82 spectra using multiple correlation analysis for the shower parameters simulated by the log - normal @xmath52 probability density functions and shower parameters obtained from the corsika eas simulations at power - law primary energy spectra ( @xmath83 ) and equivalent abundances of primary nuclei . the corresponding correlation coefficients were equal to @xmath84 , @xmath85 , @xmath86 , @xmath87 , and were in close agreement for both methods of @xmath88 and @xmath89 generations . we replicated the kascade 2-dimensional eas size spectrum @xmath90 ( and corresponding @xmath91 ) by picking out @xmath88 and @xmath92 randomly from the 2-dimensional shower spectra @xmath52 after randomly picking @xmath5 and @xmath7 parameters of a primary particle from the power - law energy spectra @xmath93 with a rigidity - dependent knee @xmath94 , the sharpness parameter @xmath95 and normalization of the all - particle spectrum @xmath96 . the relative abundance of nuclei was arbitrarily chosen to be @xmath97 and @xmath98 for primary @xmath99 and @xmath2 nuclei respectively , which approximately conforms with the expected abundance from balloon and satellite data @xcite . + the mediate values of the parameters of the probability density function @xmath100 were estimated by the corresponding log - parabolic splines . + the total number of simulated eas events was set to @xmath101 in order to replicate the corresponding statistical errors @xmath102 of the kascade data . on the basis of the obtained estimations of @xmath102 ( section 3 ) for the kascade experiment , we examined the uniqueness of unfolding ( 1 ) by @xmath62-the minimization : @xmath103 where @xmath104 represents the left - hand side of eq . ( 2 ) for 2 kinds of the empirical pseudo functions @xmath105 @xmath106 while @xmath107 , otherwise @xmath108 . the unknown @xmath109 and @xmath110 parameters in expressions ( 7,8 ) were derived from @xmath62 minimization ( 6 ) . the numbers of bins were @xmath111 and @xmath112 with the bin size @xmath113 . + in fact , the minimization of @xmath62 ( 6 ) for different representations ( 7,8 ) of the pseudo functions @xmath31 provides a solution of the corresponding parametric eq . ( 2 ) with a zero right - hand side . to avoid the trivial solutions @xmath114 and reveal the domains of the pseudo functions , the values of some of the parameters were arbitrarily fixed during the minimization of @xmath62 ( 6 ) . the magnitudes of the fixed parameters were empirically determined via optimization of conditions @xmath115 and @xmath116 for the pseudo spectra with the fixed parameters . + the true primary energy spectra @xmath4 for @xmath117 nuclei ( 5 ) and the all - particle energy spectrum @xmath118 ( lines ) along with the corresponding distorted ( pseudo ) spectra @xmath13 ( symbols ) are presented in fig . 1 respectively . and the all - particle spectrum @xmath118 for @xmath117 nuclei ( lines ) and the corresponding pseudo solutions @xmath13 for the pseudo function ( 7 ) ( symbols).,width=264,height=264 ] the parameters of the pseudo functions ( 7 ) derived for @xmath119 ( @xmath120 ) are presented in table 1 . + .parameters @xmath121 ( tev@xmath122 ) and @xmath123 of the pseudo function ( 7 ) for different primary nuclei @xmath5 and @xmath124 tev . [ cols="<,^,^",options="header " , ] simulated showers.,width=264,height=264 ] since the measurement errors are negligibly small , the significance of the mutually compensative effects is well seen . the singularity of the proton spectrum was approximately compensated by the @xmath125 and @xmath126 spectra . this is due to both the large number ( @xmath127 ) of possible mutually compensative combinations ( 3 ) and the peculiarities of eas development in the atmosphere ( kernel functions @xmath36 , section 3 ) , which are expressed by the approximately log - linear dependences of the statistical parameters @xmath128 , @xmath129 , @xmath130 and @xmath131 of shower spectra @xmath36 on energy ( @xmath132 ) and nucleon number ( @xmath133 ) of primary nuclei @xcite . the value of @xmath134 for a @xmath135 times smaller eas sample ( @xmath136 ) was equal to @xmath137 . the results from figs . 13 show that the pseudo functions with mutually compensative effects exist and belong practically to all families - linear ( 7 ) , non - liner ( 8) and even singular ( 9 ) in a logarithmic scale . + the all - particle energy spectra in figs . 13 are practically indifferent to the pseudo solutions of elemental spectra . this fact directly follows from eq . ( 2 ) for pseudo solutions and is well confirmed by the identity of the gamma @xcite and kascade @xcite all - particle energy spectra in spite of disagreements of the elemental ( @xmath138 ) primary energy spectra ( see @xcite ) . + the @xmath62 minimization ( 6 ) uses mainly the nearest pseudo energy spectra with free parameters for compensation of the pseudo spectra with fixed parameters . and @xmath2 primary nuclei ( light shaded areas ) and corresponding `` methodical errors '' of the kascade unfolding spectra ( dark shaded areas ) taken from @xcite . the solid and dotted lines resulted from pseudo functions ( 7 ) and the dashed lines stemmed from ( 8).,width=264,height=264 ] the significance of the pseudo functions @xmath51 in most cases exceeds the significance of the evaluated primary energy spectra @xmath4 and unfolding of ( 1 ) can not be effective for @xmath139 . + the unfolding of the primary energy spectra for @xmath50 will increase the number of possible combinations ( 3 ) of the pseudo solutions and the corresponding pseudo functions by a factor of two . taking into account the large values of applied @xmath140 @xcite one may conclude that the contributions of the pseudo functions in the unfolded energy spectra of @xcite have to be dominant . + the `` methodical errors '' obtained in @xcite for @xmath50 define the uncertainties of the solutions intrinsic only to the given unfolding algorithms . the existence and significance of the mutually compensative pseudo solutions follow from eqs . ( 1,2 ) and from the peculiarities of the shower spectra @xmath6 regardless of the unfolding algorithms . + comparison of the methodical errors @xmath141 for @xmath142 and @xmath143 from @xcite with corresponding errors @xmath144 due to the pseudo solutions from expressions ( 7,8 ) are shown in fig . 4 . the magnitudes of the fixed parameters were empirically determined by maximizing @xmath145 ( left panel ) and @xmath146 ( right panel ) for a given goodness - of - fit test @xmath147 from @xcite . + it is seen that the methodical errors ( dark shaded areas ) from @xcite significantly underestimate the contribution of the pseudo solutions ( light shaded areas ) from expressions(7,8 ) . moreover , the methodical errors from @xcite slightly depend on the primary energy ( or statistical errors ) , whereas the domains of the pseudo solutions strongly correlate with the statistical errors according to definition ( 2 ) . the results show that the reconstruction of primary energy spectra using unfolding algorithms @xcite can not be effective and the disagreement between the kascade @xcite and gamma @xcite data is insignificant in comparison with the large domains of the mutually compensative pseudo solutions ( fig 4 ) of the unfolded spectra @xcite . + even though the oscillating pseudo solutions @xmath148 ( section 2 ) are possible to avoid using regularization algorithms @xcite , the mutually compensative effect ( 4 ) of the arbitrary pseudo functions @xmath149 intrinsic to the expression ( 2 ) is practically impossible to avoid at @xmath150 . + the uncertainties of solutions due to the mutually compensative pseudo functions can be obtained by varying the initial values of iterations within a wide range in the frameworks of a given unfolding algorithm . + to decrease the contributions of the mutually compensative pseudo solutions one may apply a parameterization of the integral equations ( 1 ) @xcite using _ a priori _ ( expected from theories @xcite ) known primary energy spectra with a set of free spectral parameters . this transforms the eas inverse problem into a set of equations with unknown spectral parameters , and thereby the eas inverse problem is transmuted into a test of the given primary energy spectra using detected eas data @xcite . the reliability of the solutions can be determined by their stability depending on the number of spectral parameters , the agreement between the expected and detected eas data sets , and the conformity of the spectral parameters with theoretic predictions . + the all - particle energy spectra ( fig . 13 ) are practically indifferent toward the pseudo solutions for elemental spectra . + the obtained results depend slightly on the spectral representations of the shower spectra @xmath36 and the primary energy spectra @xmath4 . i thank my colleagues from the gamma experiment for stimulating this work and the anonymous referee for suggestions which considerably improved the paper . 25 r. glasstetter et al . , proc . 26@xmath151 icrc , salt lake city , * 1 * ( 1999 ) 222 . ter - antonyan , l.s . haroyan , preprint hep - ex/0003006 ( 2000 ) . h. ulrich et al . , proc . 27@xmath151 icrc , hamburg , * 1 * ( 2001 ) 97 . ter - antonyan and p.l . biermann , proc . 27@xmath151 icrc , hamburg , he054 ( 2001 ) 1051 ( astro - ph/0106091 ) . samvel ter - antonyan and peter biermann , proc . 28@xmath151 icrc , tsukuba , he1 , ( 2003 ) 235 ( astro - ph/0302201 ) . t. antoni et al . , astropart . phys . * 24 * ( 2005 ) 1 ( astro - ph/0505413 ) . martirosov et al . , 29@xmath151 icrc , pune , he1.5 , * 8 * ( 2005 ) 9 ( astro - ph/0506588 ) . m. aglietta et al . , astropart . phys . * 10 * ( 1999 ) 1 . glasmacher et al . , astropart . phys . * 10 * ( 1999 ) 291 . t. antoni et al . , nucl . @xmath152 meth . * a513 * ( 2003 ) 490 . ter - antonyan et al . , 29@xmath151 icrc , pune , he1.2 , * 6 * ( 2005 ) 105 ( astro - ph/0506588 ) . a.p . garyka et al . , astropart . ( 2007 ) , doi : 10.1016/j.astropartphys . 2007.04.004 ( arxiv:0704.3200v1 [ astro - ph ] ) . d. heck , j. knapp , j.n . capdevielle , g. schatz , t. thouw , forschungszentrum karlsruhe report , fzka 6019 ( 1998 ) . fletcher , t.k . gaisser , p. lipari , t. stanev , phys.rev . * d50 * ( 1994 ) 5710 . r. gold , anl-6984 report , argonne ( 1964 ) . b. wiebel - sooth , p.l . bierman and h. meyer , astrophys . 330 ( 1998 ) 330 . a.m. hillas , journal of physics * g31 * ( 2005 ) r95 . hrandel , astropart . phys . * 21 * ( 2004 ) 241 . t. stanev , p.l . biermann , t.k . gaisser , astron . 274 , ( 1993 ) 902 . ter - antonyan , 28@xmath151 icrc , tsukuba , he2 ( 2003 ) 239 ( astro - ph/0303658 ) . j.r.hrandel , astro - ph/0611387 ( 2006 ) .

the problem of the uniqueness of solutions during the evaluation of primary energy spectra in the knee region using an extensive air shower ( eas ) data set and the eas inverse approach is investigated . it is shown that the unfolding of primary energy spectra in the knee region leads to mutually compensative pseudo solutions . these solutions may be the reason for the observed disagreements in the elementary energy spectra of cosmic rays in the 1 - 100 pev energy range obtained from different experiments . cosmic rays , primary energy spectra , extensive air shower , inverse problem . 96.40.pq , 96.40.de , 96.40.-z , 98.70.sa

You are an expert at summarizing long articles. Proceed to summarize the following text: masslessness often arises as a consequence of the existence of a symmetry . in quantum electrodynamics the masslessness of the photon is normally attributed to gauge invariance , or symmetry under local changes of phase . in quantum chromodynamics , the theory of the strong interaction , masslessness of the gluons is likewise attributed to a gauge invariance , albeit a nonlinear one . in general relativity , the masslessness of gravitons can be traced to symmetry under active diffeomorphisms : no diffeomorphism - invariant mass term exists . in some circumstances , however , there exists an alternative reason why a field might be massless . surprisingly , this alternative explanation involves the breaking of a symmetry rather than its existence . a general result , the nambu - goldstone theorem @xcite , states under mild assumptions that there must be a massless particle whenever a continuous global symmetry of an action is nt a symmetry of the vacuum . in this talk , based on ongoing work with alan kosteleck @xcite , we show that an alternative description of gravity can be constructed from a symmetric two - tensor without the assumption of masslessness . in this picture , masslessness is a consequence of symmetry breaking rather than of exact symmetry : diffeomorphism symmetry and local lorentz symmetry are spontaneously broken , but the graviton remains massless because it is a nambu - goldstone mode . the cardinal object in the theory is a symmetric two - tensor - density , denoted by @xmath0 . starting point is the lagrange density @xmath1 here , @xmath2 is the usual quadratic kinetic operator for a massless spin-2 field , @xmath3,\end{aligned}\ ] ] and @xmath4 is a potential which is built out of the four scalars @xmath5 we will suppose that @xmath4 has a local minimum at @xmath6 , and that @xmath0 acquires an expectation value @xmath7 . note that this implies spontaneous breaking of lorentz invariance . we can now decompose @xmath8 where @xmath9 are the fluctuations of @xmath0 . at low energy , the values of the scalars @xmath10 will be approximately fixed to their values @xmath11 in the local minimum . then , the linearized form of the potential can be taken equivalent to a sum of lagrange multiplier terms that fix the values of the four scalars @xmath10 to the values @xmath11 : @xmath12 we obtain the linearized equations of motion @xmath13 together with the constraints @xmath14 noting that the left - hand side of eq . ( [ lin_eom ] ) equals the linear part of the ricci tensor , it follows that the low - energy linearized dynamics of this model is equivalent to linearized general relativity in an axial - type gauge defined by conditions ( [ gauge_conditions ] ) . the four gauge conditions ( [ gauge_conditions ] ) reduce the original 10 @xmath15 modes to 6 degrees of freedom . they can be expressed as the generators of the lorentz generators @xmath16 acting on the vacuum expectation value @xmath17 of the cardinal field : @xmath18 imposing the equations of motion imposes masslessness as well as the lorenz conditions : @xmath19 reducing the number of propagating degrees of freedom to two helicities . in order that the cardinal model correctly describe gravity it needs to be coupled to the matter energy - momentum tensor . at the linear level , this can be done by including the term @xmath20 where @xmath21 is the trace - reversed energy - momentum tensor . as a result , the equations of motion reduce to the linearized einstein equation @xmath22 ( where we have taken , for now , the values of the lagrange multipliers @xmath23 equal to zero ) . the total energy - momentum tensor consists not only of contributions of matter . there is a contribution of the gravitons themselves as well , quadratic in @xmath9 . as a consequence , the inclusion of a cubic term in ( [ cardinal_flat ] ) is required . this , in turn , implies a cubic contribution to the energy - momentum tensor , corresponding to a quartic term in the lagrangian . this process continues indefinitely , yielding in the limit the full einstein - hilbert action @xcite . it has been shown by deser @xcite that this `` bootstrap '' process can be carried out , for general relativity , in one step if one rewrites the free graviton action in first order ( palatini ) form . in order to implement this bootstrap procedure for the cardinal theory , we pass , as a first step , to the trace - reversed tensor @xmath24 as @xmath25 the kinetic term in ( [ cardinal_flat ] ) can be rewritten in palatini form as @xmath26 where the connection coefficients @xmath27 are to be considered as independent additional variables . it can be shown that the bootstrap of the kinetic term then terminates after one step , yielding @xmath28 which is equivalent to the einstein - hilbert lagrangian @xmath29 from the latter form we conclude that @xmath30 is naturally interpreted as a curved - space metric density . the bootstrap procedure can also be applied to the matter interaction to determine the form of the matter coupling for cardinal gravity . for this purpose , the interaction ( [ lin_matter ] ) is conveniently expressed in terms of the trace - reversed energy - momentum tensor @xmath31 for the matter . this tensor arises by variation of the lagrange density @xmath32 for the matter fields via @xmath33 in the usual way . we can therefore write @xmath34 for the matter interaction with the cardinal field @xmath35 . applying the bootstrap procedure can be shown to generate the lagrange density @xmath36 this expression corresponds to the usual curved - space matter lagrangian if we identify ( as above ) @xmath30 as the metric density . for example , the bootstrap procedure applied to the flat - space electromagnetic energy - momentum tensor yields @xmath37 it is interesting to note at this point that the way we implemented the bootstrap procedure above is not completely unique . instead of writing the full trace - reversed tensor @xmath35 in the palatini form ( [ palatini ] ) , one could substitute its fluctuations @xmath38 ( defined analogously to @xmath9 above ) instead . while this makes no difference in the linearized expression , when applying the bootstrap to this form one ends up with a different result , yielding @xmath39 as the curved - space metric density . applying this alternative procedure to the matter coupling yields a curved - space matter lagrangian with an explicit reference to the expectation value @xmath17 , constituting lorentz - violating terms that can be related to the standard model extension @xcite . its parameters for lorentz violation have been subject to numerous experimental measurements @xcite . for details see ref . @xcite . the most interesting application of the bootstrap is to the scalar potential v , which we now express as a function of four scalars @xmath40 ( @xmath41 ) defined analogously to the @xmath42 defined in eqs . ( [ x1])([x4 ] ) but with @xmath0 replaced by the trace - reversed tensor @xmath35 . as it turns out , for the procedure to be able to be applied to @xmath4 , the latter needs to satisfy nontrivial integrability conditions that strongly restrict its functional form . for instance , it follows that the unique lowest - order ( in terms of the total power of @xmath24 ) integrable polynomials are @xmath43 more generally , it follows that any polynomial obtained as the term at order @xmath44 in the series expansion of @xmath45|}$ ] is a solution @xmath46 . applying the bootstrap to @xmath46 yields @xmath47|}-\sum_{k=0}^{q-1}\fv_k.\ ] ] we postulate that any integrable potential can be obtained as a suitable linear combination of the @xmath46 polynomials . of particular interest are scalar potentials @xmath48 that have a taylor expansion of the form @xmath49 with @xmath50 real constants . if we impose that @xmath50 be positive definite , @xmath48 has a local minimum at @xmath51 ( @xmath41 ) , and it can represent a possibly stable vacuum . it can be shown that the integrability conditions impose constraints on the @xmath50 and on the higher order coefficients , but that a nontrivial solution space exists @xcite . we have verified numerically that there is a non - empty subset of positive definite solutions , confirming the possibility of scalar potential with stable vacua @xcite . it is reasonable to expect that , in the limit @xmath52 the integrable potentials @xmath48 of eq . [ potential - taylor ] correspond to integrable potentials in the linearized limit considered above @xmath53 fixing the value of @xmath40 to be @xmath54 . we conclude that in this limit , the bootstrapped cardinal model corresponds to general relativity in the gauge defined by the constraints @xmath51 . this is also the case for a potential with finite @xmath50 , if we only consider energy scales low enough such that any fluctuations away from the minimum in the potential can be neglected . after applying the bootstrap procedure described above we end up with a lagrangian density of the form @xmath55 the linearized equations of motion become : @xmath56 where we defined @xmath57 we see from eq . ( [ full_lin_eqs ] ) that the first term on the right - hand side naturally takes the form of a ( trace - reversed ) energy momentum tensor . explicitly , we can identify a `` vacuum energy - momentum tensor '' @xmath58 with @xmath59 it takes nonzero values whenever the scalar potential takes values away from the minimum . explicit solutions of the linearized equations of motion can be obtained with nonzero vacuum energy - momentum tensor . in the latter case , independent initial / boundary field values can be defined on maximally 4 suitably defined timelike / spacelike spacetime slices . if the matter energy - momentum tensor is known to be independently conserved ( e.g. , by symmetry arguments ) , the same has to be true for the vacuum energy - momentum tensor . in such a case , choosing @xmath60 to be zero at the initial value spacetime slices ensures it is zero throughout spacetime we showed it is possible to construct an alternative theory of gravity , the cardinal model , based on spontaneous breaking of lorentz symmetry . the massless gravitons can be interpreted as nambu - goldstone modes of the spontaneously broken lorentz symmetry . the full nonlinear form of the lagrangian is fixed by consistent coupling of gravity to the total energy - momentum tensor , and can be constructed by a bootstrap process , starting with an initial lagrangian for the cardinal field consisting of a quadratic kinetic term and a scalar potential . as it turns out , consistency of the bootstrap process imposes strong restrictions on the form of the scalar potential . nevertheless , consistent potentials exist with local minima . at low energy , the lagrangian corresponds to the einstein - hilbert action , with the possible presence of a nonzero `` vacuum energy - momentum tensor '' . at high energy , four extra massive graviton modes appear that modify the dynamics of the theory . an open problem remains the classification of all possible bootstrapped potentials and a study of their properties . other issues that merit further study are the effect of the massive modes , in particular near singularities or at high temperatures , as well as a study of the quantization of the cardinal model . it is a pleasure to thank alan kosteleck for collaboration . financial support by the fundao para a cincia e a tecnologia is gratefully acknowledged . 9 kraichnan r 1947 _ mit thesis _ ; kraichnan r 1955 _ phys . rev . _ * 98 * 1118 ; papapetrou a 1948 _ proc . irish acad . _ * 52a * 11 ; gupta s n 1952 _ proc . phys . london _ * a65 * 608 ; feynman r p 1956 _ chapel hill conference _

we present a model of gravity based on spontaneous lorentz symmetry breaking . we start from a model with spontaneously broken symmetries for a massless 2-tensor with a linear kinetic term and a nonderivative potential , which is shown to be equivalent to linearized general relativity , with the nambu - goldstone ( ng ) bosons playing the role of the gravitons . we apply a bootstrap procedure to the model based on the principle of consistent coupling to the total energy energy - momentum tensor . demanding consistent application of the bootstrap to the potential term severely restricts the form of the latter . nevertheless , suitable potentials exists that permit stable vacua . it is shown that the resulting model is equivalent , at low energy , to general relativity in a fixed gauge .

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the most profound discoveries of observational physics is that the universe is accelerating in its expansion @xcite . there have been many attempts to explain this late - time acceleration , for example , a pure cosmological constant , dark energy associated with some new scalar field and modified gravitational theories , although all current models require some level of fine - tuning and none are considered to be a complete explanation . whatever is responsible for the current acceleration may arise from some completely new physical principle . this is the possibility we consider in this paper . our goal is to construct a toy model that represents a late - time accelerating universe using a new , possibly fundamental , principle . as our guiding principle , we hypothesize the existence of a _ minimal curvature _ scale in gravity . in a friedmann , robertson - walker ( frw ) space - time , without cosmological constant @xmath0 and with only standard matter sources such as dust and radiation , the universe will always decelerate as it expands . one way to avoid this is to add matter to the system that violates the strong energy condition ( sec ) . in a cosmological context this violation constitutes the addition of matter sources satisfying the equation of state @xmath1 . a second possibility is to explicitly remove flat space - time as a solution to the theory . in this case the vacuum of the theory , which is approached at late times as the energy density in matter fields becomes more and more dilute , is not minkowski space - time , but instead an accelerating universe @xcite . to remove flat spacetime as a solution we hypothesize the existence of a minimal curvature in our underlying fundamental theory . the simplest example of this is , of course , to introduce a bare cosmological constant into general relativity . however , in principle there may exist many other ways to achieve this result . indeed , it appears that many accelerating cosmological models derived from modified gravity theories contain such a minimal curvature @xcite . the idea of a minimal curvature scale in gravity mirrors that of a maximal curvature scale . in the literature many authors have considered this possibility and used it to remove the curvature singularities of general relativity by bounding curvature invariants from above at the level of the classical action @xcite-@xcite . in the case of singularity removal , it is necessary to bound _ all _ curvature invariants in order to cover all possible physical situations in which such a singularity may occur . by contrast , in the case of a minimal curvature approached at late times in a homogeneous , isotropic universe , symmetry implies that it is only necessary to bound the ricci scalar @xmath2 from below . hence , unlike in the case of a maximal curvature hypothesis , we shall see that one may implement a minimal curvature by using a modified brans - dicke theory where the brans - dicke field couples non - minimally to the matter lagrangian . within this context we demonstrate that the existence of the minimal curvature ( mc ) produces a universe that evolves from a matter dominated period to an accelerating phase mimicking the @xmath0-cold - dark - matter ( @xmath0cdm ) model . we emphasize that the model presented here is only a _ toy construction of the late universe . the model is not intended to provide a consistent cosmology from the time of big - bang nucleosynthesis ( bbn ) until today . it is unlikely that the precise model presented here is compatible with solar system experiments and the tight constraints on the time variation of newton s constant . however , the model _ does provide an example of how the postulated existence of a minimal curvature scale in gravity can provide a new mechanism to generate cosmological acceleration of the late universe . furthermore , the model may capture features of a possibly more fundamental theory that admits a minimal curvature scale . _ _ in section [ sec : mc ] , we describe the minimal curvature construction , first by using a toy example and then by using a class of modified brans - dicke theories . we solve the equations of motion for this example and demonstrate how the universe evolves from a matter dominated phase to an accelerating period as the curvature approaches its minimal value . in section [ sec : comp ] , we compare the mc model with @xmath0cdm and to the supernovae ( sneia ) gold " sample of @xcite . finally , we comment on the possibility of constructing more realistic models that satisfy the limiting curvature hypothesis and offer our conclusions and speculations in section [ sec : conclusions ] . in appendix a , we provide a detailed analysis of the vacuum mc theory . in appendix b , we construct an einstein frame description of the vacuum theory and compare it to the mc vacuum . our goal is to construct theories in which a certain physical quantity is bounded from below . before leaping directly into our model , it is instructive to consider an example of how a similar effect may be achieved in a simpler theory - the bounding of velocities from above in special relativity by the speed of light @xcite . the newtonian action for a free particle of mass @xmath3 in motion is [ old ] s = dt m x^2 . in this classical theory the velocity of the particle is _ without bound_. now let us implement one of the fundamental consequences of special relativity : to ensure that the speed of this particle is _ limited _ by the speed of light we introduce a field @xmath4 which couples to the quantity in the action that we want to bound ( @xmath5 ) and has a potential " @xmath6 . the resulting action is [ newa ] s = m dt . the variational equation with respect to @xmath7 [ bit ] x^2 = , ensures that @xmath8 is bounded , provided @xmath9 is bounded . note the absence of a kinetic term for @xmath10 in the action , and hence , the reason the word _ potential appears in quotes above . in order to obtain the correct newtonian limit for small @xmath8 and small @xmath7 we take @xmath6 proportional to @xmath11 . in the newtonian limit the action ( [ newa ] ) reduces to ( [ old ] ) . a simple potential satisfying the above asymptotics is u ( ) = . integrating out @xmath7 yields ( up to an irrelevant constant ) the action for relativistic particle motion : s_sr = m dt . the above model provides a powerful example of how a toy construction based on a fundamental principle the existence of a universal speed limit " can capture features of a more fundamental theory . _ we now use a similar construction to model the existence of a _ minimal _ curvature ( mc ) scale in gravity . because we are interested in late time cosmology , we need only be concerned with bounding one curvature invariant , the ricci scalar @xmath2 . in direct analogy with our example from special relativity , we introduce a scalar field @xmath12 that couples to the quantity we wish to bound , @xmath2 , and a potential " function @xmath13 [ act2 ] s_mc = d^4 x ( r - r + v ( ) ) , where @xmath14 is the reduced planck mass and both @xmath12 and @xmath15 posses dimensions of mass , @xmath16 = [ \g]=m$ ] . the vacuum theory is equivalent to a brans - dicke theory with brans - dicke parameter @xmath17 . this is seen by re - writing the action ( [ act2 ] ) in terms of a new field [ newp ] = ( - ) , so that the action becomes [ actbd ] s_bd = d^4 x ( r + v ( ) ) . ordinarily , such a theory can be re - cast as a purely gravitational theory with lagrangian @xmath18 ( see , e.g. @xcite ) ; however , this is not possible for all forms of the potential @xmath19 . for reasons that will become clear shortly , we allow the field @xmath12 to couple non - minimally to matter . the non - minimal coupling yields the matter action , [ actmat ] s_m = d^4 x _ m(_i , , f(/m_pl ) | ) , where @xmath20 is the lagrangian made up of whatever matter fields @xmath21 are in the theory . in the case were @xmath22 represents a dark matter dirac spinor , the field @xmath12 couples non - minimally _ only to the dark matter sector and need not couple to baryons . in this case it is possible to avoid constraints on such a coupling from solar - system and table - top tests of gravity . in string theory , non - perturbative string loop effects do not generically lead to universal couplings , allowing the possibility that the dilaton decouples more slowly from dark matter than ordinary matter ( see , e.g. @xcite ) . _ this coupling can be used to address the coincidence problem , since the acceleration is triggered by the coupling to matter . for the purposes of our toy construction , we do not distinguish between baryonic and dark matter in the remainder of our discussion . note that in the case of @xmath23 theories which are conformally identical to models of quintessence in which matter is coupled to dark energy with a large coupling , this strong coupling induces a cosmological evolution radically different from standard cosmology @xcite . similar difficulties may arise in the model presented here , however , we have yet to investigate this issue . the matter stress - energy tensor is given by t _ - . we assume a perfect fluid t _ = ( _ m + p_m ) u_u_+p_m g _ , where @xmath24 is the fluid rest - frame four - velocity , and the energy density @xmath25 and pressure @xmath26 are related by the equation of state @xmath27 . because we are focusing on the late universe we shall ignore the presence of radiation and consider only a matter density @xmath25 , which redshifts with expansion in the usual manner ( with the exception of the non - trivial @xmath12 dependence ) _ m . in the above , @xmath28 is a function describing the non - minimal coupling of the field @xmath12 to the matter lagrangian . such couplings in the context of ordinary scalar - einstein gravity were studied in @xcite where it was found that this coupling can be made consistent with all known current observations , the tightest constraint coming from estimates of the matter density at various redshifts . this coupling plays a critical role in modified source gravity , introduced in @xcite . variation of the total action @xmath29 with respect to the metric tensor , @xmath30 yields the modified einstein equations [ eomg ] ( 1- ) r _ & - & ( 1- ) r g _ + & + & _ _ - g _ - v g _ = 8 g t _ where @xmath31 and we have introduced @xmath32 . variation with respect to the field @xmath12 gives [ eomp ] r = + . equation ( [ eomp ] ) is the key to imposing the limiting curvature construction . it is clear that the curvature @xmath2 will remain bounded and approach a constant curvature at late times ( @xmath33 ) as long as @xmath34 constant ; this is the essence of the construction . we assume a flat ( @xmath35 ) friedmann - robertson - walker metric ds^2=-dt^2+a^2(t)\{dr^2 + r^2 d^2 } , [ frwm ] where @xmath36 is the scale factor of the universe and @xmath37 is the line - element of the unit @xmath38sphere . defining the hubble parameter by @xmath39 and substituting the metric ansatz ( [ frwm ] ) into ( [ eomg ] ) and ( [ eomp ] ) gives the generalized friedmann equation ( the @xmath40-component of ( [ eomg ] ) ) and the equation of motion for @xmath12 : [ eom1 ] 3 ^ 2 h^2 - 6 h^2 - 6 h + v ( ) = , & & + 6(2h^2 + h ) - v ( ) = f ( ) , where a prime denotes differentiation with respect to @xmath12 , @xmath41 , @xmath42 and @xmath43 are the values of @xmath12 and @xmath44 today . by considering the asymptotics of our cosmology at early and late times , we can constrain the forms of the functions @xmath13 and @xmath45 . we require that the effective newton s constant for our theory remain positive definite so that gravity is always attractive . this imposes a constraint on @xmath46 in equation ( [ actbd ] ) . there are rather strong constrains on the time variation of newton s constant from the period of nucleosynthesis until today ( roughly , @xmath47 @xcite ) . for the time being , we will allow ourselves to ignore this constraint in order to produce a toy model capable of realizing the mc conjecture . furthermore , because we are only interested in the behavior of the universe from the matter dominated epoch until today , we have ignored the presence of radiation . the absolute earliest our theory is valid is up to the period of equal matter and radiation domination @xmath48 . for specificity , by _ early times we refer to times near the time of photon decoupling at a redshift of @xmath49 , during which the universe is typically already well into the matter dominated regime . _ at late times ( and low curvatures ) we want to bound the ricci scalar @xmath2 from below . this will constitute a successful example of a model obeying the minimal curvature hypothesis . to bound the curvature we use equation ( [ eomp ] ) . it is clear that the curvature @xmath2 will remain bounded if we bound @xmath50 , where the prime denotes differentiation with respect to @xmath12 . hence , we require @xmath51 at late times , where we denote the hypothesized minimal curvature scale by @xmath52 . we anticipate that , by construction , there will be a late - time attractor that is a constant curvature space - time with @xmath53 . this attractor is not an actual solution to the equations of motion . the above considerations restrict the functional forms of potential @xmath19 and the non - minimal coupling function @xmath28 . integration singles out a class of theories that must obey @xmath54 as @xmath55 . the simplest forms for @xmath19 and @xmath28 obeying the above constraint are the linear functions [ vandf ] v ( ) = ^3 , f ( ) = , where @xmath56 is , in principle , another free parameter with dimensions of mass . however , we will take @xmath57 , so that @xmath58 , and @xmath15 is the only free parameter in the theory and then take the limit that @xmath59 the theory resembles the action of the _ modified source gravity models studied in @xcite . _ ] . we now rescale time @xmath60 so that today @xmath61 and introduce the following dimensionless quantities [ dimen ] a , h , [ dimen2 ] _ m = , r = , and take @xmath62 , where @xmath63 is the value of the scale factor today . in terms of the dimensionless quantities the eom become [ eomf ] h^2 - 2 h^2 - 2 h + = , & & + ( 2h^2 + h ) - = f ( ) [ rdshift ] , the successful implementation of the minimal curvature hypothesis is now apparent . recasting equation ( [ rdshift ] ) in terms of the curvature scalar @xmath64 , and substituting in our choices for @xmath19 and @xmath28 ( [ vandf ] ) : [ simple ] r = ^2 + 3 . we see that , as the universe expands and the matter term dilutes , we asymptotically approach the minimal value of the curvature @xmath65 . it is both interesting and surprising that the solution to eq . ( [ simple ] ) reduces to the simple case of @xmath0cdm plus an arbitrary amount of _ dark radiation _ which may have either positive or negative effective energy density . most notably , this model arises in randall - sundrum brane cosmology which has been extensively studied in the literature . the friedmann equation derived from the randall - sundrum model for a flat universe is [ bwf ] h^2 = + + , where @xmath66 is the five dimensional planck mass and @xmath67 is the so called dark radiation term , since it scales like radiation , but it s origins are purely gravitational and it does not interact with standard matter @xcite . at low energies ( when the energy density is much less than the critical brane tension ) , the @xmath68 term can be safely neglected . the main observational restrictions on the dark radiation term come from the acoustic scale at recombination ( see , e.g. @xcite ) , and from the amount of total growth of density perturbations in the non - relativistic matter component from the time of equal matter and radiation until the present day @xcite . as a result , the density of dark radiation can not be significantly larger than the present cmb energy density . making use of ( [ dimen ] ) and ( [ dimen2 ] ) , eq . ( [ bwf ] ) may be recast ( neglecting the @xmath68 term ) as [ hbw ] h^2 = + + , where we have included a cosmological constant term @xmath69 , and @xmath70 includes contributions from both the dark and ordinary radiation . from ( [ hbw ] ) we find h = - . constructing the ricci scalar @xmath71 from the above expressions yields eq . ( [ simple ] ) , with @xmath72 . in [ app : vac ] , we provide a detailed analysis of the vacuum mc equations with @xmath73 . although the presence of matter plays an important role in our minimal curvature construction , an analysis of the vacuum theory provides valuable insight into the solutions we are interested in studying . in [ app : einst ] , we transform the vacuum mc theory into an einstein frame and relate quantities of physical interest in both frames . for general solutions to the equations of motion ( [ eomf ] ) and ( [ rdshift ] ) with functions ( [ vandf ] ) we solve for the hubble parameter @xmath74 and scalar field @xmath75 . we plot the relevant portion of the @xmath76 phase space in fig . [ phase1.eps ] . vs. @xmath77 . all solutions asymptotically approach a late - time accelerating phase with constant @xmath77 at the minimal curvature @xmath78 denoted by the red line.,width=5 ] to solve the equation we integrate from the past to today @xmath79 and then from today into the future and patch the solutions together . in the plots we take the conditions @xmath80 , @xmath81 , @xmath82 and let the values of @xmath42 vary . at late times , the solutions approach the constant @xmath77 attractor when the universe is well into the accelerating epoch . we now compare our model with @xmath0cdm . to do so , we must enter reasonable initial conditions into our numerical study and solve the equations of motion ( [ eomf ] ) , ( [ rdshift ] ) together with ( [ vandf ] ) and the equation @xmath83 . let us begin by considering the value of the minimal curvature . physically the minimal curvature corresponds to an emptying of the matter in the universe due to cosmological expansion . in our model the value of the minimal curvature is approached asymptotically . today , the value of the curvature is given by [ rprox ] r_0 = 6 ( 2 h_0 ^ 2 + h_0 ) 12 h_0 ^ 2 . because we observe a significant amount a matter in the universe today , we know that we have not yet reached the minimal curvature . however , because we are accelerating , we know that we are _ near the minimal curvature ( i.e. the first and second terms on the right hand side of equation ( [ simple ] ) must be comparable ) . hence , from equation ( [ rprox ] ) , we expect the value of the minimal curvature @xmath78 to be close to but less than @xmath84 . therefore , in terms of our dimensionless quantities ( [ dimen ] ) , @xmath85 and the free parameter in our theory @xmath86 . for the solutions considered below , we choose a value of @xmath81 meeting the above requirements and that follows the @xmath0cdm model to our satisfaction for our toy construction ( i.e. a value leading to a matter dominated cosmology followed by a jerk " near a redshift of @xmath87 into an accelerating phase ) . _ in the action ( [ act2 ] ) , the effective newton s constant is @xmath88 is given by ( 16 g_n_eff)^-1 = - . to ensure that the effective newton s constant remains positive definite over the history of the universe ( @xmath89 ) we must have @xmath90 . we are almost in a position to compare our model both with @xmath0cdm , which has the friedmann equation [ lcdm ] h = , and with the observational data provided by the sneia gold sample . to make a comparison with the observational data we require an understanding of the luminosity distance formula in the context of modified gravity models . an important consideration arises when using the formula for the luminosity distance in theories of the form : [ modact ] s = d^4 x , where the @xmath91 represent the different types of possible matter lagrangians present . such a theory arises as the low - energy effective action for the massless modes of dilaton gravity in string theory , and our model is an action of just this sort @xcite ; albeit , with an unusual choice for the functions @xmath92 . as we have already discussed , these theories typically lead to time variation in newton s gravitational constant . the time variation can affect the way one should compare the theory to observations @xcite . in particular , the time - evolution can alter the basic physics of supernovae . for example , the time variation in @xmath93 leads to different values of the chandrasekhar mass at different epochs , and hence , a supernova s peak luminosity will vary depending on when the supernova exploded . this makes treating the supernovae ia as standard candles difficult @xcite . specifically , the peak luminosity of sneia is proportional to the mass of nickel synthesized which is a fixed fraction of the chandrasekhar mass @xmath94 . hence , the luminosity peak of sneia varies as @xmath95 and the corresponding absolute magnitude evolves as m = m_0 + , where the subscript zero indicates the local values of the quantities . therefore , the magnitude - redshift relation of sneia in modified gravity theories of the type given by ( [ modact ] ) is related to the luminosity distance via @xcite : m(z ) = m_0 + 5 + . even if gravitational physics is described by some theory other than general relativity the standard formula for the luminosity distance applies as long as one is considering a metric theory of gravity @xcite : [ lumd ] d_l(z ; h(z ) , h_0 ) = _ 0^z . for @xmath0cdm , the luminosity distance ( [ lumd ] ) can be written @xcite : [ lumlcdm ] d_l(z ; _ m , _ , h_0 ) = & & + s ( _ 0^z ^- ) where , @xmath96 and @xmath97 for @xmath98 while @xmath99 with @xmath100 for @xmath101 while @xmath102 and @xmath103 , for @xmath104 . here and throughout , @xmath105 . given the above considerations we take the following values of parameters today : [ initc ] a_0 = 1.0 , h_0 = 1.0,_0 = 0.1 , = 3.15 , _ m^(0 ) = 0.25 . we then integrate our equations from the past to today and then from today into the future and patch the solutions together . while we do not provide an exhaustive study of the parameter space for the mc model , the parameters given above provide a successful example of the construction which fulfills our rather modest goals . ] . it is quite possible that the parameters ( [ initc ] ) can be tuned to achieve an even better agreement with @xmath0cdm . the history and future of the curvature @xmath2 together with the co - moving hubble radius are plotted in fig . [ crvandcmh2.eps ] . . the curvature decreases from the matter - dominate past ( red curve ) to the constant minimal value at @xmath106 indicated by the green line . today , @xmath107 and the blue curve indicates the future evolution of the curvature . while the universe is accelerating the co - moving hubble radius decreases . ] in the figure the past history of the curvature is plotted in red while the future is plotted in blue . today we are at the value @xmath108 . the minimal curvature is given by the green line at @xmath109 . as expected , the curvature is large in the past when the universe is matter dominated and then decreases , approaching the accelerating late - time de - sitter attractor with constant minimal curvature @xmath53 . the second plot in the figure shows the evolution of the co - moving hubble radius @xmath110 . the phenomenologically desired cosmological transition from matter domination to late - time acceleration of the universe is clearly indicated by the decreasing of the co - moving hubble radius , when < 0 . in some modified gravity models it can be difficult to achieve this transition ( see , e.g. @xcite ) . the majority of our results are presented in figures [ vslcdm3.eps ] and [ mcfit4.eps ] . in figure [ vslcdm3.eps ] we plot the scale factor @xmath111 as a function of cosmic time @xmath112 , the hubble parameter @xmath77 as function of redshift @xmath113 , where z + 1 , the deceleration parameter q(z ) = - , and the luminosity distance , @xmath114 . quantities plotted in red are the mc model while those in blue are for @xmath0cdm . today we are at the values @xmath115 , @xmath116 . plot a. ) shows the time evolution of the scale factor . plot b. ) shows the hubble parameter as a function of redshift . plot c. ) compares the deceleration parameters as a function of redshift . plot d. ) shows the luminosity distance , @xmath114 , as a function of redshift . the mc model is clearly a good fit with @xmath0cdm . ] for the particular set of parameters considered , the scale factor of our model differs from @xmath0cdm most strongly in the far future , although the difference in the expansion rates of the mc construction with @xmath0cdm is apparent in the plot of the hubble parameters at high redshifts . the transition from matter domination to acceleration ( the jerk ) occurs at a slightly lower redshift than @xmath0cdm . there is only a slight difference in @xmath114 that occurs at high redshifts ( @xmath117 ) , although the difference is not significant enough to distinguish our model from @xmath0cdm using only the current supernova data . cdm models with observational data . the supernovae data points are plotted with error bars and the data is taken from @xcite . the luminosity distance @xmath114 for the mc model is plotted by the dashed ( blue ) curve . the various theoretical predictions for @xmath0cdm are represented by the solid curves and were examined in @xcite.,width=480 ] in figure [ mcfit4.eps ] , we compare the luminosity distance predicted by our model with several versions of @xmath0cdm and with the observational data . the luminosity distances are plotted out to a redshift of @xmath118 ( the highest redshift supernova data is from @xmath119 ) . the theoretical predictions of the minimal curvature construction and @xmath0cdm are compared with the gold " supernovae sample of @xcite . the particular choice of @xmath0cdm models shown are from @xcite . the luminosity distance for the minimal curvature construction is denoted by the blue dashed line and fits the supernova data extremely well . the luminosity distance @xmath114 is virtually indistinguishable from @xmath0cdm with @xmath120 and @xmath121 ( there is a small discrepancy at high redshift as can be seen from figure [ vslcdm3.eps ] ) . we have shown that a period of late - time cosmic acceleration can follow directly from a simple minimal curvature conjecture ( mcc ) . the model fits the sneia data exceptionally well . while the specific formulation considered here is only a toy construction , unlikely to be compatible with constraints from solar system and table top test of the equivalence principle , it may capture phenomenologically interesting features of a more fundamental theory that admits a limiting minimal curvature . furthermore , the construction successfully demonstrates the possibility that a new fundamental physical principle may ultimately be responsible for the recent period of cosmological acceleration . it is possible that experimentally viable models based on the minimal curvature conjecture exist . the search for such models within the context of scalar - gauss - bonnet gravity is currently underway . despite the tight theoretical and experimental constraints on scalar - gauss - bonnet cosmologies @xcite , we remain optimistic that an experimentally and theoretically viable model based on the minimal curvature construction can be discovered . it is a pleasure to thank r. brandenberger , r. gregory , v. jejjala , i. moss , r. myers and t. underwood for helpful discussions . i am especially grateful to m. trodden for numerous useful discussions over the course of this work . this work is supported in part by pparc and by the eu 6th framework marie curie research and training network universenet " ( mrtn - 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however , once the universe becomes sufficiently large and dilute the dynamics will resemble those of the vacuum theory . as we shall see the vacuum mc theory is significantly richer than that of @xmath0cdm . the equations describing the vacuum are given by the equations ( [ eom1 ] ) with @xmath122 . the dimensionless forms are [ eomvac ] h^2 - 2 h^2 - 2 h + = 0 , & & + ( 2h^2 + h ) - = 0 , where we have re - introduced the parameter @xmath56 from equation ( [ vandf ] ) ( previously set equal to @xmath15 ) for completeness and the potential is @xmath123 . in the case of the vacuum we may solve exactly for @xmath74 . we focus on two types of solutions of particular interest . in the first , the hubble parameter is given by [ htanh ] h(t ) = , where@xmath124 is the minimal curvature and @xmath125 is a constant . in this case h = , and the scale factor evolves as a(t ) = a_0 the quantities mentioned above , along with the field @xmath12 are plotted in fig . ( [ vacplot.eps ] ) . . in the plots we take @xmath126 . ] in this case the universe undergoes a bounce as indicated in plots a. ) and b. ) in fig . ( [ vacplot.eps ] ) . after the bounce the hubble parameter is pulled up to the value at the minimum curvature @xmath127 . this is a de sitter solution of the vacuum theory with [ julia ] h_ds = , while @xmath12 continues to evolve exponentially @xmath128 . interestingly , @xmath77 and @xmath129 evolve in such a way that the curvature @xmath130 is constant throughout the evolution of the solution , fixed at the minimal value @xmath131 . this behavior is not surprising as the constraint on @xmath2 comes from the second equation in the eom ( [ eomvac ] ) . the second family of solutions are of greater relevance . they are the vacuum analogs of the solutions plotted in fig . ( [ phase1.eps ] ) and discussed at the end of section [ sec : mc ] . these solution are pulled _ down to the value of @xmath77 at the minimal curvature . the entire phase space of solutions to the vacuum theory is shown in fig . ( [ vacphase.eps ] ) . representative solutions of the two families of solutions discussed above are plotted by the solid curves . the red curve is the solution given by equation ( [ htanh ] ) . the second set of solutions ( relevant to section [ sec : mc ] ) are plotted in green in the upper right quadrant of the phase space . the system defined by the vacuum equations ( equation ( [ eomvac ] ) ) has two unstable saddle equilibrium points at the values [ eqpts ] ( , h ) = ( - , ) . the equilibrium points are marked by the purple dots in fig . ( [ vacphase.eps ] ) . _ is plotted on the vertical axis and @xmath12 is plotted on the horizontal axis . the vector field for solutions is drawn in black . solutions of particular interests are plotted by the solid curves . the constant de sitter value of @xmath77 at the minimal curvature is indicated by the dashed ( blue ) line . the two unstable saddle equilibrium points are designated by the purple dots . in the plot we take @xmath132.,width=5 ] it is not our intent to provide a complete einstein frame analysis of the full mc system with matter ; however , it is instructive to consider the vacuum theory in an einstein frame . to move to the einstein frame we begin with the brans - dicke frame defined by the action ( [ actbd ] ) . passage to the einstein frame is achieved via a conformal transformation of the form [ ctrans ] g _ = ^2 g _ , where @xmath133 is the conformal factor which must be positive to leave the signature of the metric unaltered . from this point forward a tilde shall denote a quantity built out of the einstein - frame metric tensor @xmath134 . under this transformation the infinitesimal line element and the determinant of the metric transform as ds^2 = ^2 ds^2 , = ^4 , respectively . the ricci scalar transforms as r = ^2 ( r + 6 ( ) - 6 g^ ) . omitting the ordinary divergence @xmath135 , the action in the brans - dicke frame transforms to [ actcf ] s = d^4 x ( r - 6 g^ _ _ + ) , where the potential in terms of @xmath136 is given by ( [ newp ] ) together with ( [ vandf ] ) : [ vofbdp ] v ( ) = ( - ) . here we have re - introduced the parameter @xmath56 from equation ( [ vandf ] ) ( previously set equal to @xmath15 ) for completeness . by choosing our conformal factor to be @xmath137 , and performing a field redefinition to define the einstein frame field @xmath138 , = , the action ( [ actcf ] ) becomes the einstein frame action [ acteinst ] s_ef = d^4 x ( r - g^ _ _ - ( ) ) , where we have defined the potential ( ) = - ( ) ^2 . the einstein frame potential is given by [ einstv ] ( ) = ( e^/ - 1 ) e^-2 / , where @xmath124 is the minimal curvature and @xmath139 . in this frame the field @xmath138 has a canonically normalized kinetic term making the interpretation of solutions more simple due to our familiarity with minimally coupled scalar field to ordinary einstein gravity . it is important to note , however , that so far we have only considered the vacuum theory . even in the einstein frame the einstein field @xmath138 will couple non - minimally to matter and therefore , when matter is present in significant amounts , the simple einstein frame vacuum solutions will not be an accurate description of the theory . under the conformal transformation ( [ ctrans ] ) , the cosmic time coordinate transforms as [ ttran ] d t^2 = e^/ dt^2 . taking the einstein - frame friedmann - robertson - walker flat metric ds^2 = - d t^2 + a^2(t ) d**x^2 , leads to the familiar equations of motion [ sasha ] h^2 = ( ^2 + v ( ) ) , and + 3 h + = 0 , where prime denotes differentiation with respect to the einstein frame cosmic time coordinate @xmath140 and @xmath141 . using the conformal transformation ( [ ctrans ] ) , we find the hubble parameters in the einstein and minimal curvature frames are related by [ relate ] h = e^ ( h - ) . * * let us examine the first possibility in greater detail . the unstable de sitter solution with constant @xmath144 sits at the value at the maximum of the potential ( [ einstv ] ) . the maximum is at constant value [ dspt ] = _ ds = ^-1 , from equation ( [ sasha ] ) we find the corresponding @xmath145 h_ds = . from equation ( [ relate ] ) we see that the corresponding value in the minimal curvature frame is h(h_ds , _ ds ) = , which is the exact de sitter solution we found in our analysis of the vacuum of the minimal curvature theory ( [ julia ] ) . the minimal curvature frame field is related to the einstein frame field via [ vpofphi ] ( ) = ( 1 - e^/ ) , and consequently , ( ) = . using the relation ( [ vpofphi ] ) , we find the value of @xmath12 at the de sitter point ( [ dspt ] ) : ( _ ds ) = - . hence we conclude that the unstable de sitter solution in the einstein frame is mapped to one of the unstable saddle critical points in the vacuum of the minimal curvature frame ( see equation ( [ eqpts ] ) ) . we now examine case iii ) : when @xmath138 rolls to large positive values the einstein frame potential may be approximated by [ vprox ] ( ) e^-/ , leading to the exact solutions = + h(t ) = , where @xmath146 , corresponding to power - law acceleration in the einstein frame with [ ltpl ] a(t ) = a_0 t^3 . to find the corresponding behavior in the minimimal curvature frame we use the conformal transformation ( [ ctrans ] ) along with the transformation for the cosmic time coordinate ( [ ttran ] ) . we find the late - time power law attractor in the einstein frame ( [ ltpl ] ) is mapped to the asymptotic de sitter attractor in the mc frame at the minimal curvature @xmath78 ( depicted by the dashed blue line in fig . ( [ vacphase.eps ] ) ) , a(t ) = a_0 e^ t .

we consider the hypothesis of a limiting minimal curvature in gravity as a way to construct a class of theories exhibiting late - time cosmic acceleration . guided by the minimal curvature conjecture ( mcc ) we are naturally lead to a set of scalar tensor theories in which the scalar is non - minimally coupled both to gravity and to the matter lagrangian . the model is compared to the lambda cold dark matter concordance model and to the observational data using the gold " sneia sample of riess et . al . ( 2004 ) . an excellent fit to the data is achieved . we present a toy model designed to demonstrate that such a new , possibly fundamental , principle may be responsible for the recent period of cosmological acceleration . observational constraints remain to be imposed on these models . dcpt-06/17 s l h

You are an expert at summarizing long articles. Proceed to summarize the following text: questions of how to compare hadronic observables to the apparent underlying fundamental theory of qcd lie at the heart of understanding the nature of strong interactions . thirty years after its inception , qcd in @xmath7 spacetime dimensions still stubbornly refuses to admit a global solution . the asymptotic freedom property of the theory permits the perturbative calculation of ( euclidean ) green functions involving large values of momentum transfer or energy release in terms of quarks and gluons , the fundamental objects of qcd . but at lower scales one enters the nonperturbative regime , which not only invalidates ( or at least complicates ) the standard perturbative methods of field theory developed in qed , but also leads to a dramatic change in the physical spectrum of the theory . instead of quarks and gluons , only colorless hadrons are produced as asymptotic states in any process , even at arbitrarily large energy . many nontrivial theoretical techniques respecting qcd first principles have been developed to study nonperturbative features of the theory . yet despite numerous advances , no one has been able to compute the masses , wavefunctions , or transition amplitudes of hadrons in terms of quark masses and couplings directly from the qcd lagrangian . moreover , many existing theoretical tools are expressed through various expansions in certain small parameters ; the actual range of each parameter where the expansions are applicable is often not well known . in such a situation , it is clearly advantageous to build a soluble toy field theory that incorporates as many features of the qcd lagrangian as possible . such a theory does indeed exist , the famous t hooft model @xcite , which is defined by the yang - mills lagrangian in @xmath8 spacetime dimensions in the limit of a large number of colors @xmath9 . as was shown in the original paper , the quark - antiquark sector of the theory admits an infinite tower of confined , color - singlet solutions that can be obtained , in principle , to an arbitrary degree of numerical accuracy . the reason for this solubility lies precisely in the defining features of the model . large @xmath9 eliminates all feynman diagrams with internal @xmath10 loops and nonplanar gluons . on the other hand , @xmath8 allows gluon self - couplings to be eliminated by gauging away one component of the gauge potential @xmath11 . since only two components are initially present , the commutator term @xmath12 $ ] in the covariant derivative , which gives gluon self - coupling , vanishes identically in such gauges . then the only remaining feynman diagrams to be summed for the quark - antiquark green function are `` rainbow '' and `` ladder '' diagrams , whose schwinger - dyson equations can be solved , giving rise to an integral expression called the t hooft equation ( discussed in sec . [ thm ] ) . the t hooft model provides an excellent laboratory for testing various approaches to strong interaction physics . after all , the t hooft equation provides a means to compute hadronic masses , wavefunctions , and transition amplitudes in terms of the underlying partonic degrees of freedom . in this work we are specifically interested in questions of local quark - hadron duality in the inclusive decays of heavy quarks . the notion of duality in general terms was first introduced in the early days of qcd in ref . @xcite but not pursued for quite some time . a more detailed consideration was given a few years ago by shifman @xcite and later reiterated in a number of papers ( see , e.g. , refs . @xcite ) , with applications relevant to minkowskian observables amenable to study via an operator product expansion ( ope ) . this allows the formulation of the concept of local duality in a more quantitative way , including nontrivial nonperturbative effects ; we refer the reader to these recent publications for the theoretical aspects . here the question of duality is studied concretely by comparing the weak decay width of a meson containing a heavy quark computed in two ways . in terms of partonic degrees of freedom , one has an ope depending upon the free quark diagram ( with perturbative corrections ) and a number of nonperturbative matrix elements suppressed by powers of the heavy quark mass . in terms of the hadronic degrees of freedom , one simply computes the weak decay amplitude for each allowed exclusive channel , and adds them up one by one . this comparison is especially instructive since one may consider the behavior of solution as the mass @xmath13 of the heavy decaying quark is varied . such a problem was first considered in ref . @xcite , where the main elements in numerical computations of exclusive decay rates were annunciated . the hadronic result was compared to the born - level free - partonic diagram as a function of @xmath13 . in terms of the ope , the latter is the tree - level piece of the wilson coefficient corresponding to the unit operator . the numerical agreement was seen to be remarkable , in that the onset of the asymptotic agreement was clearly visible already for relatively small values of @xmath13 . the intrinsically limited numerical accuracy for sufficiently heavy quarks , however , prohibited drawing a definite conclusion about the size of nonperturbative corrections for asymptotically large @xmath13 . additional numerical studies @xcite considered similar questions for weak decay topologies other than the simple spectator tree diagram , in particular weak annihilation ( wa ) . the validity of the ope was addressed analytically in refs . @xcite , which considered on one hand the nature of the ope for heavy quark decays , and on the other an explicit @xmath3 expansion of the decay amplitudes , which allows an analytical summation of the individual decay rates in the asymptotic regime . the agreement of the two approaches through relative order @xmath14 was obtained by means of a number of sum rules derived directly from the t hooft equation , the archetype of which first appeared in ref . @xcite . while adequate to illustrate the theoretical validity of the ope for the inclusive decay widths of heavy flavors , the analytic methods per se can not help in answering the practical question relevant to phenomenology of beauty and charm quarks : namely , how accurately do the ope - improved parton computations describe the true weak decay width of a heavy flavor meson with finite mass , only a few times larger than the typical strong interaction scale ? a purely analytic expansion can hardly be used for this purpose , since it is a priori unknown how small an expansion parameter must be for the expansion to start yielding a reasonable approximation , not to mention achieving the necessary precision . to obtain insights into the size of deviations between the actual decay widths and the expressions obtained from the ope for quarks in the intermediate mass range , one must employ real numerical computations . in this paper we focus on semileptonic decays of heavy quarks . in the contexts of both real qcd and the t hooft model , they are technically simpler than nonleptonic decays . moreover , the magnitude of local duality violation is phenomenologically most important in semileptonic decays when one extracts @xmath1 and @xmath2 . we use the techniques developed in ref . @xcite to evaluate the required decay rates , and confront the total decay width with the expansion in terms of a power series in @xmath3 of ref . moreover , by making use of a number of relations derived in the large-@xmath13 limit of the t hooft equation @xcite , members of the set of nonperturbative matrix elements involved can be related to each other , providing an economical description of the nonperturbative physics . these are the tools that allow us to study the onset of quark - hadron duality . as explained in appendix [ newmult ] , we use a scheme based on the modified multhopp method , by which the t hooft equation is converted into an infinite - dimension eigenvector system that for practical reasons must be truncated at some number @xmath15 of eigenvector modes . the asymptotic convergence of this approach has not been rigorously studied , although it apparently must yield unlimited accuracy when the number of the multhopp modes @xmath15 goes to infinity . yet the rate of convergence at large @xmath15 is not well known . additionally , large quark masses turn out to require one to use a larger @xmath15 for sufficient numerical accuracy , as discussed in sec . it therefore seems mandatory to make an independent cross - check of the numerical accuracy . we investigate this problem by comparing the numerical values of a number of static properties of heavy mesons at different values of @xmath13 , with the results of their @xmath3 expansions obtained analytically from the t hooft equation ; this is the topic of sec . [ check ] . we find that our solutions have sufficient numerical accuracy for masses @xmath13 corresponding to physical values ( in the sense explained in sec . [ check ] ) as large as @xmath16gev . the duality of the inclusive widths of heavy - flavor hadrons to the parton - level widths , including the power corrections from the ope , emerges through a set of sum rules that equate sums of weighted transition probabilities to possible final states and expectation values of the local heavy quark operators . since our main interest lies in @xmath17 transitions , which carry in practice a limited energy release , the most relevant are the so - called small velocity ( sv ) sum rules , which we study here in the heavy quark limit . the behavior of these sum rules not only shapes the semileptonic @xmath17 decays in actual qcd , but is also important for the determination of the basic parameters of the heavy quark expansion . an additional advantage of the heavy quark limit for our investigation is that we are able to compute the sv amplitudes semi - analytically , using the exact relations @xcite derived from the t hooft equations and relying for input only on a few static parameters , which can be computed with a high precision . a discussion of these relations appears in sec . [ dual ] . we find that the sv sum rules in the t hooft model are saturated to an unexpectedly high degree by the first excitation above the ground state ( which we henceforth call the `` @xmath18-wave '' excitation , despite the fact that in @xmath8 only radial excitations occur ) . its contributions to even the darwin ( @xmath19 ) and kinetic ( @xmath20 ) expectation values constitute over @xmath21 and @xmath22 of the totals , respectively , while it saturates the `` optical '' sum rule for @xmath23 to a @xmath24 accuracy . this appears to be an intriguing dynamical feature of the model . a similar high - saturation effect has been observed in a quark flux - tube model @xcite , for the contribution from the `` valence '' quarkonium states . we study the size of violations of local duality in the semileptonic decays @xmath25 assuming vectorlike weak currents and massless leptons . these assumptions are important for comparison with qcd far beyond the obvious parallel of closely resembling the actual world : the strength of the resonance - related duality violation crucially depends on the threshold behavior in the decay probabilities , which is completely different in two and four dimensions . the two - body phase space , while @xmath26 in @xmath7 , is @xmath27 , that is , infinite at threshold , in @xmath8 . on the other hand , the situation is special for massless leptons : their invariant mass is always zero if they are produced by a vectorlike source , and the weak vertex is then proportional to the momentum . as a result , in this case the threshold behavior of the decay rate becomes @xmath26 much in the same way as in real qcd . this is a crucial detail if one tries to draw practical lessons from the t hooft model . the need for a vectorlike coupling in @xmath8 is even more stark for the parton - level calculation . there one finds that the integrated three - body phase space actually diverges for massless leptons , and only the behavior of the weak decay amplitude renders the width finite . we provide more arguments in favor of such a choice in sec . [ wid ] , which is dedicated to the inclusive decay widths . in sec . [ vac ] we briefly illustrate how well the duality works for the vacuum correlator of light quarks in the timelike domain . in the context of the heavy quark expansion this is relevant for the nonleptonic decay widths , including spectator - dependent effects like wa . section [ concl ] summarizes our investigation and discusses the conclusions that can be drawn for actual qcd . appendices describe the computational technique employed and contain a number of relations for the heavy quark limit of the t hooft equation employed in these numerical studies . we first review some well - known properties of the t hooft model both as a reminder and to establish notation . confinement is manifest in 1 + 1 spacetime dimensions with large @xmath9 , and the quark(@xmath28)-antiquark(@xmath29 ) two - particle irreducible green function , i.e. , the meson wavefunction @xmath30 , is given by the t hooft equation : @xmath31 where @xmath32 is the momentum fraction in light - cone coordinates carried by the quark , and @xmath33 since @xmath34 is finite in the large-@xmath9 limit , it provides a natural unit of mass . thus , all masses in this paper are understood as multiples of @xmath34 . indeed , as pointed out in ref . @xcite , @xmath34 fills the role in 1 + 1 dimensions of served by @xmath35 in 3 + 1 . we discuss the estimation of @xmath34 as a particular number in sec . [ check ] . the singularity of the qcd coulomb interaction in eq . ( [ the ] ) is regularized using a principal value prescription , indicated by p in eq . ( [ the ] ) . solutions @xmath36 of the t hooft equation alternate in parity , with the lowest being a pseudoscalar . the general analytic solution in closed form is not known . as the eigenvalue index @xmath37 increases , the eigenvalues @xmath38 asymptotically approach @xmath39 $ ] . the static limit @xmath40 is most easily studied @xcite by employing the `` nonrelativistic '' variables @xmath41 , @xmath42 and @xmath43 , in terms of which eq . ( [ the ] ) assumes the form @xmath44 we solve the finite - mass t hooft equation using a numerical method called the multhopp technique @xcite , a venerable system for solving integral equations with singular kernels . it was first applied to the t hooft equation in ref . the idea is to expand the wavefunction in a series of modes , not unlike fourier analysis , and then turn the equations for the mode coefficients into an equivalent infinite - dimension eigenvector problem . in practice , one then truncates at some point where the higher modes are deemed to have little effect upon the wavefunction solutions , which is of course strongly dependent on the highest value of @xmath37 used . the detailed formulas for applying the _ standard _ multhopp technique to mesons with unequal quark masses in the t hooft model appear in appendix a of ref . @xcite . intrinsic to the original multhopp technique is the evaluation of the wavefunction at a discrete set of points called `` multhopp angles , '' which in the current problem are equivalent to @xmath45 \ ; , \hspace{1em } \;\ ; k = 1 , \ldots , n \ ; , \ ] ] where @xmath15 is the number of modes retained in the numerical solution . the mode coefficients are then obtained by the use of a discrete inversion formula [ ( a7 ) in @xcite ] . however , the multhopp solutions can be seen to vanish as @xmath46 and @xmath47 at the endpoints @xmath48 and @xmath49 , respectively [ see ( a10)(a11 ) in @xcite ] , while the exact solutions are known to vanish as @xmath50 and @xmath51 , respectively , where @xmath52 leading to a type of gibbs phenomenon in the multhopp solutions . since the multhopp angles cease to sample the wavefunction at some finite distance from the endpoints , it may be expected that the wavefunctions thus obtained are numerically inaccurate there . this shortcoming led brower , spence , and weis @xcite to improve the multhopp technique by eliminating the multhopp angles and using instead a continuous inversion formula . the algebraic details are presented in appendix [ newmult ] , and it is this improved numerical technique that is used in obtaining our results . let us first establish a bit of notation . the mass of a heavy quark of flavor @xmath53 is labeled as @xmath54 ; in the weak transitions considered in subsequent sections , the final - state quark @xmath55 is assigned the mass @xmath56 . the spectator antiquark mass @xmath57 is labeled by @xmath58 , or @xmath59 if there is any chance of confusion . as explained in the previous section and appendix [ newmult ] , we use the modified multhopp technique to find numerical solutions of the t hooft eigenstate problem . since the heavy meson wavefunctions are peaked near the end of the interval , the accuracy deteriorates with increasing @xmath13 . the same , in principle , applies to the high excitations of light hadrons . a more appropriate strategy for heavy quarks is to start with a solution of the infinite - mass ( static ) equation . this has been done analytically @xcite , and full consistency created by the weak current have @xmath60 , this agreement was shown up to and including @xmath61 terms in the weak decay width in @xcite , while terms up to and including @xmath62 were shown to coincide with those in the ope in @xcite . in the current work we take @xmath63 . ] with the ope was demonstrated . however , our practical interest lies in the properties of heavy hadrons with @xmath13 lying in the intermediate domain , specifically for @xmath13 one order of magnitude larger than @xmath34 . the convergence of the @xmath3 expansion in this case is too difficult to quantify analytically . this is just the situation where the numerical computations are best employed . therefore , an important element of the analysis is to check the accuracy of the numerical computations of both the heavy hadron masses and wavefunctions at different values of @xmath13 . to this end , we compute the masses and certain moments for the ground and first excited states , and compare them to the analytic @xmath3 expansion . in general , the terms in the @xmath3 expansion depend on a number of expectation values in the static limit , like the kinetic one @xmath64 , etc . however , one can show @xcite that the parameters appearing here through high order in @xmath3 can be expressed in terms of just the asymptotic value @xmath65 and the corresponding decay constant . these quantities are the ones most accessible to numerical evaluation ; in particular , @xmath66 is expected to be the most accurately determined quantity . our main computations refer to the case of @xmath67 , as chosen in ref . it corresponds ( see sec . [ dual ] ) to a mass of the strange quark in qcd . the choice of a noticeable light quark mass may be motivated by an attempt to mimic the effect of the transverse gluons absent in @xmath8 , which in a certain respect supply some effective mass to the light quark . clearly , this can be only a rather crude approximation , since the bare quark mass breaks chiral invariance . one can suppose , nevertheless , that this side effect is not too important for our purposes . the chiral symmetry is spontaneously broken anyway , and the presence of a massless versus a massive pion does not seem to be essential for the range of problems we address here . on the other hand , the effect of the transverse degrees of freedom is known to soften the @xmath68 singularity of the heavy quark distribution function @xcite , similar to the impact of the light quark mass in the t hooft model . the behavior of the distribution function affects the inclusive decays of the heavy quarks in an essential way . we also present some results for @xmath69 , partly to explore light quark dependences of matrix elements and partly to investigate the beginnings of failure of the numerical solutions as @xmath70 . the number @xmath15 of multhopp modes used is @xmath71 ; we considered smaller @xmath15 as well to study this dependence , but since the behavior was found to be stable , we do not dwell on it further here . the masses of heavy hadrons obey @xcite @xmath72 where @xmath73 and these expectation values refer to the infinite mass limit . in the @xmath74 expression , @xmath55 is the momentum variable conjugate to @xmath32 , and diagonal transitions within the correlator have been removed . since @xmath66 in qcd traditionally denotes the mass difference between a ground - state pseudoscalar meson and its corresponding heavy quark in the large @xmath13 limit [ as it is defined in eq . ( [ 50 ] ) ] , and we need it for a number of the excited states @xmath75 as well , we assign the notation @xmath76 and use @xmath77 and @xmath66 throughout the paper on equal footing . equations ( [ 50])([58 ] ) hold for each state @xmath75 with @xmath78 , so that an implicit superscript @xmath79 is to be understood in these expressions . according to ref . @xcite , the following relations hold in the t hooft model : @xmath80 \;. \label{52}\ ] ] here @xmath81 is the scaled decay constant in the heavy quark limit , i.e. , @xmath82 where the superscript is suppressed if there is no ambiguity , @xmath83 and the exact relation between @xmath84 and the decay constant of the @xmath37th excitation is given in eq . ( [ dec ] ) . in the heavy quark limit one has @xmath85 but there are @xmath86 corrections to these relations . for further applications to the decay widths we also consider the scalar expectation value @xcite @xmath87 then the following expansions hold : [ 58 ] c_n & = & ( 1 - ) f^(n ) + o ( ) , + m_q & = & - + + o ( ) , + m_q^2(-^2 ) & = & _ ^2- ( 8 _ ^2 + ^2 f^2 ) + o ( ) , + & = & 1- - + o ( ) . we note that values of @xmath20 , @xmath19 , or @xmath74 determined from the expansions eqs . ( [ 58 ] ) suffer degraded numerical accuracy compared to those taken directly from eqs . ( [ 52 ] ) since @xmath66 and @xmath81 are determined from more stable expansions ( in particular , they do not depend upon close numerical cancellations ) . therefore , we use eqs . ( [ 52 ] ) as primary information and relegate eqs . ( [ 58 ] ) to numerical checks . our method of determining @xmath66 from the @xmath88 expression , designed to minimize the influence of potentially large uncertainties at large @xmath13 , is described in appendix [ calc ] . values of @xmath88 and and the averages in eqs . ( [ 58 ] ) as functions of @xmath13 from @xmath89 to @xmath90 are presented in table [ t1 ] for both the ground and first excited states . similar results for just the ground state with @xmath91 are presented in table [ t2 ] . based upon the 10 data points presented in table [ t1 ] for the ground state , one may fit to a polynomial in @xmath3 , obtaining @xmath92 the corresponding expressions using the approach of appendix [ calc ] ( neglecting the one for @xmath88 , which is used as input and hence is identical through @xmath93 ) read @xmath94 this agreement between the two approaches is quite excellent and is exhibited in figs . [ m56][mx2 ] for @xmath88 and the quantities in eqs . ( [ 58 ] ) ; in general , the exact results are presented as points on a solid line , while each fit using eqs . ( [ 1mfits ] ) is presented as a dashed line . in fig . [ m56p ] the analogous expression @xmath88 for the @xmath95 first excited state is presented , while fig . [ m26 ] uses the same methods and values from table [ t2 ] to present @xmath88 for the @xmath96 ground state . in fig . [ m56 ] and especially in fig . [ m26 ] , the quality of numerical results is seen ( as expected ) to begin breaking down at large @xmath13 and small @xmath58 , since @xmath97 is fixed . we conclude that the numerical routine we rely upon is sufficiently accurate for @xmath97 up to @xmath98 . the critical value of @xmath13 also depends , however , on the meson s light quark mass , decreasing for small @xmath58 . this is expected since at small @xmath58 the sharpness of the wavefunction as @xmath99 becomes stronger , and more multhopp functions are required to approximate it : each multhopp function vanishes like @xmath47 . likewise , the required @xmath15 increases for the excited states . still , one can check that it is possible to go as high as @xmath100 even for @xmath58 as small as @xmath101 . it turns out that a numerically significant cancellation occurs in the value of @xmath102 in @xmath103 corrections and , in particular , at the @xmath104 level between @xmath19 and @xmath74 , for the ground state just around our primary value @xmath89 . such a numerical suppression of the power corrections is accidental and does not occur for the excited states , nor for @xmath105 . let us note that the expectation value of the _ light - quark _ scalar density in the heavy meson turns out very close to unity for the ground state , which may be seen by taking @xmath106 and @xmath107 in eq . ( [ 56 ] ) and referring to table [ light ] ; this is a characteristic feature of a nonrelativistic ( with respect to the light quark ) bound - state system . it implies an almost simple additive dependence of @xmath66 on the light quark mass @xmath58 , @xmath108 and indeed one can verify this feature by comparing @xmath88 values between table [ t1 ] ( @xmath95 ) and table [ t2 ] ( @xmath96 ) . while this pattern is expected when the spectator quark is heavy , it a priori needs not hold when it is light this supports the naive expectation that the chiral symmetry breaking may lead to a description in some aspects resembling the nonrelativistic constituent quark model . the above expectation value , however , decreases for the excited states , as expected from such a picture . drawing semi - quantitative conclusions for qcd requires a translation rule between the mass parameters in the two theories , that is , an estimate of the value of @xmath34 in @xmath109 . different dimensionful quantities can be taken as the yardstick ; since the theories are not identical , this translation rule must be introduced with some care . as follows from the heavy quark sum rules , the physics of duality in the decay widths of heavy flavors crucially depends on the properties of the lowest excited heavy - quark states , in particular the @xmath18-wave excitations with opposite parity to the ground - state multiplet . it will become evident from the next section that they are of primary importance for the @xmath3 expansion of static properties as well . therefore , we choose the mass difference between the lowest parity - even ( @xmath18-wave ) state and the parity - odd ground - state meson to gauge the translation between the mass scales . in the t hooft model the mass difference @xmath110 for light spectators amounts to about @xmath111 . real charm spectroscopy suggests that the first @xmath18-wave excitations are between @xmath112 and @xmath113 above the ground state . taking the larger value for sake of illustration , we arrive at the estimate @xmath114 which is adopted in our analysis . this falls rather close to the estimate of ref . @xcite , which relied on a quite different type of effects in the light - quark systems . assuming a value for the `` bare '' @xmath115 quark mass in qcd ( normalized at the appropriate scale @xmath116 ) of about @xmath117 , we conclude that mesons with quarks of masses @xmath118 represent in the t hooft model the actual beauty and charm mesons . the value of @xmath119 seems to be in a reasonable correspondence with the size of this difference in qcd when it is normalized at a low hadronic scale , @xmath120 @xcite . it should be noted , however , that the kinetic expectation value in the t hooft model turns out to be rather small , @xmath121 . this is not surprising , since the chromomagnetic field is absent in two dimensions , while it was shown @xcite to be crucial in the real case . indeed , the comparison is better justified for the difference @xmath122 in actual qcd versus the value of @xmath20 in the t hooft model . these questions were discussed in detail in ref . @xcite , and can be easily understood using the sum rule representation . due to the absence of spin in two dimensions , there is no difference between the would - be spin-@xmath123 and spin-@xmath124 light degrees of freedom . in particular , labeling the `` oscillator strengths '' @xmath125 defined in the next section by spin rather than excitation number , @xmath126 and @xmath127 effectively hold . then the sum @xmath128 and the latter sum is just the general expression for @xmath129 : @xmath130 accepting such an identification suggested in ref . @xcite and the estimate @xmath131 , we again observe a reasonable agreement with the findings of the t hooft model . a useful theoretical limit the so - called small velocity ( sv ) regime was suggested in the mid 80 s @xcite as a theoretical tool for studying semileptonic heavy quark decays . this refers to kinematics where both @xmath115 and @xmath132 quarks are heavy , but the energy release is limited , so that the velocity of the final charm hadron is small . at large energy release the ope for the width must converge rapidly to the actual hadronic width . still , at fixed energy release the deviations , although @xmath3 suppressed , are present regardless of the absolute values of masses . in the sv regime the semileptonic decays proceed either to the ground - state charm final state , @xmath133 or @xmath134 ( the semi - elastic transitions ) , or to excited `` @xmath18-wave '' states of the opposite parity . other decays are suppressed by higher powers of velocity , or by heavy quark masses . the equality of the sum of partial decay widths and its ope expansion is achieved through the sum rules that relate the sums of the @xmath18-wave transition probabilities , weighted with powers of the excitation energies , to the static characteristics of the decaying heavy hadron . the onset of convergence of the ope expansion for the widths is then directly related to the pattern of saturation of the sum rules by the lowest excitations . if higher states contribute significantly , they delay the onset of duality , while their absence leads to a tight quark - hadron duality after the first @xmath18-wave channel is open . knowledge of degree of saturation of the heavy quark sum rules is also important for another reason : it determines the hadronic scale above which one can apply the perturbative treatment to compute corrections or account for evolution of the effective operators . the lower this scale , the more predictive in turn is the treatment of the nonperturbative effects in the ope . a recent review of the sv sum rules can be found in ref . @xcite ( the perturbative aspects are discussed in more detail in ref . @xcite ) . for most practical purposes addressed here , one can consider the perturbative effects to be absent in the t hooft model . in particular , the heavy quark parameters do not depend perturbatively on the normalization point , and there is no need in the explicit ultraviolet cutoff to introduce a normalization point . the sum rules we address are _ k^2- & = & _ n [ oscsum1 ] _ nk^2 , + _ k & = & _ n ( _ n-_k ) _ nk^2 , + ( _ ^2)_k & = & _ n ( _ n-_k)^2 _ nk^2 , + ( _ d^3)_k & = & _ n ( _ n-_k)^3 _ nk^2 . [ oscsum4 ] here @xmath135 and @xmath37 denote excitation indices for the initial and final states , respectively ( in practice only transitions from the ground state are interesting , so we limit ourselves to @xmath136 ; in this case the index @xmath135 is omitted ) . the so - called `` oscillator strengths '' @xmath125 parameterize the transition amplitudes into the opposite - parity states in the sv limit , = _ nk _ v^+ o(^3 ) , [ 102 ] where @xmath137 is the velocity of the final state hadron . in the diagonal transition @xmath138 is the slope of the isgur - wise ( iw ) function of state @xmath139 : = 1-_k^2 + o ( ^4 ) . [ 104 ] the expressions for @xmath140 and @xmath141 in terms of the light - cone wavefunctions are _ nk & = & _ 0^ dt _ k(t ) = -_m_q _ 0 n(x ) ( 1-x ) _ k(x ) , + ^2_k & = & _ 0^ dt |(t + ) _ k(t)|^2 = _ m_q _ 0 ^ 1 dx |_k(x)|^2 . + [ 106 ] the finite-@xmath13 corrections to the @xmath32-integral forms turn out to be rather significant , leading to significant problems in precision numerical studies . to avoid this problem we use the analytic expression for the inelastic amplitudes obtained in ref . @xcite : @xmath142 where @xmath143 are the asymptotic values of the decay constants @xmath84 scaled up by the factor @xmath144 , as in eq . ( [ 112 ] ) . the constants @xmath143 are computed as the values of @xmath145 at @xmath146 ( see table [ t1 ] ) augmented by the @xmath3 corrections detailed in the first of eqs . ( [ 58 ] ) , while values of @xmath147 are computed using the procedure described in appendix [ calc ] . the results of the computations for the case @xmath100 , @xmath148 are presented in table [ osc ] . our central result is a surprisingly good saturation of the sum rules : the first @xmath149 excitation generates @xmath150 of @xmath151 , @xmath152 of @xmath66 , @xmath22 of @xmath20 , and even @xmath153 of @xmath19 . the rest is almost completely saturated by the second @xmath18-wave state ( @xmath154 ) , where the cumulative values for the same quantities read @xmath155 , @xmath156 , @xmath157 , and @xmath158 , respectively . in terms of absolute numbers , the sum rules eqs . ( [ oscsum1])([oscsum4 ] ) would give @xmath159 , @xmath160 , @xmath161 , and @xmath162 , the last of which gives @xmath163 , in fine agreement with the values obtained from the values obtained in the previous section via the methods described in appendix [ calc ] . the few - percent discrepancy corresponds to the accuracy in determinations of squared decay constants . the level of saturation by the lowest open channels is extraordinary . the explicit reason for such a perfect saturation of the sum rules involving even rather high , @xmath164 powers of the excitation energy can be read off eq . ( [ 110])@xmath165 s are inversely proportional to the third power of the excitation energy . with the asymptotics @xmath166 , @xmath167 , the first excitation energy @xmath110 is notably smaller than the next one @xmath168 including three energy gaps . the general peculiarity of the t hooft model leading to such a saturation is not understood completely . with this pattern of saturation of the sv sum rules for the ground - state meson , one expects an early onset of the accurate duality for the inclusive widths in the @xmath17 transitions , only slightly above the threshold of the first excitation . demonstrating this result through direct evaluation of the decay widths is one of the purposes of the next section . the semileptonic widths described in this work were considered in detail in ref . @xcite . here we recapitulate a few basic points . the weak decay lagrangian is @xmath169 in terms of the previous notation , @xmath170 , @xmath171 ( or , later in this section , @xmath172 ) , and @xmath173 . the key property of all @xmath8 vectorlike currents is that for @xmath174 , the invariant mass @xmath175 of the lepton pair is always zero . for all computational purposes decays into this massless lepton pair are equivalent to decays into a single massless pseudoscalar particle @xmath176 weakly coupled to quarks according to @xmath177 several arguments favor our choice of a vectorlike weak decay interaction in the t hooft model . one is of course the simplicity of eq . ( [ 122 ] ) . another is that for @xmath178 some difficult problems of renormalization are absent , as we now discuss . the central problem in applying the ope in practice is disentangling perturbative and nonperturbative effects . more precisely , this refers to the separation of short - distance effects attributed to the coefficient functions from long - distance effects residing in the matrix elements of the effective heavy - quark operators . the perturbative corrections , for example those that renormalize the weak quark current , are generally rather nontrivial , even in the t hooft model . however , according to the nonrenormalization theorem of ref . @xcite , such vertex corrections are absent from the decays with @xmath178 . this allows one to isolate the problem of renormalization of the underlying current from the question of interest in our study : possible deviations of the full decay widths due to the presence of thresholds in the production of the hadronic resonances . in reality , from the ope viewpoint some short - distance corrections still remain even in this special kinematic region due to the high - momentum tails in the meson wavefunctions . these tails come from the hard gluon exchanges between the constituents . in principle , these `` hard '' components can also be separated from the `` soft '' bound - state dynamics explicitly . however , in practice this is not necessary : these effects are completely contained in the meson wavefunctions . another advantage of vectorlike currents is apparent when one notes that the @xmath8 three - body `` semileptonic '' phase space diverges logarithmically for massless leptons . explicitly , for the decay @xmath179 ( equal lepton masses are assumed to render the expressions simpler ) , the three - body phase space turns out to be @xmath180}{(m\!-\!m)^2 \left [ ( m\!+\!m)^2 \!-\!4m_\ell^2 \right ] } \right],\end{aligned}\ ] ] where @xmath181 is the complete elliptic integral of the first kind . as @xmath182 , one regains eqs . ( 4.1)(4.3 ) of @xcite , while as @xmath183 , the argument of the elliptic integral goes to unity , and @xmath184 . this is a manifestation of the logarithmic infrared divergence of the massless scalar green function at large distance in @xmath8 . a detailed calculation shows that the vector nature of the weak coupling regularizes the phase space integral , preventing the partonic rate from diverging in the limit of massless leptons . furthermore , as discussed in the introduction , this also removes the @xmath185 singularity in the threshold behavior for hadronic two - body decays . as a final advantage of vectorlike currents and the special kinematic point @xmath178 , note that at @xmath178 the @xmath186 transition amplitudes are directly expressed in terms of the overlap between the initial and the final wavefunctions : @xmath187 where @xmath188 , so that the partial decay width for @xmath189 is given by @xmath190 the threshold suppression mentioned above is manifested in the explicit factor @xmath191 : the reciprocal of this factor in the phase space is removed by @xmath192 from the matrix element . it is also possible to derive this result directly using the methods of ref . @xcite ; note , however , that these expressions are much simpler than those of ref . @xcite , because the vectorlike current with massless leptons restricts @xmath175 to 0 . the sum of these widths over all open channels is to be compared to the ope prediction . the remarkable speed of saturation in @xmath37 , anticipated in the last section , is illustrated for one sample case in table [ satur ] . turning to the ope , we mention one more problem associated with an accurate understanding of local duality violation . apart from the purely theoretical aspect that ope power series are generally only asymptotic and , thus have a formally zero radius of convergence in @xmath3 , one normally has additional practical limitations . only a limited number of the terms , as well as the associated expectation values , are usually known , which places additional theoretical uncertainties that dominate in practice at sufficiently large @xmath13 . this feature can be naturally incorporated in the analysis of our concrete model . we account completely only through terms that scale like @xmath14 , the highest order that emerges from the ope free from the four - fermion operators @xcite . the rest , although calculable in principle term - by - term in the t hooft model , are taken to represent the ope `` tails '' discarded by the unavoidable truncation . using the sum rules of the t hooft model , ref . @xcite established the following exact representation for the total decay width : @xmath193 where @xmath194 at @xmath195 are understood as given by eq . ( [ 124 ] ) without the explicit @xmath196-function singling out the open channels ; such @xmath194 are therefore all negative . on the other hand , the ope yields the result @xmath197 \ ; , \label{132}\ ] ] with @xmath198 generically denoting the ope expansion parameter ; we do not specify here if it is @xmath199 or @xmath200 , or some other combination . it was shown in ref . @xcite that the @xmath194 term in eq . ( [ 130 ] ) is dual to the order term in eq . ( [ 132 ] ) ; however we do not use this here and rather treat the latter as an intrinsic uncertainty in the `` practical '' version of the ope . thus , our strategy is to compare the exact width @xmath201 to @xmath202 the expectation value @xmath203 above can either be evaluated numerically , or in the spirit of the ope , computed in the form of a @xmath204 expansion , the last of eqs . ( [ 58 ] ) . it turns out that the expansion converges very rapidly to the exact result , so that this does not significantly affect the observed pattern of local duality at the quantitative level . the born - term partonic rate is simply given by @xmath205 with this expectation value set to unity , @xmath206 the main practical interest of these calculations lies in the @xmath207 width with its limited energy release @xmath208 . in general , @xmath208 can be small either if @xmath199 is not large enough , or even at large @xmath199 if @xmath209 ( or @xmath210 if @xmath175 is nonzero ) is insufficient due to a significant @xmath132 quark mass . the latter case falls into the sv category , and the violations of duality are suppressed here even at the maximal @xmath175 by heavy quark symmetry , as was pointed out in the mid-80 s @xcite . therefore , one a priori expects a different pattern in the two cases . we try to separate the possible effects by considering different choices for @xmath199 and @xmath211 rather than by only taking them close to their realistic values . with these arguments in mind , one can expect to find significant effects of duality violation in the cases where @xmath212 or @xmath204 effects are important . as suggested in ref . @xcite , in this case it is advantageous to fix @xmath199 close to its actual value , and vary @xmath211 from near @xmath199 down to smaller values , changing in this way the energy release . at one end of the interval the local duality is supported by the heavy quark symmetry with large quark masses and sv kinematics , while at another end it rests on the large energy release . we start from the sv case when @xmath199 is fixed and large and @xmath211 is large as well , varying the energy release by increasing @xmath211 towards @xmath199 . since the violation of local duality is expected to be suppressed for all values of @xmath211 , high numerical accuracy is vital . we fix @xmath213 ( @xmath214 ) , and vary @xmath211 from @xmath215 up to @xmath199 . the results are given in table [ dual15 ] and fig . [ dual15fig ] . we note that the difference between the two widths is so small that one must plot @xmath216 rather than the widths themselves . this is expected since the sv sum rules are very well saturated , as detailed in the previous section the higher thresholds are then strongly suppressed numerically at finite energy release . but for @xmath211 approaching @xmath199 , where they could be noticeable , the heavy quark symmetry works efficiently since both quarks are very heavy . in fact , the only prominent features on the plot occur when thresholds to the first few @xmath133 states of opposite parity to the ground - state @xmath217 meson are crossed , for example between @xmath218 and @xmath219 . the deviation is extremely small also for smaller @xmath211 where the @xmath132 quark velocity is rather large yet there the energy release is significant , and a large number of excited states ( up to @xmath220 at @xmath221 ) are produced . table [ dual10 ] and fig . [ dual10fig ] show analogous results for @xmath222 , @xmath148 . to render the duality violation more apparent , we consider ( table [ dual5 ] and fig . [ dual5fig ] ) the same decay widths for a @xmath115 quark with half the mass , @xmath223 . even here the deviation is below per mill as soon as the first excitation can appear with sufficient phase space . the duality - violating component at last exhibits the proper oscillating behavior ( note the decrease between @xmath224 and @xmath225 or 1 and @xmath226 ) , but this effect is too small to be extracted reliably at larger energy release where this property becomes an asymptotic rule . as follows from our computations , local duality is violated at a tiny level in the @xmath17 decays in the t hooft model whenever it is a priori meaningful to apply ope . a possible reason behind this might be that for unidentified reasons the heavy quark symmetry works for the inclusive widths too effectively , down to relatively low masses and velocities of order @xmath227 . this was conjectured in the early papers on the subject @xcite . therefore , our final attempt in the quest for the sizeable duality violation in beauty is considering the @xmath228-type transitions , where the heavy quark symmetry per se does not constrain the individual transition form factors . we fix in our expressions @xmath229 or @xmath230 ( but still keep the two quarks flavor - distinguished ) and vary @xmath199 from @xmath231 to @xmath232 . the results are shown in table [ dualmc56 ] and fig . [ dual_mc56 ] , and table [ dualmc26 ] and fig . [ dual_mc26 ] , respectively . although the difference between the actual width and its ope approximation is larger , it still is very small and approaches a percent level for @xmath199 as low as @xmath233 . the total decay width is no longer saturated to such a high degree by transitions to the ground state , especially for larger @xmath199 . nevertheless , the duality is amazingly well satisfied when just the first few open channels are summed . again , the only prominent features in the plots appear when crossing kinematic thresholds due to the lightest @xmath133 mesons of opposite parity to the ground - state @xmath217 . the extraordinary agreement between @xmath234 and @xmath235 may be underscored by instead plotting ( fig . [ part_mc56 ] , final column of table [ dualmc56 ] ) the difference between @xmath234 and the born - term partonic rate @xmath236 given in eq . ( [ born ] ) . from an algebraic point of view , @xmath234 and @xmath235 differ generically at @xmath237 , while @xmath234 and @xmath236 begin to differ already at @xmath238 . thus , we find local duality between the actual semileptonic decay width and its ope expansion to be very well satisfied in all cases . before concluding this section , let us briefly address duality in the differential distribution @xmath239 . in the heavy quark limit the shape of the final - state hadronic mass distribution follows the heavy quark distribution function in the decaying meson ; for the @xmath240 decays under consideration , this is the light - cone distribution function @xmath241 . in decays with @xmath178 the recoil energy of the lepton pair @xmath242 is directly related to the final state mass @xmath243 : e = . [ 150 ] since @xmath178 , these decays are analogous to @xmath244 in the standard model . in the large-@xmath199 limit one has = f ( ) . [ 152 ] at finite @xmath199 in a theory with narrow resonances the actual distribution is given by the comb of @xmath245-functions with spacing in the argument of eq . ( [ 152 ] ) of order @xmath246 . in order to obtain a continuous result , we adopt the simple ansatz of averaging over the peaks . using eq . ( [ 150 ] ) to define the energy @xmath247 of the @xmath37th state @xmath243 , we integrate the @xmath245-function for the @xmath37th state evenly over the energy range @xmath248 to @xmath249 , i.e. , the midpoints between energy eigenvalues . letting @xmath15 be the maximum number of kinematically allowed @xmath243 values , we establish the endpoint bins by defining @xmath250 and @xmath251 . we find that our numerical computations yield a distribution resembling the light - cone distribution function @xmath252 ; specifically , f(y)=^2 ( ( 1-y ) ) = _ m_q ^2 ( 1-(1-y ) ) . [ 154 ] recalling that @xmath253 and combining eqs . ( [ 152 ] ) and ( [ 154 ] ) yields @xmath254 the two sides of this expression are plotted in fig . [ dgde_25v10 ] , using @xmath255 to represent the limit @xmath256 , while the actual distribution is considered at @xmath257 . the agreement is quite remarkable . the continuous distribution appears to pass approximately through the midpoint of each bin ; owing to the near - equal spacing of t hooft model eigenvalues in @xmath38 , eq . ( [ 150 ] ) shows that these bin midpoints are very close to the values @xmath247 themselves . it is also interesting to consider integration over a range of @xmath242 . in particular , define @xmath258 as the cumulative fractional width from maximum energy @xmath259 down to the given @xmath242 ; then @xmath260 and @xmath261 . while the exact result for @xmath262 amounts to an integration of the @xmath245-function differential widths renormalized so that the cumulative result approaches unity , the integral of the continuous distribution gives @xmath263 these two curves are presented in fig . [ inc10 ] . two features particularly stand out in this plot . first , even for @xmath199 as large as @xmath264 , the overwhelming part of the decay probability falls into the transitions to at most four lowest states . second , the continuous curve seems to provide a nearly optimal description possible for the step - like exact distribution . the point - to - point deviation for all plotted values with @xmath265 does not exceed half of the contribution of the nearest threshold . in this section we briefly illustrate the onset of duality for the absorptive part of the vector current correlator with light quarks , of the type that determines the normalized cross section @xmath266 as a function of energy . in the context of the heavy quark decays this is relevant in nonleptonic decay widths in two kinds of processes : in spectator - independent decays , where @xmath267 determines the weight with which the semileptonic width at given @xmath175 must be integrated over @xmath175 ( see ref . @xcite ) , and in the effects of wa decays . in either case , at @xmath268 the cross section appears as a comb - like collection of @xmath245-functions : r(q^2)= _ n c_n^2 ( q^2-m_n^2 ) ; c_n= _ 0 ^ 1 dx _ [ 200 ] the above expression for the residues refers to the case where a vector current is considered . we suppress here the factor of @xmath269 relating @xmath84 to @xmath270 [ eq . ( [ dec ] ) ] . we also assume in what follows that the light quark masses are equal , @xmath271 , and are @xmath272 or less , in order to reach asymptotic @xmath175 more quickly . in the extreme situation of infinitely narrow resonances one can not , of course , discuss a point - to - point equality of the cross section @xmath267 with its ope in the form of @xmath273 expansion . a meaningful comparison is possible if each resonant peak is somehow averaged over an interval no smaller than the distance between adjacent peaks , the latter being approximately given by @xmath274 @xcite . it is worth recalling that @xmath267 is proportional to @xmath275 , so one must consider nonvanishing masses for the vector current , and address the ope terms formally suppressed by @xmath276 . this question was first addressed in the context of nonleptonic decays in ref . @xcite using the numerical approach . duality for the average cross section in the same manner as above , i.e. , using sum rules derived from the t hooft equation and analytically matching terms in the @xmath3 expansion , was obtained in ref . yet establishing the asymptotics per se can not tell us beforehand how early one can expect the onset of duality . here we study this question numerically , in the domain of intermediate @xmath175 . the concrete amount of the deviation between @xmath267 and @xmath277 in the case of direct resonances may depend in an essential way on the chosen smearing procedure . interested in the qualitative features only , we choose a rather simplified , crude method : we spread the integral of @xmath267 evenly over the interval between the successive resonances . more precisely , we put @xmath278 , with @xmath279 , the partonic pair production threshold . here we use the fact that @xmath84 vanish for odd @xmath37 when @xmath280 . this smearing is very similar to that described for the differential width in the last section , except that averaging is performed in @xmath175 rather than @xmath242 . the free quark loop @xmath267 , which is of course the leading term of the ope , is given by r_0(q^2)= . [ 212 ] table [ vaccor ] and fig . [ vacfig ] show the results for our reference case @xmath148 . the agreement of the average hadronic cross section with the parton - computed probability again turns out to be very good . apparently , this can be related to two facts : the heavy suppression of power corrections to @xmath267 in the ope ( see eqs . ( 34)(35 ) in @xcite ) , and an early onset of the asymptotics in the spectrum , @xmath281 which even at @xmath282 is satisfied to about @xmath283 . the main motivation behind the present study has been to assess the magnitude of local duality violations in the inclusive semileptonic decays of beauty particles . we considered this question using the t hooft model as a toy theory in which all relevant decay amplitudes can be evaluated numerically . the t hooft model , while retaining certain key features of full @xmath7 qcd that shape the spectrum of hadrons ( quark confinement , chiral symmetry breaking ) , still differs from @xmath7 in many respects . yet using it as a lab for exploration carries an important advantage it allows no `` wiggle room '' for interpretation of the results . there are no ad hoc parameters to choose or adjust , and as soon as the underlying weak decay lagrangian is fixed , the numerical results are unambiguous and must be accepted at face value . this positively distinguishes this approach from various models where often the conclusions , even qualitatively , depend on the arbitrary choice of parameters according to one s preferences . the question of a particular model being compatible with the general dynamical properties of qcd underlying the ope approach , often quite problematic in simplified quark models , does not arise for the t hooft model . although the simplest illustration of the asymptotic nature of the decay width @xmath3 expansion and related violations of local duality @xcite follows just from the existence of hadronic thresholds ( see , e.g. , @xcite ) , violation of local duality is a more universal phenomenon that is _ not _ directly related to existence of hadronic resonances nor even confinement itself . this has been illustrated in ref . @xcite by the example of soft instanton effects that do not lead , at least at small density , to quark confinement but do indeed generate computable oscillating duality - violating contributions to the total decay rates . nevertheless , there is a widespread opinion that decays with manifest resonance structure in the final state are most difficult for if compatible at all with the standard ope . even the possibility that the ope does not fully apply in the case of `` hard '' confinement has been occasionally voiced in the literature . the analytic studies performed in refs . @xcite , which explicitly demonstrate in the t hooft model the applicability of the ope to the total widths , should help to allay such conceptual concerns . nevertheless , the intuition remains that resonance dominance is not `` favorable '' for the ope , and problems might show up , for instance , through a delayed numerical onset of duality , in that the approximate equality of the ope predictions and the actual decay widths may set in only after a significant number of thresholds has been passed . to address such issues , the t - hooft model seems to represent the most certain testing ground for local duality in the domain of decays of moderately heavy quarks . contrary to naive expectations , we found surprisingly accurate duality between the ( truncated ) ope series for @xmath284 and the actual decay widths . the deviations are suppressed to a very high degree almost immediately after the threshold for the first excited final state hadron is passed . no suspected delay in the onset of duality was found . the key property that governs the onset of the @xmath3 expansion for the semileptonic widths is the pattern of saturation of the heavy quark sum rules . we examined a particular class , the sv sum rules in the heavy quark limit , that has the most transparent quantum mechanical meaning . we found them saturated to an amazing degree by the very first excitation . the contribution of the remaining , higher states to the slope of the iw function , @xmath66 , and @xmath20 does not exceed a few percent . even in the darwin operator sum rule , the first excitation accounts for @xmath21 of the whole expectation value , despite the fast - growing weight , @xmath285 of higher - order contributions . this peculiarity underlies the early onset of duality for the case when initial- and final - state quarks are both heavy . some of the duality - violating features observed in these studies have natural explanations . at fixed energy release @xmath286 the magnitude of the deviations is smaller if @xmath13 , @xmath56 are both large ( as in @xmath287 ) than if they are both small . this is expected , since in the former case the heavy quark symmetry for the elastic amplitude additionally enforces approximate duality even when no expansion in large energy release can be applied . it is interesting , however , that at fixed @xmath199 the duality violation decreases rapidly as @xmath211 decreases , in full accord with the ope where the higher order terms are generally suppressed by powers of @xmath288 . this is clearly a _ dynamical _ feature that goes beyond heavy quark symmetry per se , the quality of which deteriorates as @xmath211 decreases . it is also instructive to note that including the calculated power - suppressed ope terms significantly reduces the difference between the actual decay width and its purely partonic evaluation . moreover , the seeds of oscillations inherent to duality violation ( as functions of quark masses ) , can be seen . since we adopted the truncated ope expansion to mirror the existing implementation of the ope in qcd , the deviations do not average to zero but rather oscillate around the ( rapidly dissipating ) contributions attributed to discarded higher - order terms . the numerical effects of duality violation we study turn out to be typically quite small . partially this can be attributed to moderate size of the corresponding expectation values multiplying @xmath289 corrections in the ope . yet certainly not all power corrections in heavy quarks are suppressed in the model . it is well known from ordinary quantum mechanics that masses ( eigenvalues ) typically are much more robust against perturbations than wavefunctions themselves ( or transition amplitudes ) . we observe a similar pattern in the t hooft model . for example , @xmath3 corrections to the meson decay constants turn out very significant even at the scale of the @xmath115 quark mass . apparently , the inclusive decay rates fall into the class of `` robust '' observables , although , as explained above , this was difficult to anticipate beforehand . we note here another `` fragile '' observable , the light - cone heavy quark distribution function , which can be measured in decays of the type @xmath290 . in @xmath8 the scaled spread @xmath291 of the @xmath32 distribution approaches @xmath20 at large @xmath13 . yet , as seen in fig . [ mx2 ] , even at the @xmath115 quark mass one would obtain from this distribution only about @xmath292 of the actual value of @xmath20 , due to significant @xmath3 corrections . this caveat may be important for existing analyses of the decay distributions in @xmath217 decays , where such effects routinely are not included . we also briefly addressed the inclusive differential decay distributions in the analogues of @xmath293 or @xmath294 decays . generally , we find good agreement ( at the scale corresponding to the physical @xmath115 mass ) with the parton - based prediction incorporating effects of the `` fermi motion , '' and in particular for the partially integrated probability ( x ) = _ 0^xm_b^2 dm_h^2 . [ c4 ] this distribution , following refs . @xcite , is examined in real @xmath217 decays in the quest for @xmath2 @xcite . however , the point - to - point deviations are clearly still significant , for the decays to only the 4 or 5 lowest final states saturate the overwhelming fraction of the total decay probability . it is quite conceivable , though , that such deviations are less pronounced in actual qcd owing to the significant resonance widths and to a richer resonance structure . the vacuum current correlator also turns out to be especially robust ; even neglecting all ope corrections except the leading partonic contribution leads to excellent agreement with the hadronic result . turning to the direct phenomenological conclusions that can be inferred from our studies , we see that , to the extent our findings can be transferred to real qcd , violation of local duality in the total semileptonic widths of @xmath217 mesons is not an issue . the scale of duality violation lies far below the phenomenologically accessible limits , and can not affect the credibility of @xmath1 or @xmath2 extractions . in reality there are , of course , essential conceptual differences between the two theories , including those aspects that are expected to be essential for local duality ( for a discussion , see ref . although many seem to optimistically suggest that duality violation is more pronounced in the t hooft model than for actual heavy flavor hadrons , some differences may still work in the opposite direction . in @xmath8 there are no dynamical gluons , nor a chromomagnetic field that in @xmath7 provides a significant scale of nonperturbative effects in heavy flavor hadrons . likewise , there is no spin in @xmath8 , and no corresponding @xmath18-wave excitations of the light degrees of freedom ( the so - called @xmath295 states ) , which seem to play an important role in @xmath7 . two - dimensional qcd neither has long perturbative `` tails '' of actual strong interactions suppressed weakly ( by only powers of @xmath296s of the energy scale ) . in @xmath8 the perturbative corrections are generally power - suppressed , as follows from the dimension of the gauge coupling . as discussed in ref . @xcite , it is conceivable that the characteristic mass scale for freezing out the transverse gluonic degrees of freedom is higher than in the `` valence '' quark channels . this would imply a possibly higher scale for onset of duality in @xmath297 corrections to various observables . regardless of these differences , we conclude that presence of resonance structure per se is not an obstacle for fine local quark - hadron duality tested in the context of the ope . as we see in the t hooft model , resonances themselves do not seem to demand a larger duality interval . as soon as the mass scale of the states saturating the sum rules in a particular channel ( quark or hybrid ) has been passed , the decay width can be well approximated numerically by the expansion stemming from the ope . the ground states of heavy mesons in the t hooft model exhibit relatively small expectation values of nonperturbative operators ( @xmath20 , @xmath19 , but not @xmath66 ) compared to real qcd , if our identification @xmath298 is adopted . this may be regarded as a reason for small duality violation for @xmath299 in the model . however , even if we scale @xmath34 up to @xmath300@xmath301 to make up for smallness of the nonperturbative ope effects , the duality violation is still very small , and superficially rather insignificant even in charm . we note , however , that the specific choice eq . ( [ 120 ] ) of the weak interaction effectively requires decays to occur only at @xmath178 , and therefore the effects of four - fermion operators of the type @xmath302 are totally absent , at least in the lowest orders of perturbation theory ( cf.ref . @xcite , sec . iii.b.3 ) . as was suggested in ref . @xcite , it is conceivable that the apparent excess in @xmath303 is simply related to a noticeable magnitude of the non - valence ( nonfactorizable ) expectation values @xmath304 . if this conjecture is true , similar effects in @xmath305 are still suppressed but possibly detectable in future precision experiments . in the context of the present study , it suffices to say that this would be a legitimate ope effect rather than a manifestation of a significant local duality violation in the strict sense . + * acknowledgments : * r.l . thanks the department of energy for support under contract no.de-ac05-84er40150 ; n.u . acknowledges the support of the nsf under grant number phy96 - 05080 , by nato under the reference pst.clg 974745 , and by rffi under grant no . 99 - 02 - 18355 . we are grateful to n. isgur for inspiring interest and discussions , and to m. burkardt for invaluable insights . n.u . also thanks i. bigi , m. shifman and a. vainshtein for encouraging interest and collaboration on related issues , and a. zhitnitsky for useful comments . n.u . enjoyed the hospitality of physics department of the technion and the support of the lady davis grant during completion of this paper . the brower - spence - weis @xcite ( bsw ) improvement of the multhopp technique avoids the need for evaluating the wavefunction at a discrete set of points called `` multhopp angles , '' thus improving the behavior of the solutions in the endpoint regions , as described in sec . [ thm ] . here we exhibit the expressions used by bsw , correcting along the way some minor typographical errors in their work . starting with the t hooft equation ( [ the ] ) with bare quark masses @xmath28 and @xmath57 , one converts the kinematic variables @xmath306 to angular variables : @xmath307 in terms of which the t hooft equation reads @xmath308 \varphi_p ( \theta ) \nonumber \\ & & + \int_0^\pi { \rm d}\theta^\prime \ , \varphi_p ( \theta^\prime ) \ , { \rm p } \frac{1}{(\cos { \theta } \!-\ ! \cos { \theta^\prime})^2 } .\end{aligned}\ ] ] expanding @xmath309 and using the continuous inversion identity ( contrast with eq . ( a7 ) of ref . @xcite ) @xmath310 one obtains the infinite - dimensional eigenvector system @xmath311 where @xmath312\ , \sin { n\theta } \,\sin { m\theta } , \\ v_{nm } & = & -\frac{4}{\pi } \beta^2 \int_0^\pi { \rm d}\theta \ , \sin { n\theta}\ , \int_0^\pi { \rm d}\theta^\prime \,\sin{\theta^\prime } \sin { m\theta^\prime } \ , { \rm p } \frac{1}{(\cos { \theta } \!-\ ! \cos{\theta^\prime})^2 } \;. \label{vmn1}\end{aligned}\ ] ] both of these integrals can be evaluated , with the result @xmath313 , \\ v_{nm } & = & v_{n-1,m-1 } \left ( \frac{m}{m\!-\!1 } \right ) + \frac{8m}{n\!+\!m\!-\!1 } \left [ \frac{1+(-1)^{n+m}}{2 } \right ] , \end{aligned}\ ] ] where @xmath314 for @xmath315 . this recursive form for @xmath316 is most convenient for numerical calculations ; however , one may also write the closed - form solution , @xmath317 \left [ \psi \left ( \frac{1\!-\!n\!-\!m^{\vphantom{2}}}{2 } \right ) - \psi \left ( \frac{1\!-\ ! @xmath318 in eq . ( [ vmn2 ] ) is real but not symmetric , owing to the extra @xmath319 in eq . ( [ vmn1 ] ) ; therefore , the `` hamiltonian '' @xmath320 is not hermitian , and the eigenvectors @xmath321 are not orthogonal . this is a direct result of converting the exact wavefunctions , which are eigenfunctions of a hermitian hamiltonian when written in terms of the variable @xmath32 ( and therefore orthogonal in @xmath32 ) , into orthogonal functions of the variable @xmath196 . this transformation is nonunitary because the number of modes used is not infinite ; therefore , the overlap of different eigenvector solutions should be small when a large number of modes are used . indeed , this turns out to be empirically true ; nevertheless , we take the further step of orthogonalizing the numerical eigenvector solutions recursively by means of the standard gram - schmidt procedure , i.e. , @xmath322 for @xmath323 modes , this typically changes expectation values by one part in @xmath324 . the expressions for these overlaps and other matrix elements in terms of the mode coefficients @xmath325 are presented in appendix [ ovlap ] . a number of useful overlaps and other integrals are straightforward to evaluate in terms of the mode coefficients , using the expressions ( [ exp ] ) . solving them amounts to evaluating a number of trigonometric integrals . such expressions are especially convenient since they permit a number of integrations that introduce no numerical uncertainties ( except due to machine precision ) beyond those of solving the original multhopp - bsw eigenvector equation ( [ mulbsw ] ) . in particular , denote the @xmath326th eigenstate wavefunction presented in eq . ( [ exp ] ) by @xmath327 and that for some other set of masses in the @xmath55th eigenstate by @xmath328 ; the latter wavefunction then has an expansion like ( [ exp ] ) with mode coefficients @xmath329 . truncating after @xmath15 modes , one then finds @xmath330 \frac{1}{\left [ 1 \!-\ ! ( m\!-\!n)^2 \right ] \left [ 1 \!-\ ! ( m\!+\!n)^2 \right ] } . \nonumber \\ & & \end{aligned}\ ] ] indeed , the normalization integral @xmath331 is just the case @xmath332 and @xmath333 , in agreement with eq . ( a9 ) of ref . @xcite . other useful expectation values include @xmath334 ^ 2 \nonumber \\ & = & -\sum_{m=1}^n m a_m^{(p ) } \sum_{n=1}^n n a_n^{(p ) } \left [ \frac{1 \!-\ ! ( -1)^{m+n}}{2 } \right ] \frac{1}{\left [ 4 \!-\ ! ( m\!-\!n)^2 \right ] \left [ 4 \!-\ ! ( m\!+\!n)^2 \right ] } , \nonumber \\ & & \end{aligned}\ ] ] @xmath335 ^ 2 \nonumber } \\ & = & -\frac 1 2 \sum_{m=1}^n m a_m^{(p ) } \sum_{n=1}^n n a_n^{(p ) } \left [ \frac{1 \!+\ ! ( -1)^{m+n}}{2 } \right ] \left [ 21 \!-\ ! 6(m^2\!+\!n^2 ) + ( m^2\!-\!n^2)^2 \right ] \nonumber \\ & & \times \left [ \left ( 1 \!-\ ! ( m\!-\!n)^2 \right ) \left ( 1 \!-\ ! ( m\!+\!n)^2 \right ) \left ( 9 \!-\ ! ( m\!-\!n)^2 \right ) \left ( 9 \!-\ ! ( m\!+\!n)^2 \right ) \right]^{-1 } .\end{aligned}\ ] ] note that the spread of the wavefunction may be computed about any convenient point in @xmath32 , viz . , @xmath336 so that the additive constants of @xmath337 above are irrelevant . also , @xmath338 ^ 2 \:=\ : \sum_{m=1}^n a_m^{(p ) } \,\sum_{n=1}^n a_n^{(p ) } \ , i_{mn } , \ ] ] where @xmath339 one also finds @xmath340 ^ 2 \,=\ , \sum_{m=1}^n \,a_m^{(p ) } \,\sum_{n=1}^n \,a_n^{(p ) } \ , j_{mn } \;,\ ] ] where , using the notation of eq . ( [ imn ] ) , one finds @xmath341 for @xmath342 even , and @xmath343 for @xmath342 odd . finally , the decay constant of the @xmath326th excitation [ cf.eqs . ( [ 112])([cn ] ) ] is given by @xmath344 the numerical calculation of large-@xmath13 matrix elements with acceptable accuracy relies on achieving a balance between competing effects . on one hand , multhopp solutions to the t hooft equation with @xmath345 tend to suffer degraded numerical accuracy since they are highly concentrated into the small kinematic region @xmath346 . as discussed in sec . [ thm ] , the endpoint regions @xmath347 and @xmath227 are where the multhopp solutions or more precisely , their derivatives tend to break down . this effect is compounded when @xmath348 , since lighter quark masses force sharper endpoint behavior in the wavefunction . although the bsw solution ameliorates this behavior , as @xmath13 is increased one eventually faces the problem of attempting to represent a function with only a very small region of support in @xmath32 by a finite number of modes with support over the full range @xmath349 $ ] . in practice , we gauge the errors committed through such `` lattice spacing '' effects by computing a given quantity with @xmath350 and noting the amount by which its value shifts if one uses instead @xmath351 , and as expected , such errors become substantial ( as much as a few percent ) by the time one reaches @xmath352 or @xmath353 . we adopt an intermediate strategy of employing certain exact relations that hold for the t hooft solutions . to determine the relevant static expectation values , we solve the finite-@xmath13 heavy hadron mass expansion for @xmath356 [ eq . ( [ 50 ] ) ] : @xmath357 neglecting the order term and using the relations [ eq . ( [ 52 ] ) ] @xmath358 , \label{c52}\ ] ] we thus arrive at an equation cubic in @xmath66 that depends on @xmath359 . we solve it at @xmath360 . the asymptotic value of the scaled decay constant @xmath361 must also be evaluated at a finite value of @xmath13 , thus including @xmath3-suppressed pieces . we account for them explicitly using the expansion @xcite [ the first of eqs . ( [ 58 ] ) ] @xmath362}{3m_q } \right ) f^{(n ) } + o\left(\frac{\beta^{5/2}}{m_q^2}\right)\;. \label{c58}\ ] ] we likewise solve this equation for @xmath143 at @xmath360 . turning to the analysis of the sv sum rules eqs . ( [ oscsum1])([oscsum4 ] ) in sec . 4 , we note that their rapid saturation demands an exceptionally high precision in evaluating both the oscillation strengths @xmath125 in the r.h.s . and the expectation values in the l.h.s . reaching such an accuracy through direct computation seems impossible . therefore , we use a number of identities to get meaningful results . first , we employ the expression for @xmath140 in terms of @xmath147 , @xmath363 , and the corresponding decay constants : @xmath364 then we make use of the fact that the discussed sum rules , being completeness sums , are exact when summation includes all excitations ( see ref . therefore , one has & = & _ = n+1^_k^2 , [ csv1 ] + & = & _ = n+1^(_-_k ) _ k^2 , + & = & _ = n+1^(_-_k)^2 _ k^2 , + & = & _ = n+1^(_-_k)^3 _ k^2 . the sums on the r.h.s . can be accurately evaluated since the higher contributions fall off in magnitude very quickly . in practice , we truncate the sum at @xmath365 . a similar approach was used to evaluate the duality - violating difference @xmath366 as a function of @xmath13 . we use the exact relation @xcite [ eqs . ( [ 124 ] ) , ( [ 130 ] ) , ( [ 141 ] ) ] @xmath367 and therefore , @xmath368 the summation runs over all final excited states kinematically _ forbidden _ in the decay . once again , the sum converges rapidly and is dominated by the lowest couple of states . n. uraltsev , preprint und - hep-98-big1 , in _ heavy flavour physics : a probe of nature s grand design _ , proceedings of the international school of physics `` enrico fermi , '' course cxxxvii , varenna , july 7 - 18 , 1997 , eds . i. bigi and l. moroni ( ios press , amsterdam , 1998 ) p. 329 [ hep - ph/9804275 ] . 0.56 & 1.21918 & 0.7300 & 0.280 & 0.017 & @xmath374 + 1.0 & 1.24633 & 0.9534 & 0.432 & 0.048 & @xmath375 + 3.0 & 1.28764 & 1.4210 & 0.791 & 0.211 & @xmath376 + 5.0 & 1.29904 & 1.6061 & 0.944 & 0.333 & @xmath377 + 7.0 & 1.30423 & 1.7503 & 1.029 & 0.417 & @xmath378 + 10.0 & 1.30820 & 1.7901 & 1.102 & 0.500 & @xmath379 + 15.0 & 1.31131 & 1.8633 & 1.166 & 0.582 & @xmath380 + 25.0 & 1.31375 & 1.9271 & 1.222 & 0.661 & @xmath381 + 35.0 & 1.31475 & 1.9560 & 1.248 & 0.700 & @xmath382 + 50.0 & 1.31545 & 1.9783 & 1.268 & 0.732 & @xmath383 + + 0.56 & 2.82831 & 0.0000 & 0.280 & 0.032 & @xmath384 + 1.0 & 2.77888 & 0.0922 & 0.476 & 0.091 & @xmath385 + 3.0 & 2.66569 & 0.4429 & 1.094 & 0.457 & @xmath386 + 5.0 & 2.61977 & 0.6427 & 1.437 & 0.775 & @xmath387 + 7.0 & 2.59522 & 0.7648 & 1.649 & 1.014 & @xmath388 + 10.0 & 2.57436 & 0.8775 & 1.848 & 1.267 & @xmath389 + 15.0 & 2.55644 & 0.9812 & 2.033 & 1.529 & @xmath390 + 25.0 & 2.54088 & 1.0765 & 2.205 & 1.797 & @xmath391 + 35.0 & 2.53382 & 1.1213 & 2.287 & 1.933 & @xmath392 + 50.0 & 2.52833 & 1.1566 & 2.351 & 2.046 & @xmath393

we address numerical aspects of local quark - hadron duality using the example of the exactly solvable t hooft model , two - dimensional qcd with @xmath0 . the primary focus of these studies is total semileptonic decay widths relevant for extracting @xmath1 and @xmath2 . we compare the exact channel - by - channel sum of exclusive modes to the corresponding rates obtained in the standard @xmath3 expansion arising from the operator product expansion . an impressive agreement sets in unexpectedly early , immediately after the threshold for the first hadronic excitation in the final state . yet even at higher energy release it is possible to discern the seeds of duality - violating oscillations . we find the `` small velocity '' sum rules to be exceptionally well saturated already by the first excited state . we also obtain a convincing degree of duality in the differential distributions and in an analogue of @xmath4 . finally , we discuss possible lessons for semileptonic decays of actual heavy quarks in qcd . pacs numbers : 12.38.aw , 11.10.kk , 13.20.-v 15.2 true cm 22.0 true cm 0 cm 0 cm 0.15 true in 0.4 true in 0.25 true in .7ex .7ex 255=255 by 60 255 by-60255 by jlab - thy-00 - 21 + und - hep-00-big05 richard f. lebed@xmath5 and nikolai g. uraltsev@xmath6 + ( june , 2000 )

You are an expert at summarizing long articles. Proceed to summarize the following text: coding for cooperative wireless relay networks has attracted considerable attention recently . distributed space time coding was proposed as a coding strategy to achieve full cooperative diversity in @xcite assuming that the signals from all the relay nodes arrive at the destination at the same time . but this assumption is not close to practicality since the relay nodes are geographically distributed . in @xcite , a transmission scheme based on orthogonal frequency division multiplexing ( ofdm ) at the relay nodes was proposed to combat the timing errors at the relays and a high rate space time code ( stc ) construction was also provided . however , the maximum likelihood ( ml ) decoding complexity for this scheme is prohibitively high especially for the case of large number of relays . several other works in the literature propose methods to combat the timing offsets but most of them are based on decode and forward at the relay node and moreover fail to address the decoding complexity issue . in @xcite , a simple transmission scheme to combat timing errors at the relay nodes was proposed . this scheme is particularly interesting because of its associated low ml decoding complexity . in this scheme , ofdm is implemented at the source node and time reversal / conjugation is performed at the relay nodes on the received ofdm symbols from the source node . the received signals at the destination after ofdm demodulation are shown to have the alamouti code structure and hence single symbol maximum likelihood ( ml ) decoding can be performed . however , the alamouti code is applicable only for the case of two relay nodes and for larger number of relays , the authors of @xcite propose to cluster the relay nodes and employ alamouti code in each cluster . but this clustering technique provides diversity order of only two and fails to exploit the full cooperative diversity equal to the number of relay nodes . the main contributions of this report are as follows . * the li - xia transmission scheme is extended to a more general transmission scheme that can achieve full asynchronous cooperative diversity for any number of relays . * the conditions on the stc structure that admit its application in the proposed transmission scheme are identified . the recently proposed full diversity four group decodable distributed stcs in @xcite for synchronous wireless relay networks are found to satisfy the required conditions for application in the proposed transmission scheme . * it is shown how differential encoding at the source node can be combined with the proposed transmission scheme to arrive at a transmission scheme that can achieve full asynchronous cooperative diversity in the absence of channel knowledge and in the absence of knowledge of the timing errors of the relay nodes . moreover , an existing class of four group decodable distributed differential stcs @xcite for synchronous relay networks with power of two number of relays is shown to be applicable in this setting as well . in section [ sec2 ] , the basic assumptions on the relay network model are given and the li - xia transmission scheme is briefly described . section [ sec3 ] describes the transmission scheme proposed in this report and also provides four group decodable codes for any number of relays . section [ sec4 ] briefly explains how differential encoding at the source node can be combined with the proposed transmission scheme and four group decodable distributed differential stcs applicable in this scenario are also proposed . simulation results and discussion on further work comprise sections [ sec5 ] and [ sec6 ] respectively . + + * notation : * vectors and matrices are denoted by lowercase and uppercase bold letters respectively . @xmath0 denotes an @xmath1 identity matrix and @xmath2 denotes an all zero matrix of appropriate size . for a set @xmath3 , the cardinality of @xmath3 is denoted by @xmath4 . a null set is denoted by @xmath5 . for a matrix , @xmath6 , @xmath7 and @xmath8 denote transposition , conjugation and conjugate transpose operations respectively . for a complex number , @xmath9 and @xmath10 denote its in - phase and quadrature - phase parts respectively . in this section , the basic relay network model assumptions are given and the li - xia transmission scheme in @xcite is briefly described . the transmission scheme in @xcite is based on the use of ofdm at the source node and the alamouti code implemented in a distributed fashion for a @xmath11 relay system . essentially , the transmission scheme in @xcite is applicable mainly for the case of @xmath11 relays but by forming clusters of two relay nodes , it can be extended to more number of relays at the cost of sacrificing diversity benefits . consider a network with one source node , one destination node and @xmath12 relay nodes @xmath13 . this is depicted in fig . [ fig_network ] . every node is assumed to have only a single antenna and is half duplex constrained . the channel gain between the source and the @xmath14-th relay @xmath15 and that between the @xmath16-th relay and the destination @xmath17 are assumed to be quasi - static , flat fading and modeled by independent and complex gaussian distributed with mean zero and unit variance . the transmission of information from the source node to the destination node takes place in two phases . in the first phase , the source broadcasts the information to the relay nodes using ofdm . the relay nodes receive the faded and noise corrupted ofdm symbols , process them and transmit them to the destination . the relay nodes are assumed to have perfect carrier synchronization . the overall relative timing error of the signals arrived at the destination node from the @xmath14-th relay node is denoted by @xmath18 . without loss of generality , it is assumed that @xmath19 , @xmath20 . the destination node is assumed to have the knowledge of all the channel fading gains @xmath21 and the relative timing errors @xmath22 . the source takes @xmath23 complex symbols @xmath24 and forms two blocks of data denoted by @xmath25^t , j=1,2 $ ] . the first block @xmath26 is modulated by @xmath27-point inverse discrete fourier transform ( idft ) and @xmath28 is modulated by @xmath27-point discrete fourier transform ( dft ) . then a cyclic prefix ( cp ) of length @xmath29 is added to each block , where @xmath29 is not less than the maximum of the overall relative timing errors of the signals arrived at the destination node from the relay nodes . the resulting two ofdm symbols denoted by @xmath30 and @xmath31 consisting of @xmath32 complex numbers are broadcasted to the two relays using a fraction @xmath33 of the total average @xmath34 consumed by the source and the relay nodes together . if the channel fade gains are assumed to be constant for @xmath35 ofdm symbol intervals , the received signals at the @xmath14-th relay during the @xmath16-th ofdm symbol duration is given by @xmath36 where , @xmath37 is the additive white gaussian noise at the @xmath14-th relay node during the @xmath16 the ofdm symbol duration . the two relay nodes then process and transmit the resulting signals as shown in table [ table_alamouti ] using a fraction @xmath38 of the total power @xmath34 . the notation @xmath39 denotes the time reversal operation , i.e. , @xmath40 . .alamouti code based transmission scheme [ cols="^,^,^",options="header " , ] this code is @xmath41 real symbol decodable and achieves full diversity for appropriately signal sets @xcite . example [ eg_5relay ] illustrates how the proposed transmission scheme can be extended to odd number of relays as well . in this section , it is shown how differential encoding can be combined with the proposed transmission scheme described in section [ sec3 ] and then the codes in @xcite are proposed for application in this setting . for the proposed transmission scheme in section [ sec3 ] , at the end of one transmission frame , we have in the @xmath42-th sub carrier @xmath43 . note that the channel matrix @xmath44 depends on @xmath45 . thus the destination node needs to have the knowledge of these values in order to perform ml decoding . now using differential encoding ideas which were proposed in @xcite for non - coherent communication in synchronous relay networks , we combine them with the proposed asynchronous transmission scheme . supposing the channel remains approximately constant for two transmission frames , then differential encoding can be done at the source node in each sub carrier @xmath46 as follows : @xmath47^t,~ \mathbf{s_k^t}=\frac{1}{a_t-1}\mathbf{c_t}\mathbf{s_k^{t-1 } } , \mathbf{c_t}\in\mathscr{c}\ ] ] where , @xmath48 denotes the vector of complex symbols transmitted by the source during the @xmath14-th transmission frame in the @xmath42-th sub carrier and @xmath49 is the codebook used by the source which consists of scaled unitary matrices such that @xmath50=1 $ ] . if for all @xmath51 , @xmath52 then we have : @xmath53 from which @xmath54 can be decoded as in each sub carrier @xmath46 . note that this decoder does not require the knowledge of @xmath55 at the destination . it turns out that the four group decodable distributed differential space time codes constructed in @xcite for synchronous relay networks with power of two number of relays meet all the requirements for use in the proposed transmission scheme as well . the following example illustrates this fact . let @xmath56 . the codebook at the source is given by + where @xmath57 , @xmath58 , @xmath59 , @xmath60 and . differential encoding is done at the source node for each sub carrier @xmath46 as follows : @xmath47^t,~ \mathbf{s_k^t}=\frac{1}{a_t-1}\mathbf{c_t}\mathbf{s_k^{t-1 } } , \mathbf{c_t}\in\mathscr{c}.\ ] ] once we get @xmath61 from the above equation , the @xmath27 length vectors @xmath62 can be obtained . then idft / dft is applied on these vectors as shown below and broadcasted to the relay nodes . @xmath63 , @xmath64 , @xmath65 and @xmath66 . the relay nodes process the received ofdm symbols as given in table [ table_4relay ] for which @xmath67 , @xmath68 , , @xmath69 $ ] and . it has been proved in @xcite that @xmath70 and for all @xmath51 . at the destination node , decoding for @xmath57 , @xmath58 , @xmath59 and @xmath71 can be done separately in every sub carrier due to the four group decodable structure of @xmath49 . relay system with and without channel knowledge , width=5 ] in this section , we study the error performance of the proposed codes using simulations . we take @xmath56 , @xmath72 and the length of cp as @xmath73 . the delay @xmath18 at each relay is chosen randomly between @xmath74 to @xmath75 with uniform distribution . two cases are considered for simulation : ( 1 ) with channel knowledge at the destination and ( 2 ) without channel knowledge at the destination . for the case of no channel information , differential encoding at the source as described in section [ sec4 ] is done using the distributed differential space time in @xcite . when channel knowledge is available at the destination , rotated qpsk is used as the signal set @xcite . the transmission rate for the both the schemes is @xmath76 bit per channel use ( bpcu ) if the rate loss due to cp is neglected . the error performance curves for both the cases is shown in fig . [ fig_simulation ] . it can be observed from fig . [ fig_simulation ] that the error performance of the no channel knowledge case performs approximately @xmath77 db worser than that with channel knowledge at the destination . this is due to the differential transmission / reception technique in part and also in part because of the change in signal set from rotated qpsk to some other signal set @xcite in order to comply with the requirement of scaled unitary codeword matrices . the change in signal set for the sake of scaled unitary codeword matrices results in a reduction of the coding gain . a general transmission scheme for arbitrary number of relays that can achieve full cooperative diversity in the presence of timing errors at the relay nodes was proposed . it was then pointed out that the four group decodable distributed space time codes in @xcite can be applied in the proposed transmission scheme for any number of relay nodes . finally it was shown how the proposed scheme can be combined with differential encoding at the source node to end up with a transmission scheme that is robust to timing errors and also does not require the knowledge of the channel fading gains as well as the timing errors at any of the nodes . for this differential scheme , it was pointed out that the four group decodable distributed differential space time codes in @xcite are applicable for power of two number of relays . a drawback of the proposed transmission scheme is that it requires a large coherence interval spanning over multiple ofdm symbol durations . moreover there is a rate loss due to the use of cp , but this loss can be made negligible by choosing a large enough @xmath27 . some of the interesting directions for further work are listed below : 1 . constructing single symbol decodable distributed space time codes for the proposed transmission scheme . 2 . the codes in @xcite are applicable only for power of two number of relay nodes . constructing four group decodable distributed differential space time codes for all even number of relay nodes that are applicable in asynchronous relay networks without channel knowledge is an important direction for further work . 3 . in this work , we have assumed that there are no frequency offsets at the relay nodes . extending this work to asynchronous relay networks with frequency offsets is an interesting direction for further work . this problem has been addressed in @xcite for the case of two relay nodes . the authors sincerely thank prof . xiang gen xia and prof . hamid jafarkhani for sending us preprints of their recent works @xcite . g. susinder rajan and b. sundar rajan , `` algebraic distributed space - time codes with low ml decoding complexity , '' proceedings of _ ieee international symposium on information theory _ , nice , france , june 24 - 29 , 2007 , pp . 1516 - 1520 . - , `` a non - orthogonal distributed space - time coded protocol , part - ii : code construction and dm - g tradeoff , '' proceedings of _ ieee information theory workshop _ , chengdu , china , october 22 - 26 , 2006 , pp.488 - 492 . kiran t. and b. sundar rajan , `` partially - coherent distributed space - time codes with differential encoder and decoder , '' _ ieee journal on selected areas in communications _ , vol . 25 , no . 2007 , pp . 426 - 433 . frdrique oggier , babak hassibi , " cyclic distributed space - time codes for wireless relay networks with no channel information , submitted for publication . available online http://www.systems.caltech.edu/~frederique/submitdstcnoncoh.pdf zheng li and x .- g . xia , `` an alamouti coded ofdm transmission for cooperative systems robust to both timing errors and frequency offsets , '' to appear in _ ieee transactions on wireless communications_. private communication .

recently li and xia have proposed a transmission scheme for wireless relay networks based on the alamouti space time code and orthogonal frequency division multiplexing to combat the effect of timing errors at the relay nodes . this transmission scheme is amazingly simple and achieves a diversity order of two for any number of relays . motivated by its simplicity , this scheme is extended to a more general transmission scheme that can achieve full cooperative diversity for any number of relays . the conditions on the distributed space time code ( dstc ) structure that admit its application in the proposed transmission scheme are identified and it is pointed out that the recently proposed full diversity four group decodable dstcs from precoded co - ordinate interleaved orthogonal designs and extended clifford algebras satisfy these conditions . it is then shown how differential encoding at the source can be combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full cooperative diversity in asynchronous wireless relay networks with no channel information and also no timing error knowledge at the destination node . finally , four group decodable distributed differential space time codes applicable in this new transmission scheme for power of two number of relays are also provided .

You are an expert at summarizing long articles. Proceed to summarize the following text: * .. * the geometry on cayley s surface and the geometry in the ambient space of cayley s surface has been investigated by many authors from various points of view . see , among others , @xcite , @xcite , @xcite , @xcite , and @xcite . in these papers the reader will also find a lot of further references . as a by - product of a recent publication @xcite , it turned out that the cayley surface ( in the real projective @xmath0-space ) carries a one - parameter family of twisted cubics which have mutually contact of order four . these curves belong to a well - known three - parameter family of twisted cubics @xmath1 on cayley s surface ; cf . formula ( [ eq : abc_proj ] ) below . all of them share a common point @xmath2 with a common tangent @xmath3 , and a common osculating plane @xmath4 , say . however , according to @xcite such a one - parameter family of twisted cubics with contact of order four should not exist : _ zwei kubiken dieser art , die einander in @xmath2 mindestens fnfpunktig berhren , sind identisch . _ the aim of the present communication is to give a complete description of the order of contact ( at @xmath2 ) for the twisted cubics mentioned above . in particular , it will be shown in theorem [ thm:1 ] that the twisted cubics with parameter @xmath5 play a distinguished role , a result that seems to be missing in the literature . furthermore , since the order of contact is not a self - dual notion , we also investigate the order of dual contact for twisted cubics @xmath1 . somewhat surprisingly , in the dual setting the parameters @xmath6 and @xmath7 are exceptional ; see theorem [ thm:2 ] . in section [ para:2.4 ] we show that certain results of theorem [ thm:1 ] have a natural interpretation in terms of the _ twofold isotropic geometry _ which is based on the absolute flag @xmath8 , and in terms of the _ isotropic geometry _ in the plane @xmath4 which is given by the flag @xmath9 . section [ para:3.2 ] is devoted to the interplay between theorem [ thm:1 ] and theorem [ thm:2 ] . * .. * the calculations which are presented in this paper are long but straightforward . hence a computer algebra system ( maple v ) was used in order to accomplish this otherwise tedious job . nevertheless , we tried to write down all major steps of the calculations in such a form that the reader may verify them without using a computer . * .. * throughout this paper we consider the three - dimensional real projective space @xmath10 . hence a point is of the form @xmath11 with @xmath12 being a non - zero vector in @xmath13 . we choose the plane @xmath4 with equation @xmath14 as _ plane at infinity _ , and we regard @xmath10 as a projectively closed affine space . for the basic concepts of projective differential geometry we refer to @xcite and @xcite . * .. * the following is taken from @xcite , although our notation will be slightly different . _ cayley s _ ( _ ruled cubic _ ) _ surface _ is , to within collineations of @xmath10 , the surface @xmath15 with equation @xmath16 the line @xmath17 is on @xmath15 . more precisely , it is a torsal generator of second order and a directrix for all other generators of @xmath15 . the point @xmath18 is the cuspidal point on @xmath3 . in figure [ abb1 ] a part of the surface @xmath15 is displayed in an affine neighbourhood of the point @xmath2 . in contrast to our general setting , @xmath19 plays the role of the plane at infinity in this illustration . on the surface @xmath15 there is a three - parameter family of cubic parabolas which can be described as follows : each triple @xmath20 with @xmath21 gives rise to a function @xmath22 if moreover @xmath23 then @xmath24 yields the mapping @xmath25 its image is a _ cubic parabola _ @xmath26 . all these cubic parabolas have the common point @xmath2 , the common tangent @xmath3 and the common osculating plane @xmath4 . we add in passing that for @xmath27 we have @xmath28 for all @xmath29 , whereas the points of the form @xmath30 comprise the affine part of a _ parabola _ , @xmath31 say , lying on @xmath15 . each curve @xmath1 ( @xmath21 ) is on the _ parabolic cylinder _ with equation @xmath32 the mapping @xmath33 is injective , since different triples @xmath34 yield different parabolic cylinders ( [ eq : zylinder ] ) . figure [ abb2 ] shows some generators of @xmath15 , and five cubic parabolas @xmath35 together with their corresponding parbolic cylinders , where @xmath36 ranges in @xmath37 and @xmath5 . 1.0 cm ( 4.0,5.18 ) ( 0.0 , 0.0 ) ( 1.75,4.5)@xmath3 ( 2.1,2.1)@xmath2 ( 3.75,4.5)@xmath15 [ abb1 ] figure [ abb1 ] . ( 4.0,5.18 ) ( 0.0 , 0.0 ) [ abb2 ] figure [ abb2 ] . * .. * our first goal is to describe the order of contact at @xmath2 of cubic parabolas given by ( [ eq : abc_proj ] ) . since twisted cubics with contact of order five are identical @xcite , we may assume without loss of generality that the curves are distinct , and that the order of contact is less or equal four . [ thm:1 ] distinct cubic parabolas @xmath1 and @xmath38 on cayley s ruled surface have 1 . second order contact at @xmath2 if , and only if , @xmath39 or @xmath40 ; 2 . third order contact at @xmath2 if , and only if , @xmath39 and @xmath41 , or @xmath42 ; 3 . fourth order contact at @xmath2 if , and only if , @xmath42 and @xmath41 . we proceed in two steps : \(i ) first , we consider the quadratic forms @xmath43 which determine a hyperbolic paraboloid and a quadratic cone , respectively . their intersection is the cubic parabola @xmath44 , given by @xmath45 and the line @xmath46 . the tangent planes of the two surfaces at @xmath2 are different . next , let @xmath47 be a lower triangular matrix , i.e. , @xmath48 for all @xmath49 . the collineation which is induced by such a matrix @xmath50 fixes the point @xmath2 , the line @xmath3 , and the plane @xmath4 ; it takes @xmath44 to a cubic parabola , say @xmath51 . in order to determine the order of contact of @xmath44 and @xmath51 we follow @xcite . as @xmath52 , so we expand for @xmath53 the functions @xmath54 in terms of powers of @xmath55 and obtain @xmath56 the remaining coefficients @xmath57 will not be needed . note that the matrix entry @xmath58 does not appear in ( [ eq : koeffizienten ] ) . \(ii ) we consider the collineation of @xmath10 which is induced by the regular matrix @xmath59 where @xmath20 and @xmath60 . obviously , it fixes the point @xmath2 and takes @xmath44 to @xmath1 , since @xmath61 the ( irrelevant ) scalar factor in the definition of @xmath62 enables us to avoid fractions in the matrix @xmath63 the order of contact at @xmath2 of the cubic parabolas @xmath1 and @xmath38 coincides with the order of contact at @xmath2 of @xmath44 and that cubic parabola which arises from @xmath44 under the action of the matrix @xmath64 this matrix takes over the role of the matrix @xmath50 from the first part of the proof . ( its entry in the south - west corner has a rather complicated form and will not be needed ) . therefore @xmath1 and @xmath38 have contact of order @xmath65 at @xmath2 if , and only if , in ( [ eq : expand ] ) the coefficients @xmath66 , @xmath67 , @xmath68 vanish for @xmath53 . by ( [ eq : koeffizienten ] ) , this leads for @xmath69 to the single condition @xmath70 which proves the assertion in ( a ) . by virtue of ( a ) , for @xmath71 there are two cases . if @xmath39 then @xmath72 vanishes and we obtain the condition @xmath73 whereas @xmath40 yields @xmath74 altogether this proves ( b ) . finally , for @xmath75 there again are two possibilities : if @xmath39 and @xmath41 then @xmath76 vanishes , whence we get @xmath77 note that here @xmath78 , since @xmath79 . on the other hand , if @xmath42 then the conditions read @xmath80 this completes the proof . alternatively , the preceding results could be derived from ( * ? ? ? * theorem 1 ) which describes contact of higher order between curves in @xmath81-dimensional real projective space . * .. * in the following pictures we adopt once more the same alternative point of view like in figure [ abb1 ] , i.e. , the plane with equation @xmath19 is at infinity . in figure [ abb3 ] two curves @xmath1 and @xmath38 are displayed . as @xmath82 and @xmath83 , they have contact of second order at @xmath2 . a family of curves @xmath35 with @xmath84 and @xmath5 is shown in figure [ abb4 ] . all of them have mutually contact of order four at @xmath2 . these curves are , with respect to the chosen affine chart ( @xmath85 ) , cubic hyperbolas for @xmath86 , a cubic parabola for @xmath87 , and cubic ellipses for @xmath88 ; the corresponding values of @xmath36 are written next to the images of the curves . see also figure [ abb2 ] for another picture of this family , although with different values for @xmath36 and @xmath14 as plane at infinity . 1.0 cm ( 4.0,5.59 ) ( 0.0,0.25 ) ( 1.5,4.9)@xmath3 ( 1.9,2.5)@xmath2 ( 3.0,5.0)@xmath15 ( 0.8,3.55)@xmath1 ( 3.05,2.45)@xmath38 [ abb3 ] figure [ abb3 ] . ( 4.0,5.59 ) ( 0.0 , 0.0 ) ( 2.15,5.35)@xmath3 ( 4.0,4.6)@xmath15 ( 0.2,3.05)@xmath89 ( 2.8,1.35)@xmath89 ( -0.2,4.2)@xmath90 ( 3.7,1.0)@xmath90 ( 3.5,-.15)@xmath91 ( 0.2,5.44)@xmath91 ( 0.8,5.40)@xmath92 ( 2.9,-.11)@xmath92 ( 4.05,3.9)@xmath93 ( 3.75,2.6)@xmath93 ( -0.05,1.68)@xmath93 ( 2.25,0.5)@xmath94 ( 2.4,0.7)(-3,4)0.3 ( 1.5,0.7)(3,4)0.4 ( 1.35,0.55)@xmath95 ( 3.5 , 5.2)@xmath94 ( 3.42,2.15)@xmath94 ( 2.15,1.1)@xmath93 ( 2.9 , 5.24)@xmath95 ( 3.2,1.8)@xmath95 ( 0.11,0.22)@xmath94 ( 0.71 , 0.18)@xmath95 ( 0.1,3.6)@xmath93 ( 0.365,4.28)(1,-1)0.4 ( 0.065,4.34)@xmath94 ( 0.5,4.55)(3,-4)0.4 ( 0.2,4.61)@xmath95 [ abb4 ] figure [ abb4 ] . * .. * [ para:2.4 ] it follows from theorem [ thm:1 ] that cubic parabolas @xmath1 with @xmath5 play a special role . in order to explain this from a geometric point of view we consider the _ tangent surface _ of a cubic parabola @xmath1 and , in particular , its intersection with the plane at infinity . it is well known that this is a conic @xmath96 together with the line @xmath3 . in fact , via the first derivative of the local parametrization @xmath97 of @xmath1 we see that @xmath98 is given by @xmath99 the plane at infinity carries in a natural way the structure of an _ isotropic _ ( or _ galileian _ ) _ plane _ with the absolute flag @xmath9 . each point @xmath100 can be identified with the point @xmath101 . in this way the standard basis of @xmath102 determines a unit length and a unit angle in the isotropic plane @xcite . from this point of view each @xmath96 is an _ isotropic circle_. by ( [ eq : p_lokal ] ) , its _ isotropic curvature _ @xcite equals @xmath103 ; this bound is attained for @xmath5 . it is well known that two isotropic circles @xmath96 and @xmath104 have second order contact at the point @xmath2 if , and only if , their isotropic curvatures are the same @xcite , i.e. for @xmath39 or for @xmath40 . from this observation one could also derive the assertion in theorem [ thm:1 ] ( a ) as follows : we introduce an auxiliary euclidean metric in a neighbourhood of @xmath2 , and we take into account that the ratio of the euclidean curvatures at @xmath2 of the curves @xmath1 and @xmath96 ( the curves @xmath38 and @xmath104 ) equals @xmath105 ; see @xcite for this theorem of e. beltrami . the flag @xmath8 turns @xmath10 into a _ twofold isotropic _ ( or _ flag _ ) _ space_. the definition of metric notions in this space is based upon the identification of @xmath106 with @xmath107 , and the canonical basis of @xmath108 ; see @xcite . by @xcite , each cubic parabola @xmath1 has the _ twofold isotropic conical curvature _ @xmath109 . hence the following characterization follows . among all cubic parabolas @xmath1 on the cayley surface @xmath15 , the cubic parabolas with @xmath5 are precisely those with maximal twofold isotropic conical curvature . yet another interpretation is as follows : the regular matrix @xmath110 yields a _ homothetic transformation _ of @xmath10 which maps the cubic parabola @xmath111 to @xmath112 , since @xmath113 as all points at infinity are invariant , the corresponding isotropic circles @xmath114 and @xmath115 coincide . this homothetic transformation is identical if , and only if , @xmath5 . the cayley surface @xmath15 admits a @xmath0-parameter collineation group ; see @xcite formula ( 9 ) . the action of this group on the family of all cubic parabolas @xmath1 is described in @xcite , formula ( 12 ) . ( in the last part of that formula some signs have been misprinted . the text there should read @xmath116 ) . by virtue of this action , our previous result on homothetic transformations can be generalized to other cubic parabolas on @xmath15 . * .. * the question remains how to distinguish between cubic parabolas @xmath1 and @xmath38 satisfying the first condition ( @xmath39 ) in theorem [ thm:1 ] ( a ) , and those which meet the second condition ( @xmath40 ) . a similar question arises for the two conditions in theorem [ thm:1 ] ( b ) . we shall see that such a distinction is possible if we consider the _ dual curves _ which are formed by the osculating planes ( i.e. cubic developables ) . recall that @xmath1 and @xmath38 have , by definition , _ dual contact of order @xmath65 _ at a common osculating plane @xmath117 , if their dual curves have contact of order @xmath65 at the `` point '' @xmath117 of the dual projective space . we shall identify the dual of @xmath13 with the vector space @xmath118 in the usual way ; so planes ( i.e. points of the dual projective space ) are given by non - zero _ row vectors_. thus , for example , a plane @xmath119 is tangent to the cayley surface ( [ eq : cayley ] ) if , and only if , @xmath120 we note that all these tangent planes comprise a cayley surface in the dual space . for each twisted cubic there exists a unique null polarity ( symplectic polarity ) which takes each point of the twisted cubic to its osculating plane . in particular , the null polarity of the cubic parabola @xmath44 is induced by the linear bijection @xmath121 1 . second order dual contact at @xmath4 if , and only if , @xmath39 ; 2 . third order dual contact at @xmath4 if , and only if , @xmath39 and @xmath41 , or @xmath122 ; 3 . fourth order dual contact at @xmath4 if , and only if , @xmath123 and @xmath41 . the matrix @xmath124 determines a duality of @xmath10 which maps the set of points of @xmath44 onto the set of osculating planes of @xmath1 . since the product of a duality and the inverse of a duality is a collineation , we obtain the following : the order of dual contact at @xmath4 of the given curves @xmath1 and @xmath38 coincides with the order of contact at @xmath2 of the cubic parabola @xmath44 and that cubic parabola which arises from @xmath44 under the collineation given by the matrix @xmath125 here @xmath126 denotes an entry that will not be needed . we now proceed as in the proof of theorem [ thm:1 ] . by substituting the entries of the matrix above into ( [ eq : koeffizienten ] ) , we read off necessary and sufficient conditions for dual contact of order @xmath65 at the plane @xmath4 of @xmath1 and @xmath38 . for @xmath69 we get the single condition @xmath127 which proves the assertion in ( a ) . by ( a ) , we let @xmath39 for the discussion of @xmath71 . then @xmath72 vanishes and we arrive at the condition @xmath128 from which ( b ) is immediate . finally , for @xmath75 we distinguish two cases : if @xmath39 and @xmath41 then @xmath76 vanishes and we are lead to the condition @xmath129 note that here @xmath78 , since @xmath79 . the proof of ( c ) will be finished by showing that the case @xmath122 does not occur . from the assumption @xmath122 follows the first condition @xmath130 now , letting @xmath41 , the second condition @xmath131 is obtained . however , both conditions can not be satisfied simultaneously , since the first condition and @xmath79 together imply that @xmath78 . * .. * by combining the results of theorem [ thm:1 ] and theorem [ thm:2 ] , it is an immediate task to decide whether or not two ( not necessarily distinct ) cubic parabolas @xmath1 and @xmath38 have contact at @xmath2 and at the same time dual contact at @xmath4 of prescribed orders . in particular , we infer that two cubic parabolas of this kind , with fourth order contact at @xmath2 and fourth order dual contact at @xmath4 , are identical . let us choose a _ fixed _ real number @xmath60 . we consider the local parametrization @xmath132 of @xmath15 ; its image is @xmath133 , i.e. the affine part of @xmath15 . for our fixed @xmath134 and @xmath135 the affine parts of the parabolic cylinders ( [ eq : zylinder ] ) form a partition of @xmath136 ; see figure [ abb2 ] . hence @xmath137 is injective so that through each point @xmath138 there passes a unique curve @xmath35 . consequently , we can define a mapping @xmath139 of @xmath133 into the dual projective space by @xmath140 the image of the affine part of the cayley surface @xmath15 under the mapping @xmath139 described in _ ( [ eq : sigma ] ) _ consists of tangent planes of a cayley surface for @xmath141 , and of tangent planes of a hyperbolic paraboloid for @xmath142 . as the null polarity of @xmath35 arises from the matrix @xmath143 so the @xmath139-image of a point @xmath144 is the plane which is described by the non - zero row vector @xmath145 in discussing @xmath146 there are two cases : \(i ) suppose that @xmath147 . then a duality of @xmath10 is determined by the regular matrix @xmath148 letting @xmath149 the transpose of @xmath150 is easily seen to equal the row vector in ( [ eq : polar ] ) . hence @xmath146 is part of a cayley surface in the dual space which in turn , by ( [ eq : dual_cayley ] ) , is the set of tangent planes of a cayley surface in @xmath10 . \(ii ) if @xmath151 then the row vector ( [ eq : polar ] ) simplifies to @xmath152 thus the set @xmath146 is part of the non - degenerate ruled quadric in the dual space with equation @xmath153 ( in terms of dual coordinates ) . in other words , @xmath154 consists of tangent planes of a hyperbolic paraboloid in @xmath10 . let us add the following remark . the linear fractional transformation @xmath155 is an involution such that our fixed @xmath141 goes over to @xmath156 , as defined in ( [ eq : betastrich ] ) , whereas @xmath157 . in particular , if @xmath7 then @xmath158 . this explains the relation between theorem [ thm:1 ] ( c ) and theorem [ thm:2 ] ( c ) . also the fixed values of @xmath159 are noteworthy : for @xmath160 the curves @xmath161 are _ asymptotic curves _ of @xmath15 , i.e. , the osculating plane of @xmath161 at each point @xmath162 is the tangent plane of @xmath15 at @xmath163 . this means that the planes of the set @xmath146 are tangent planes of @xmath15 rather than tangent planes of another cayley surface . for @xmath164 it is immediate form ( [ eq : n_allgemein ] ) that the matrix @xmath165 does not depend on the parameter @xmath166 , whence in this particular case the mapping @xmath139 is merely the restriction of a null polarity of @xmath10 to the affine part of the cayley surface @xmath15 . h. havlicek and k. list . a three - dimensional laguerre geometry and its visualization . in g. wei , editor , _ proceedings dresden symposium geometry : constructive & kinematic ( dsg.ck)_ , pages 122129 , institut fr geometrie , technische universitt dresden , dresden , 2003 .

cayley s ( ruled cubic ) surface carries a three - parameter family of twisted cubics . we describe the contact of higher order and the dual contact of higher order for these curves and show that there are three exceptional cases . _ 2000 mathematics subject classification : _ 53a20 , 53a25 , 53a40 . _ keywords : _ cayley surface , twisted cubic , contact of higher order , dual contact of higher order , twofold isotropic space . _ dedicated to gunter wei on the occasion of his 60th birthday , in friendship _

You are an expert at summarizing long articles. Proceed to summarize the following text: the prediction of the yield stress for electrorheological ( er ) fluids is the main concern in theoretical investigations of er fluids . early studies failed to derive the experimental yield stress data @xcite because these studies were almost based on a point - dipole approximation @xcite . the point - dipole approximation is routinely adopted in computer simulation because it is simple and easy to use . since many - body and multipolar interactions between particles have been neglected , the strength of er effects predicted by this model is of an order lower than the experimental results . hence , substantial effort has been made to sort out more accurate models . klingenberg and coworkers developed empirical force expression from numerical solution of laplace s equation @xcite . davis used the finite - element method @xcite . clercx and bossis developed a full multipolar treatment to account for multipolar polarizability of spheres up to 1,000 multipolar orders @xcite . yu and coworkers developed an integral equation method which avoids the match of complicated boundary conditions on each interface of the particles and is applicable to nonspherical particles and multimedia @xcite . although the above methods are accurate , they are relatively complicated to use in dynamic simulation of er fluids . alternative models have been developed to circumvent the problem : the coupled - dipole model @xcite and the dipole - induced - dipole model @xcite , which take care of mutual polarization effects when the particles approach and finally touch . the did model accounts for multipolar interactions partially and is simple to use in computer simulation of er fluids @xcite . as an illustration , we employed the did model to simulate the athermal aggregation of particles in er fluids both in uniaxial and rotating fields . we find that the aggregation time is significantly reduced . in the next section , we review the multiple image method and establish the dipole - induced dipole ( did ) model . in section iii , we apply the did model to the computer simulation of er fluids in a uniaxial field . in section iv , we extend the simulation to athermal aggregation in rotating fields . discussion on our results will be given . here we briefly review the multiple images method @xcite and extend the method slightly to handle different dielectric constants . consider a pair of dielectric spheres , of radii @xmath0 and @xmath1 , dielectric constants @xmath2 and @xmath3 respectively , separated by a distance @xmath4 . the spheres are embedded in a host medium of dielectric constant @xmath5 . upon the application of an electric field @xmath6 , the induced dipole moment inside the spheres are respectively given by ( si units ) : @xmath7where the dipolar factors @xmath8 are given by : @xmath9 from the multiple image method @xcite , the total dipole moment inside sphere @xmath0 is : @xmath10 , \label{trans - a - dielectric } \\ p_{al } & = & ( \sinh \alpha)^3 \sum_{n=1}^\infty \left [ { p_{a0 } b^3 ( 2\beta)^{n-1}(2\beta')^{n-1 } \over ( b\sinh n\alpha + a\sinh ( n-1)\alpha)^3 } + { p_{b0 } a^3 ( 2\beta)^{n}(2\beta')^{n-1 } \over ( r \sinh n\alpha)^3 } \right ] , \label{long - a - dielectric } \end{aligned}\]]where the subscripts @xmath11 ( @xmath12 ) denote a transverse ( longitudinal ) field , i.e. , the applied field is perpendicular ( parallel ) to the line joining the centers of the spheres . similar expressions for the total dipole moment inside sphere @xmath1 can be obtained by interchanging @xmath0 and @xmath1 , as well as @xmath13 and @xmath14 . the parameter @xmath15 satisfies : @xmath16 in ref.@xcite , we checked the validity of these expressions by comparing with the integral equation method . we showed that these expression are valid at high contrast . our improved expressions will be shown to be good at low contrast as well ( see below ) . the force between the spheres is given by @xcite : @xmath17 for monodisperse er fluids ( @xmath18 , @xmath19 and @xmath20 ) , klingenberg defined an empirical force expression @xcite : @xmath21being normalized to the point - dipole force @xmath22 , where @xmath23 and @xmath24 ( all tending to unity at large separations ) are three force functions being determined from the numerical solution of laplace s equation . the klingenberg s force functions can be shown to relate to our multiple image moments as follow ( here @xmath18 , @xmath19 and @xmath25 ) : @xmath26where @xmath27 and @xmath28 are the reduced multiple image moments of each sphere . we computed the numerical values of these force functions separately by the approximant of table i of the second reference of ref.@xcite and by eq.([klingen ] ) . in fig.1 , we plot the multiple image results and the klingenberg s empirical expressions . we show results for the perfectly conducting limit ( @xmath29 ) only . for convenience , we define the reduced separation @xmath30 . for reduced separation @xmath31 , simple analytic expressions were adopted by klingenberg . as evident from fig.1 , the agreement with the multiple image results is impressive at large reduced separation @xmath32 , for all three empirical force functions . however , significant deviations occur for @xmath33 , especially for @xmath34 . for @xmath35 , alternative empirical expressions were adopted by klingenberg . for @xmath36 , the agreement is impressive , although there are deviations for the other two functions . from the comparison , we would say that reasonable agreements have been obtained . thus , we are confident that the multiple image expressions give reliable results . the analytic multiple image results can be used to compare among the various models according to how many terms are retained in the multiple image expressions : ( a ) point - dipole ( pd ) model : @xmath37 term only , ( b ) dipole - induced - dipole ( did ) model : @xmath37 to @xmath38 terms only , and ( c ) multipole - induced - dipole ( mid ) model : @xmath37 to @xmath39 terms . in a previous work @xcite , we examine the case of different size but equal dielectric constant ( @xmath19 ) only . here we focus on the case @xmath18 and study the effect of different dielectric constants . in fig.2 , we plot the interparticle force in the longitudinal field case against the reduced separation @xmath40 between the spheres for ( a ) @xmath41 ( @xmath42 ) and ( b ) @xmath43 ( @xmath44 ) and various @xmath45 ratios . at low contrast , the did model almost coincides with the mid results . in contrast , the pd model exhibits significant deviations . it is evident that the did model generally gives better results than pd for all polydispersity . the multiple image expressions [ eqs.(3)(4 ) ] allows us to calculate the correction factor defined as the ratio between the did and pd forces : @xmath46where @xmath47 , @xmath48 and @xmath49 are the point - dipole forces for the transverse , longitudinal and @xmath50 cases respectively . these correction factors can be readily calculated in computer simulation of polydisperse er fluids . the results show that the did force deviates significantly from the pd force at high contrast when @xmath13 and @xmath14 approach unity . the dipole induced interaction will generally decrease ( increase ) the magnitude of the transverse ( longitudinal ) interparticle force with respect to the point - dipole limit . for simplicity , we consider the case of two equal spheres of radius @xmath0 , initially at rest and at a separation @xmath51 . an electric field is applied along the line joining the centers of the sphere . the equation of motion is given by : @xmath52where @xmath53 is the displacement of one sphere from the center of mass . the separation between the two spheres is thus @xmath54 and the initial condition is @xmath55 at @xmath56 . eq.(11 ) is a dimensionless equation . we have chosen the following natural scales to define the dimensionless variables : @xmath57 where @xmath58 is the field strength , @xmath59 is the masss , @xmath60 is the coefficient of viscosity . using typical parameters , we find @xmath61 is of the order milliseconds . we have followed klingenberg @xcite to ignore the inertial effect , captured by the parameter @xmath62 . @xmath63 the neglect of @xmath62 can be justified as follows . for values common to er suspension : @xmath64 pa s , @xmath65 kg , @xmath66 m @xmath67 , the inertial term @xmath62 is of the order @xmath68 . we also neglect the thermal motion of the particles which is a valid assumption at high fields . we should remark that the initial separation @xmath51 is related to the volume fraction @xmath69 , defined as the ratio of the volume of the sphere to that of the cube which contains the sphere @xcite , i.e. @xmath70 , and @xmath71 for the pd approximation , eq.(11 ) admits an analytic solution : @xmath72^{1/5}.\]]we integrate the equation of motion by the 4th order runge - kutta algorithm , with time steps @xmath73 and 0.001 for small and large volume fractions respectively . in fig.3(a ) , we plot the displacement @xmath74 versus time graph for the pd case and find excellent agreement between analytic and numerical results . for the did model , we have to integrate the equation of motion numerically . in fig.4 , we plot the displacement @xmath74 versus time graph for the aggregation of two spheres in uniaxial fields . at small volume fractions , i.e. , when the initial separation is large , the time for aggregation is large and the did results deviate slightly from the pd results . however , at large volume fractions , the did results are significantly smaller than the pd calculations . the effect becomes even more pronounced at large @xmath13 . in fig.5(a ) , we plot the ratio of aggregation time of the did to pd cases . the results showed clearly that the aggregation time has been significantly reduced when mutual polarization effects are considered . the reduction in aggregation time becomes even pronounced for small initial separations . recently , martin and coworkers @xcite demonstrated athermal aggregation with the rotating field . when a rotating field is applied in the @xmath75-@xmath76 plane at a sufficiently high frequency that particles do not move much in one period , an average attractive dipolar interaction is created . the result of this is the formation of plates in the @xmath75-@xmath76 plane . consider a rotating field applied in the @xmath75-@xmath76 plane : @xmath77 . the dimensionless equation of motion for the two sphere case becomes : @xmath78where @xmath79 is the displacement of one sphere from the center of mass . for large @xmath80 , we may safely neglect the @xmath76 component of the motion . in the pd approximation , @xmath81 and @xmath82 , we find the analytic result : @xmath83^{1/5}.\]]the separation between the two spheres is just @xmath84 , with the initial separation @xmath55 at @xmath56 . in the rotating field case , we also integrate the equation of motion by the 4th order runge - kutta algorithm , but with @xmath85 and @xmath86 as the time steps . note that @xmath87 should be the largest time step which can be used because we must at least go through a cycle consisting of the transverse and longitudinal field cases . the oscillating effect of a rotating field is less observable when the time step is smaller than this maximum value . in fig.3(b ) , we plot the displacement versus time graph for the pd case in rotating field and find an excellent agreement between analytic and numerical results . it is evident that the aggregation time is 4 times of that of the uniaxial field case . in fact , eq.(14 ) reduces to eq.(12 ) as @xmath88 . at large @xmath80 , eq.(14 ) becomes : @xmath89^{1/5}.\ ] ] that is , in the pd approximation , the time average force becomes 1/4 of that of the uniaxial field case . it is because the two dipole moments spend equal times in the transverse and longitudinal orientations , while @xmath90 in the pd case , leading to an overall attractive force that is 1/4 of the force of the uniaxial field case . when the multiple image force is included , we expect that the magnitude of @xmath91 increases while that of @xmath92 decreases and we expect an even larger attractive force when the spheres approach . in this case , the aggregation time must be reduced even more significantly . in fig.5(b ) , we plot the ratio of aggregation time of the did to pd cases for @xmath29 and several @xmath80 . the @xmath93 curve is just for the uniaxial field case . the results showed clearly that the aggregation time has been significantly reduced when mutual polarization effects are considered . the reduction in aggregation time becomes even pronounced for small initial separations . it is observed that fluctuations exist when the initial separation between the spheres is 2.4@xmath0 or less . it is because the motion is sensitive to the initial orientation of the dipoles when the spheres are too close . similarly , we consider the aggregations of 3 and 4 equal spheres , arranged in a chain , an equilateral triangle and a square . for a chain of 3 spheres in a rotating field , the central sphere does not move , while the two spheres at both ends move towards the central sphere . for 3 spheres in an equilateral triangle , the center of mass ( cm ) will not move while each sphere moves towards the cm , subject to the force of the other two spheres . the same situation occurs for 4 spheres in a square , in which each sphere moves towards the cm , subject to the force of the other 3 spheres . in the pd approximation , we report the analytic results as follows . for 3 spheres in a chain , @xmath94^{1/5}.\]]for 3 spheres in an equilateral triangle , @xmath95^{1/5}.\]]for 4 spheres in a square , @xmath96^{1/5}.\]]in each of the above cases , @xmath75 is the distance of one sphere from the center of mass . in the case of 3 spheres in a chain , the separation between the spheres is same as @xmath97 . in the case of 3 spheres in an equilateral triangle , the separation between spheres is @xmath98 . in the case of 4 spheres in a square , the separation between spheres is @xmath99 . again , we integrate the equation of motion by the 4th order runge - kutta algorithm . we find excellent agreements between the analytic and numerical results ( not shown here ) . it has been found that the displacement in the @xmath76-direction is about 0.5% for @xmath100 and a larger @xmath80 has been used in the simulation . on the other hand , it is time consuming for simulations with @xmath101 . it is evident from the displacement - time graph that the results are correct . in fig.6 , the oscillation amplitude is reduced when the rotating frequency increases in the simulation . this is consistent with the assumption made in our analytic expressions . in fig.7 , it is observed that fluctuations exist when the initial separation between the spheres is 2.4@xmath0 or less in all three graphs . again , it is because the motion is sensitive to the orientation of the dipoles when the spheres are close . from the simulation , the reduction effects become even pronounced for the rotating electric field case than the uniaxial field case . here a few comments on our results are in order . in this work , we studied the aggregation time for several particles . we should also examine the morphology of aggregation , due to multiple image forces . in this connection , we can also examine the structural transformation by applying the uniaxial and rotating fields simultaneously @xcite . we have done simulation in the monodisperse case . real er fluids must be polydisperse in nature : the suspending particles can have various sizes or different permittivities . polydisperse electrorheological ( er ) fluids have attracted considerable interest recently because the size distribution and dielectric properties of the suspending particles can have significant impact on the er response @xcite . we should extend the simulation to polydisperse case by using the did model . in summary , we have used the multiple image to compute the interparticle force for a polydisperse electrorheological fluid . we apply the formalism to a pair of spheres of different dielectric constants and calculate the force as a function of the separation . the results show that the point - dipole approximation is oversimplified . it errs considerably because many - body and multipolar interactions are ignored . the dipole - induced - dipole model accounts for multipolar interactions partially and yields overall satisfactory results in computer simulation of er fluids while it is easy to use . this work was supported by the research grants council of the hong kong sar government under grant cuhk4284/00p . p. m. adriani and a. p. gast , phys . fluids * 31 * , 2757 ( 1988 ) . d. j. klingenberg , f. van swol and c. f. zukoski , j. chem . phys . * 91 * , 7888 ( 1989 ) . d. j. klingenberg , f. van swol and c. f. zukoski , j. chem . phys . * 94 * , 6160 ( 1991 ) . d. j. klingenberg and c. f. zukoski , langmuir * 6 * 15 ( 1990 ) ; d. j. klingenberg , f. van swol and c. f. zukoski , j. chem . phys . * 94 * , 6170 ( 1991 ) . l. c. davis , appl . . lett . * 60 * , 319 ( 1992 ) . h. j. h. clercx and g. bossis , phys . e * 48 * , 2721 ( 1993 ) . k. w. yu , hong sun and jones t. k. wan , in _ proceedings of the 5th international conference on electrical transport and optical properties of inhomogeneous media _ , physica b * 279 * , 78 ( 2000 ) . z. w. wang , z. f. lin and r. b. tao , int . phys . * 10 * , 1153 ( 1996 ) . k. w. yu and jones t. k. wan , in _ proceedings of the 9th international conference on discrete simulation of fluid dynamics _ , . comm . * 129 * , 177 ( 2000 ) . j. d. jackson , _ classical electrodynamics _ ( wiley , new york 1975 ) . j. e. martin , r. a. anderson and c. p. tigges , j. chem . phys . * 108 * , 3765 ( 1998 ) ; * 108 * , 7887 ( 1998 ) . c. k. lo and k. w. yu , phys . e , to be published . m. ota and t. miyamoto , j. appl . 76 * , 5528 ( 1994 ) .

we have employed the multiple image method to compute the interparticle force for a polydisperse electrorheological ( er ) fluid in which the suspended particles can have various sizes and different permittivites . the point - dipole ( pd ) approximation being routinely adopted in computer simulation of er fluids is shown to err considerably when the particles approach and finally touch due to multipolar interactions . the pd approximation becomes even worse when the dielectric contrast between the particles and the host medium is large . from the results , we show that the dipole - induced - dipole ( did ) model yields very good agreements with the multiple image results for a wide range of dielectric contrasts and polydispersity . as an illustration , we have employed the did model to simulate the athermal aggregation of particles in er fluids both in uniaxial and rotating fields . we find that the aggregation time is significantly reduced . the did model accounts for multipolar interaction partially and is simple to use in computer simulation of er fluids .

You are an expert at summarizing long articles. Proceed to summarize the following text: su(2 ) einstein - yang - mills - higgs ( eymh ) theory , with the higgs field in the adjoint representation , possesses globally regular gravitating magnetic monopole solutions and corresponding magnetically charged black hole solutions @xcite . for small gravitational constant , the gravitating fundamental monopole solution smoothly emerges from the corresponding flat space solution , the t hooft - polyakov monopole @xcite . with increasing gravitational constant , the mass of the gravitating fundamental monopole solution decreases , and it ceases to exist beyond a maximal value of the gravitational constant . the corresponding magnetically charged eymh black hole solutions represent counterexamples to the `` no - hair '' conjecture . distinct from embedded reissner - nordstrm ( rn ) black holes with unit magnetic charge , they emerge from the globally regular magnetic monopole solutions when a finite regular event horizon is imposed . consequently , they have been characterized as `` black holes within magnetic monopoles '' @xcite . besides the fundamental monopole solution there are excited monopole solutions @xcite . the gauge field function of the @xmath0-th excited monopole solution possesses @xmath0 nodes , whereas the gauge field function of the fundamental monopole solution decreases monotonically to zero @xcite . the excited monopole solutions are related to the globally regular einstein - yang - mills ( eym ) solutions , found by bartnik and mckinnon @xcite , and , like these solutions , have no flat space counterparts . in flat space also dyon solutions exist , carrying both electric and magnetic charge @xcite . here we show that , like the monopole solutions , these dyon solutions persist in the presence of gravity , up to some maximal value of the gravitational constant . beside the fundamental gravitating dyon solutions , we also construct excited gravitating dyon solutions and dyonic black hole solutions . distinct from the embedded rn solutions with the same electric and magnetic charge , these `` black holes within dyons '' again represent counterexamples to the `` no - hair '' conjecture . in contrast , pure su(2 ) eym theory possesses neither regular dyon solutions @xcite , nor dyonic black holes other than embedded rn solutions @xcite . we consider the su(2 ) eymh action @xmath1 with @xmath2 and @xmath3 where @xmath4 @xmath5 @xmath6 is the gauge coupling constant , @xmath7 is the higgs coupling constant and @xmath8 is the higgs field vacuum expectation value . variation of the action eq . ( [ action ] ) with respect to the metric @xmath9 , the gauge field @xmath10 and the higgs field @xmath11 leads to the einstein equations and the matter field equations . to construct static spherically symmetric globally regular and black hole solutions we employ schwarzschild - like coordinates and adopt the spherically symmetric metric @xmath12 with @xmath13 for the gauge and higgs field we employ the spherically symmetric ansatz @xcite @xmath14 @xmath15 and @xmath16 with unit vectors @xmath17 , @xmath18 and @xmath19 . for @xmath20 , gravitating monopole solutions are obtained @xcite . we now introduce the dimensionless coordinate @xmath21 and the dimensionless mass function @xmath22 , @xmath23 as well as the coupling constants @xmath24 and @xmath25 , @xmath26 the @xmath27 and @xmath28 components of the einstein equations then yield the equations for the metric functions , @xmath29 and @xmath30 where the prime indicates the derivative with respect to @xmath21 . for the matter functions we obtain the equations @xmath31 @xmath32 and @xmath33 a special solution of these equations is the embedded rn solution with mass @xmath34 , unit magnetic charge and arbitrary electric charge @xmath35 , @xmath36 @xmath37 the corresponding extremal rn solution has horizon @xmath38 , @xmath39 let us first consider the globally regular particle - like solutions of the su(2 ) eymh system . requiring asymptotically flat solutions implies that the metric functions @xmath40 and @xmath22 both approach a constant at infinity . we here adopt @xmath41 and @xmath42 represents the dimensionless mass of the solutions . the matter functions also approach constants asymptotically , @xmath43 where for magnetic monopole solutions @xmath44 . the asymptotic fall - off of the function @xmath45 determines the dimensionless electric charge @xmath35 ( see eq . ( [ q ] ) ) . regularity of the solutions at the origin requires @xmath46 and @xcite @xmath47 the globally regular dyon solutions have many features in common with the globally regular monopole solutions . in the prasad - sommerfield limit , @xmath48 , the dyon solutions in flat space are known analytically @xcite , whereas for finite @xmath25 they are obtained numerically @xcite . in the presence of gravity , the corresponding gravitating dyon solutions extend up to a maximal value of the coupling constant @xmath24 . beyond this value no dyon solutions exist . for small values of @xmath25 , the fundamental dyon branch does not end at the maximal value @xmath49 . instead it exhibits a small spike there and bends backwards , up to the critical coupling constant @xmath50 . since variation of the coupling constant @xmath51 can be considered in two different ways , either as changing @xmath52 and keeping @xmath8 fixed , or vice versa , for small @xmath25 the fundamental dyon branch can be interpreted as obtained by first varying @xmath52 up to the maximal value @xmath49 , and then varying @xmath8 up to the critical value @xmath50 . at the critical value @xmath50 the fundamental dyon branch reaches a limiting solution and bifurcates with the branch of extremal rn solutions of unit magnetic charge and electric charge @xmath35 . the fundamental dyon branch is thus completely analogous to the fundamental monopole branch @xcite . this is demonstrated for the normalized mass @xmath53 of the dyon solutions with electric charge @xmath54 and @xmath48 in fig . 1 , where for comparison also the monopole solutions ( @xmath55 , @xmath48 ) are shown , together with the extremal rn solutions of unit magnetic charge and electric charge @xmath54 as well as @xmath55 . the normalization in fig . 1 is chosen to obtain the finite mass of the flat space solutions in the limit @xmath56 . the adm mass @xmath57 can be read off the figure . for the dyon branch we find a critical coupling of @xmath58 , while for the monopole branch it is @xmath59 @xcite . along the fundamental branch the dyon functions approach limiting functions , when @xmath60 . as for the monopole solutions , the metric function @xmath61 of the dyon solutions develops a minimum , which decreases monotonically along the fundamental branch . in the limit @xmath60 , the minimum approaches zero at @xmath62 . the limiting metric function then consists of an inner part , @xmath63 , and an outer part , @xmath64 . for @xmath64 , the limiting metric function corresponds to the metric function @xmath61 of the extremal rn black hole with @xmath50 , unit magnetic charge and electric charge @xmath35 . likewise the other functions approach limiting functions , when @xmath60 , which for @xmath64 correspond to those of the extremal rn black hole with @xmath50 , unit magnetic charge and electric charge @xmath35 . the limit @xmath60 is demonstrated in fig . 2 for the matter function @xmath45 of the dyon solution with electric charge @xmath54 and @xmath48 . in the figure the occurrence of the spike is seen , since the function @xmath45 does not reach the limiting function at the maximal value @xmath49 , but instead at the critical value @xmath50 . the limiting function is identically zero for @xmath63 and coincides with the rn function for @xmath64 . the limiting behaviour of the other functions of the dyon solutions is analogous to those of the monopole solutions , shown in @xcite . beside the branch of fundamental dyon solutions there are branches of excited dyon solutions . the gauge field function @xmath65 of the @xmath0-th excited dyon solution has @xmath0 nodes , whereas the gauge field function of the fundamental dyon solution decreases monotonically to zero . the excited dyon solutions also exist only below some maximal value of the coupling constant @xmath24 . since these excited solutions have no flat space counterparts , the variation of @xmath24 along a branch of excited solutions must be interpreted as a variation of @xmath8 while @xmath52 is kept fixed . in the limit @xmath56 the higgs field vacuum expectation value therefore vanishes , while @xmath52 remains finite . because of the particular choice of dimensionless variables ( [ xm ] ) , in this limit the solutions shrink to zero size and their mass @xmath22 diverges . the coordinate transformation @xmath66 @xcite leads to finite limiting solutions in the limit @xmath56 . in fact , the transformed excited monopole solutions approach the bartnik - mckinnon solutions @xcite , and so do the transformed excited dyon solutions . this is seen in fig . 3 , where we show the appropriately normalized mass @xmath67 as a function of @xmath24 for the first excited dyon branch with electric charge @xmath54 and @xmath48 . for comparison also the first excited monopole branch is shown . further details will be given elsewhere @xcite . we now turn to the dyonic black hole solutions of the su(2 ) eymh system . imposing again the condition of asymptotic flatness , the black hole solutions satisfy the same boundary conditions at infinity as the regular solutions . the existence of a regular event horizon at @xmath38 requires @xmath68 and @xmath69 , and the matter functions must satisfy @xmath70 @xmath71 and @xmath72 again , the su(2 ) eymh `` black holes within dyons '' have many features in common with the `` black holes within monopoles '' . in particular , for a given coupling constant @xmath24 , the black hole solutions corresponding to the fundamental dyon branch emerge from the globally regular solution in the limit @xmath73 and persist up to a critical maximal value of the horizon radius . in fig . 4 we exhibit the mass of `` black holes within dyons '' with electric charge @xmath54 and @xmath48 as a function of the horizon radius for several values of @xmath24 , together with the corresponding branches of rn solutions of unit magnetic charge and electric charge @xmath54 . for smaller values of @xmath24 , the black hole solutions merge into non - extremal rn solutions at a critical value of the horizon radius . for larger values of @xmath24 , the black hole solutions show a critical behaviour analogous to the globally regular solutions , and bifurcate with an extremal rn solution . here , with increasing horizon radius the limiting solution is reached at a critical value of the horizon radius smaller than the horizon radius of the extremal rn black hole with the same @xmath24 , unit magnetic charge and electric charge @xmath35 . this critical behaviour is demonstrated for the function @xmath45 of the dyon black hole solutions with electric charge @xmath54 and coupling constants @xmath74 and @xmath48 in fig . further details will be given elsewhere @xcite . the globally regular dyon solutions have many features in common with the globally regular monopole solutions . like the fundamental monopole branch , the fundamental dyon branch starts from the corresponding flat space solution and extends up to a critical value of the coupling constant @xmath24 . at the critical value , both the fundamental monopole branch and the fundamental dyon branch bifurcate with the corresponding branch of extremal rn solutions . the critical coupling constant depends slightly on the electric charge @xmath35 of the dyons . likewise , beside the fundamental dyon branch there are branches of excited dyon solutions , which extend up to a maximal value of @xmath24 . in the limit @xmath56 , the branches of excited dyon solutions tend to the corresponding bartnik - mckinnon solutions , like their monopole counterparts . starting from the stable monopole solutions in flat space , the solutions on the fundamental monopole branch remain stable for @xmath75 @xcite . in contrast , the classical dyon solution in flat space is unstable , since the mass can be lowered continuously , by lowering the electric charge , as long as there is no charge quantization @xcite . this indicates , that the gravitating fundamental dyon solutions are also unstable . in analogy to `` black holes within monopoles '' also `` black holes within dyons '' exist . for a given value of the coupling constant @xmath24 , the `` black holes within dyons '' emerge from the globally regular solutions in the limit @xmath73 and persist up to a maximal value of the horizon radius . for small @xmath24 the black hole solutions merge into the corresponding non - extremal rn solutions at a critical value of the horizon radius , whereas for larger @xmath24 ( but @xmath76 ) the black hole solutions bifurcate at a critical value of the horizon radius with the corresponding extremal rn solutions . the static spherically symmetric `` black holes within monopoles '' and `` black holes within dyons '' provide counterexamples to the `` no - hair conjecture '' . eymh theory also possesses counterexamples of a different type , namely static aspherical black holes , which represent `` black holes within multimonopoles '' @xcite . beside static axially symmetric black holes , whose eym counterparts have recently been obtained non perturbatively @xcite , there are static black holes with only discrete symmetries @xcite . it presents a challenge to construct such black holes with only crystal symmetries non - perturbatively . 000 k. lee , v.p . nair and e.j . weinberg , black holes in magnetic monopoles , phys . d45 ( 1992 ) 2751 . p. breitenlohner , p. forgacs and d. maison , gravitating monopole solutions , nucl . b383 ( 1992 ) 357 ; + gravitating monopole solutions ii , nucl . b442 ( 1995 ) 126 . aichelburg and p. bizon , magnetically charged black holes and their stability , phys . d48 ( 1993 ) 607 . g. t hooft , nucl . b79 ( 1974 ) 276 ; + a.m. polyakov , jetp lett . 20 ( 1974 ) 194 . r. bartnik , and j. mckinnon , particlelike solutions of the einstein - yang - mills equations , phys . 61 ( 1988 ) 141 . poles with both magnetic and electric charges in non - abelian gauge theory , phys . d11 ( 1975 ) 2227 ; + m.k . prasad and c.m . sommerfield , exact classical solution for the t hooft monopole and the julia - zee dyon , phys . ( 1975 ) 760 . ershov and d.v . galtsov , phys . lett . 150a ( 1990 ) 159 . galtsov and a.a . ershov , non - abelian baldness of colored black holes , phys . lett . a138 ( 1989 ) 160 . p. bizon and o.t . popp , no - hair theorem for spherical monopoles and dyons in su(2 ) einstein - yang - mills theory , class . quantum grav . 9 ( 1992 ) 193 . y. brihaye , b. kleihaus and d.h . tchrakian , dyon - skyrmion lumps , preprint hep - th/9805059 . b. hartmann , j. kunz and y. brihaye , in preparation . volkov , and d.v . galtsov , black holes in einstein - yang - mills theory , sov . ( 1990 ) 747 ; + p. bizon , colored black holes , phys . 64 ( 1990 ) 2844 ; + h. p. knzle and a. k. m. masoud - ul - alam , spherically symmetric static su(2 ) einstein - yang - mills fields , j. math . ( 1990 ) 928 . h. hollmann , on the stability of gravitating nonabelian monopoles , phys . b338 ( 1994 ) 181 . s. a. ridgway and e. j. weinberg , static black hole solutions without rotational symmetry , phys . d52 ( 1995 ) 3440 . b. kleihaus and j. kunz , static black hole solutions with axial symmetry , phys . 79 ( 1997 ) 1595 ; static axially symmetric einstein - yang - mills - dilaton solutions : ii . black hole solutions , phys . d57 ( 1998 ) 6138 .

we study static spherically symmetric gravitating dyon solutions and dyonic black holes in einstein - yang - mills - higgs theory . the gravitating dyon solutions share many features with the gravitating monopole solutions . in particular , gravitating dyon solutions and dyonic black holes exist only up to a maximal coupling constant , and beside the fundamental dyon solutions there are excited dyon solutions . = 6.125truein = 8.125truein preprint hep - th/9807169

You are an expert at summarizing long articles. Proceed to summarize the following text: the next generation of neutrino oscillation experiments has the potential to not only determine the remaining unknowns in the pmns matrix but also to measure its parameters with unprecedented precision . this will mark the beginning of a period of high - precision neutrino physics , where the standard paradigms describing the neutrino sector will be put to proof and theoretical ideas about the origins of neutrino mass and leptonic flavour can be confronted with data . one of the more popular beyond the standard model ideas applied to the neutrino sector is the introduction of a discrete flavour symmetry . models based on this principle have been shown to be able to derive the observed structure of the pmns matrix from a small set of assumptions . these models generally propose a discrete symmetry ( _ e.g. _ @xmath2 or @xmath3 ) which is broken spontaneously , leaving residual symmetries amongst the leptonic mass terms . these symmetries reduce the degrees of freedom amongst the mixing parameters , generating a pattern of falsifiable predictions . by hypothesizing which symmetries of the leptonic mass terms are residual , this idea can be used to reconstruct the flavour group independently of many model specific assumptions . in ref . @xcite , we have shown that a quite general construction of this type ( first presented in ref . @xcite ) leads to only @xmath4 viable models in light of the current global oscillation data . these models fix a column of the pmns matrix which , under the assumption of unitarity , can be expressed in terms of two constraints on the pmns parameters : the _ atmospheric sum rule _ , relating @xmath5 to @xmath6 and @xmath0 , and the _ solar prediction _ , an expression for @xmath7 in terms of @xmath6 alone . in this contribution , we shall discuss the parameter correlations of these models , and how they can be constrained by the next generation of high - precision oscillation experiments . we shall employ the notation @xmath8 , @xmath9 and @xmath10 @xcite throughout . .[tab : solar_pred]the solar predictions for the @xmath4 viable models identified in ref . the model label denotes the flavour group and the pattern of breaking ; for details , see ref . [ cols="^,^,^",options="header " , ] the atmospheric sum rule can be written in a linearized form by @xmath11 where @xmath12 and @xmath13 are constants expressible in terms of the group theoretic parameters of each model . to test these relations , we require a strong precision on the parameter @xmath0 , which necessitates the consideration of the next generation of long - baseline experiments . these proposals seek to make accurate measurements of the appearance channels @xmath14 and @xmath15 , which are sensitive to the value of @xmath0 at a subdominant level . although it remains a challenging measurement , two leading designs have been shown to offer significant sensitivity to @xmath0 : superbeams and neutrino factories . such facilities would be able to constrain the atmospheric sum rules over a significant fraction of the available parameter space . for example , an on - axis superbeam with a detector mass of @xmath16 kton ( @xmath17 kton ) and a baseline of @xmath18 km would be capable of excluding models with @xmath19 and @xmath20 for over @xmath21@xmath22 ( @xmath23@xmath24 ) of the parameter space , depending on the true value of @xmath5 @xcite . in this section , we shall consider a circa @xmath1 km reactor experiment based on the jiangmen underground neutrino observatory ( juno ) @xcite and reactor experiment for neutrino oscillations ( reno-50 ) designs @xcite . these facilities will be capable of high precision measurements of the @xmath25 disappearance probability . the main goal of such experiments is to observe the subdominant oscillations whose phase depends upon the mass hierarchy . however , they will also significantly increase the precision on the oscillation parameters @xmath7 , @xmath26 and @xmath27 , reducing their uncertainty to the sub - percent level . the @xmath4 viable models identified in ref . @xcite make @xmath28 distinct solar predictions , which are shown in table [ tab : solar_pred ] . such precision will have a significant impact on the viability of the correlations predicted by flavour symmetric models . to understand the impact of these high precision measurements , we have performed a simulation based on the juno design to determine its ability to test the correlations shown in table [ tab : solar_pred ] . in our simulation , we assume a @xmath29 kton liquid scintillator detector with a linear energy uncertainty of @xmath30 . the juno facility will detect neutrinos from @xmath31 nearby reactors ; however , we model this by a single source at a baseline distance given by the power weighted average of @xmath32 km and a reactor power of @xmath33 gw @xcite . we have normalised our spectrum to produce @xmath34 events , including a @xmath35 normalisation uncertainty . in fig . [ fig : juno_sr ] , we show the allowed regions at @xmath36 significance for the models shown in table [ tab : solar_pred ] . we see that only two of the @xmath36 intervals overlap , which allows for a strong model discrimination . the ability for juno to exclude these models independently of their atmospheric sum rules provides a great complementarity between the reactor and long - baseline programmes . furthermore , the two indistinguishable models for juno predict very different atmospheric sum rules , @xmath37 where @xmath38 is the golden ratio , and we expect these to be distinguishable with a superbeam for most of the parameter space @xcite . the @xmath36 allowed regions for the solar predictions shown in table [ tab : solar_pred ] after @xmath28 years of data taking by juno.,width=359 ] the next generation of neutrino oscillation experiments , with their focus on precision measurements of the underlying parameters , will allow certain classes of models with discrete flavour symmetries to be thoroughly tested . in ref . @xcite , the role of a long - baseline superbeam experiment ( modelled after lbno or lbne ) has been shown to be able to exclude these correlations for a large fraction of parameter space . in this contribution , we have highlighted the potential for experimental exclusion of these models at a circa @xmath1 km reactor experiment based on the juno facility . by testing the solar predictions to high accuracy , such a facility will be able to independently distinguish between almost all models under consideration . the complementarity between reactor and long - baseline experiments will provide a stringent test of the idea that residual symmetries are responsible for the structure of the pmns matrix .

models of leptonic flavour with discrete symmetries can provide an attractive explanation of the pattern of elements found in the leptonic mixing matrix . the next generation of neutrino oscillation experiments will allow the mixing parameters to be tested to a new level of precision , crucially measuring the cp violating phase @xmath0 for the first time . in this contribution , we present results of a systematic survey of the predictions of a class of models based on residual discrete symmetries and the prospects for excluding such models at medium- and long - term oscillation experiments . we place particular emphasis on the complementary role that a future circa @xmath1 km reactor experiment , _ e.g. _ juno , can play in constraining these models .

You are an expert at summarizing long articles. Proceed to summarize the following text: main sequence stars with mass in the range 0.9 - 9 m@xmath2 evolve through a double shell burning phase , refered to as the asymptotic giant branch ( agb ) phase of evolution . this phase is characterized by carbon dredge up of the core to the surface after each thermal pulse - helium shell flash - ( iben & renzini 1983 ) . the temperatures of these objects are very badly known . although they are highly variable , their determination from static models such as assumed in the basel library can be justified as a first approximation . in order to explore the capabilities of the basel library ( lejeune , cuisinier & buser 1997 , 1998 and references therein , see also lastennet , lejeune & cuisinier , these proceedings ) to predict correct temperatures for such cool agb stars , we compare our results from synthetic infrared photometry of the stellar photosphere with the detailed study of lorenz - martins & lefvre ( 1994 ) of the agb carbon star r fornacis . their work is based on a modelling of the spectral energy distribution of the dust envelope , where they put tight constraints on the temperature of the heating source . table 1 gives the jhklm photometry of r for ( hip 11582 ) that we used ( le bertre , 1992 ) . the photometric errors in the individual jhklm magnitudes are not provided so we assume an error of 0.2 on each magnitude , according to the maximum uncertainty estimated from fig . 1 of le bertre ( 1988 ) . ccccccc j & h & k & l & m & t@xmath0@xmath3 & t@xmath0@xmath4 + & & & & & ( k ) & ( k ) + 5.76 & 3.97 & 2.32 & 0.21 & @xmath50.28 & 2650 & 2440 - 2520 + @xmath3 lorenz - martins & lefvre ( 1994 ) ; + @xmath4 basel jhkm synthetic photometry ( this work , see text for details ) . although the dust may have a significant contribution in the ir _ bands _ of this star , especially l and m , it should only have a secondary influence on the photospheric _ colours_. we intend of course to correct for the predicted differences by a dust model ( lorenz - martins & lefvre , 1993 ) due to the envelope . however in a first step we merely compare the observed colours of r fornacis with the photospheric predictions of the basel library ( basel-2.2 version , with spectral corrections ) by minimizing their @xmath6 differences . + this @xmath6-minimization method is similar to the one applied in lastennet et al . ( 2001 ) : we derived the t@xmath0 and log g values matching simultaneously the observed jhklm photometry listed in tab . 1 , assuming a solar metallicity ( [ fe / h]@xmath70 ) . we have tested various colour combinations of the j ( 1.25 @xmath8 ) , h ( 1.65 @xmath8 ) , k ( 2.2 @xmath8 ) , l ( 3.4 @xmath8 ) , and m ( 5.0 @xmath8 ) magnitudes : ( j@xmath5h ) , ( h@xmath5k ) , ( k@xmath5l ) , ( j@xmath5k ) and ( k@xmath5 m ) . they all give t@xmath0 estimates in agreement with the work of lorenz - martins & lefvre ( 1994 ) . + since better constraints should be obtained by matching more than 1 colour , we chose the ( j@xmath5h ) and ( k@xmath5 m ) colours which give the best @xmath6-scores . the solutions we get to match simultaneously the observed ( j@xmath5h ) and ( k@xmath5 m ) are presented in fig . our best basel - infrared solution is t@xmath0@xmath72440k , but all the solutions inside the 1-@xmath9 contour are good fits to the observed photometric data . the effective temperature of the central star of r for found by lorenz - martins & lefvre is t@xmath0@xmath72650 k ( shown as a vertical line on fig . 1 ) . this is larger by @xmath1100k than the 1-@xmath9 basel contour but still inside the 2-@xmath9 contour . additionally the basel models show that this star has a surface gravity log g @xmath1@xmath50.5@xmath100.4 , which is what one expects for carbon stars . we reported a preliminary study to determine the t@xmath0 and surface gravity of the central star of r fornacis by exploring the best @xmath6-fits to the infrared photometric data . these results are in a surprising good agreement - given the approximation we made ( no envelope absorption / emission correction ) - with the detailed study of lorenz - martins & lefvre ( 1994 ) . therefore , while detailed spectra studies are obviously highly preferred ( see e.g. loidl , lanon & jrgensen , 2001 ) , our method may provide a good starting point . if our r fornacis result is confirmed with other agb stars , this would mean that the basel jhklm synthetic photometry is suited to derive ( teff - log g ) estimates for cool agb stars . iben i. , renzini a. , 1983 , ara&a , 21 , 271 lastennet e. , lignires f. , buser r. , lejeune th . , lftinger th . , cuisinier f. , vant veer - menneret c. , 2001 , , 365 , 535 le bertre t. , 1988 , , 190 , 79 le bertre t. , 1992 , , 94 , 377 lejeune th . , cuisinier f. , buser r. , 1997 , , 125 , 229 lejeune th . , cuisinier f. , buser r. , 1998 , , 130 , 65 loidl r. , lanon a. , jrgensen u.g . , 2001 , , 371 , 1065 lorenz - martins s. , lefvre j. , 1993 , , 280 , 567 lorenz - martins s. , lefvre j. , 1994 , , 291 , 831

we discuss the possibilities of the basel models in its lowest temperature boundary ( t@xmath0@xmath12500 k for cool giants ) to provide the t@xmath0 of agb stars . we present the first step of our work , by comparing our predictions for the agb star r fornacis with the results of lorenz - martins & lefvre ( 1994 ) based on the dust spectral energy distribution .

You are an expert at summarizing long articles. Proceed to summarize the following text: q - balls are nontopological solitonic solutions of a self - interacting complex scalar field theory carrying a conserved global @xmath0 charge . introduced by coleman in 1985 @xcite , their properties have been extensively studied since then @xcite . if on the one hand , volkov and whnert @xcite showed , in the context of some theories with nonrenormalizable scalar potentials , that there exist particular q - ball configurations possessing nonvanishing angular momentum , now known as `` spinning q - balls '' , on the other hand , dvali , kusenko and shaposhnikov @xcite proved in the framework of supersymmetric extensions of the standard model that gauge - singlet combinations of squarks and sleptons corresponding to some flat direction of the supersymmetric potential can give rise to q - balls whose charge @xmath1 is some combination of baryon and lepton numbers . the attractive feature of theses `` supersymmetric q - balls '' is that they could represent the dark matter component of the universe @xcite ( for reviews on dark matter see , e.g. , ref . @xcite ) . motivated by this fact , experimental searches for q - balls are being carried out @xcite , although no compelling evidence for their existence has been reported so far . the aim of this paper is to present novel configurations of a charged scalar field describing nonspherically - symmetric supersymmetric q - balls with nonvanishing angular momentum . to our knowledge , _ spinning supersymmetric q - balls _ are the first example of analytical solution in field theory in minkowski spacetime representing a soliton possessing angular momentum . as we will see , spinning supersymmetric q - balls are excitations of spherically - symmetric supersymmetric q - balls , since their energy spectrum lies above the ground state represented by nonspinning q - balls . however , it is highly probable that these exited states could form during collisions between supersymmetric q - balls and , most importantly , during the process of fragmentation of the affleck - dine condensate , which is a plausible process that can lead to a copious production of q - balls at the end of inflation @xcite . the plan of the paper is as follows . in section ii we review the general properties of spherically - symmetric supersymmetric q - ball in order to make clearer the derivation and the study of nonspherically - symmetric supersymmetric q - balls configurations which will be tackled in section iii . in section iv we show that spinning supersymmetric q - balls are stable against small perturbations about their classical configurations . finally , in section v we draw our conclusions . let us consider a charged scalar field @xmath2 whose dynamics is described by lagrangian density @xmath3 being the theory invariant under a global @xmath0 transformation , there exists a conserved noether charge , @xmath4 , which we normalize as @xmath5 where a dot indicates a derivative with respect to time . the energy - momentum tensor associated to a given field configuration @xmath6 , reads @xmath7 so that the total energy is @xmath8 \!.\ ] ] in general , q - balls are solitonic solutions of the field equations carrying a definite value of the charge , let us say @xmath1 . an elegant way to construct such a type of solution @xcite is to introduce a lagrange multiplier @xmath9 associated to @xmath4 , and require that the physical configuration @xmath6 makes the functional @xmath10 \equiv e + \omega \left[q - \frac{1}{i } \int \ ! d^3x \left(\phi^ * \dot{\phi } - \phi \dot{\phi}^ * \right ) \right]\ ] ] stationary with respect to independent variations of @xmath2 and @xmath9 . noticing that the choice @xcite @xmath11 assures that the total energy @xmath12 is independent on the time , one finds that q - balls solutions have to satisfy the constraints @xmath13 the first two constraints lead to the equations of motion of the fields @xmath14 and @xmath15 , respectively , @xmath16 while the second one is equivalent to the requirement that the charge corresponding to the solution of the equation of motion is equal to @xmath1 : @xmath17 taking into account the equations of motion , the functional @xmath18 can be conveniently re - written as : @xmath19 + \omega q.\ ] ] a spherically - symmetric q - ball is defined as the solution @xmath20 of eq . satisfying , at fixed charge @xmath1 , the boundary conditions @xcite @xmath21 where @xmath22 . in particular , a ( spherically - symmetric ) supersymmetric q - ball is a q - ball configuration arising in a supersymmetric model of particle physics where supersymmetry is broken via low - energy gauge mediation @xcite . in this kind of model the coupling of the massive vector - like messenger fields to the gauge multiplets , with coupling constant @xmath23 , leads to the breaking of supersymmetry @xcite . the coupling itself gives rise to an effective potential for the flat direction @xmath24 whose lowest order ( two - loops ) contribution has been calculated in ref . @xcite : @xmath25}{[z^{-2}-x(1-x)]^2 } \ , .\ ] ] here , @xmath26 and @xmath27 , with @xmath28 the messenger mass scale . the value of the mass parameter @xmath29 is constrained as ( see , e.g. , ref . @xcite ) : @xmath30 where the gravitino mass , @xmath31 , is in the range @xmath32 @xcite . the asymptotic expressions of @xmath33 , for small and large @xmath34 are @xcite : @xmath35 a widely used approximation in constructing q - ball solutions , whose validity has been ascertained in ref . @xcite , consists in replacing the full potential @xmath33 with its asymptotic expansions ( [ potentialapprox ] ) in which a plateau plays the role of the logarithmic rise for large values of @xmath34 . more precisely , the approximate supersymmetric potential has the form @xmath36 where @xmath37 is the soft breaking mass and is of order @xmath38 @xcite . within this approximation , it has been shown that the potential @xmath39 allows spherically - symmetric q - ball solutions as the nonperturbative ground state of the model @xcite . the profile of the supersymmetric q - ball is easily found from eqs . and : @xmath40 where @xmath41 and @xmath42 are constants of integration , @xmath43 is the zeroth - order spherical bessel function of first kind , @xmath44 is the zeroth - order spherical bessel function of second kind , @xmath45 is the zeroth - order modified spherical bessel function of first kind , and @xmath46 is the zeroth - order modified spherical bessel function of second kind @xcite . here , we have introduced the `` thickness '' of the q - ball , @xmath47 we can now define the `` radius '' of the q - ball , @xmath48 , as the solution of the equation @xmath49 . in the limit @xmath50 we have @xmath51 , from which it follows that @xmath52 ( we will se in the following that the condition @xmath50 will correspond to have large values of the charge @xmath1 . ) if the thickness of the q - ball is much smaller than its radius ( we will see , below , that indeed large charges @xmath1 implies that @xmath53 ) , we can write @xmath54 which is the solution found in ref . @xcite . inserting the above solution in eq . and minimizing with respect to @xmath9 [ see last equation in eq . ] , we find the parameter @xmath9 as a function of the charge @xmath1 : @xmath55 where we have introduced the `` critical charge '' @xmath56 , whose meaning will be clear in the following , as @xmath57 inserting eq . in eq . , we find the q - ball radius as a function of the charge : @xmath58 from the above equation and taking into account eqs . and , we find that for large charges , @xmath59 , it results @xmath60 , and this justifies our approximation to neglect the thickness of the q - ball in computing its profile . inserting eq . in eq . , and observing that , at fixed charge @xmath1 , the energy coincides with the functional @xmath18 , we find @xmath12 as a function of the charge : @xmath61 finally , inserting eq . in eq . and taking into account eq . , we find the value of @xmath41 as a function of the charge : @xmath62 the above relation clarifies the meaning of the critical charge : the q - ball solution we found in the limit @xmath50 , corresponds indeed to the case of large charges compared to @xmath56 . if the energy @xmath12 of the q - ball at fixed charge @xmath1 is less then @xmath63 , the soliton decays into @xmath1 quanta of the field ( the perturbative spectrum of the theory ) , each of them with mass @xmath64 . instead , if @xmath65 the q - ball is said to be classically stable , and then represents the ground state of the theory . using eq . , we find classical stability , @xmath66 , for @xmath67 , with @xmath68 in general , spherically - symmetric q - balls have zero angular momentum . in fact , the total angular momentum for a scalar field configuration is given by @xmath69 where @xmath70 is the total angular momentum tensor @xcite @xmath71 with @xmath72 being the energy - momentum tensor given by eq . . now , using spherical coordinates , @xmath73 , we obtain @xmath74 \ ! , \\ \label{j2 } & & \!\!\!\!\!\!\!\!\!\ ! } = \int \ ! d^3x \ ! \left [ -\cos(2\theta ) \cos \varphi \ , t^0_\theta + \cot \ ! \theta \sin \varphi \ , t^0_\varphi \right ] \ ! , \\ \label{j3 } & & \!\!\!\!\!\!\!\!\!\ ! j^{3 } = - \int \ ! d^3x \ , t^0_\varphi \ , , \end{aligned}\ ] ] so that for a spherically - symmetric q - ball , @xmath75 , we get @xmath76 . on the other hand , for a nonspherically - symmetric q - ball ( if it ever exists ) , @xmath77 , we could have in principle a nonvanishing angular momentum ( for supersymmetric q - balls this will be indeed the case ) . in particular , if one makes use of the `` axially - symmetric ansatz '' , @xmath78 @xmath79 being a real constant , one easily finds @xmath80 where @xmath4 is given by eq . . since single - valuedness of the scalar field requires @xmath81 , the constant @xmath79 must be an integer . therefore , for this particular configuration , the third component of the angular momentum is quantized and proportional to the charge . it is useful for the following discussion to observe that if a q - ball configuration is such that the field @xmath2 is given by @xmath82 with @xmath83 a real function , then ( as it easy to verify ) it results @xmath84 , so that the nonspherically - symmetric q - ball is indeed a spinning q - ball with total angular momentum directed along the @xmath34-axis . we now return to the supersymmetric case to find nonspherically - symmetric supersymmetric q - ball configurations . we start by writing the equation of motion for the field @xmath15 [ eq . ] in spherical coordinates : @xmath85 using the technique of separation of variables , @xmath86 we easily find [ using the approximate form of the supersymmetric potential , eq . ] the solution of eq . : @xmath87 where @xmath88 are the usual spherical harmonics of degree @xmath89 and order @xmath79 , with @xmath90 being the associated legendre polynomials of degree @xmath89 and order @xmath79 @xcite , and @xmath91 here , @xmath92 , @xmath93 , @xmath42 , @xmath94 are constants of integration , the q - ball thickness @xmath95 is the same as in eq . , @xmath96 and @xmath97 are the spherical bessel function of order @xmath89 of first and second kind respectively , and @xmath98 and @xmath99 are the modified spherical bessel function of order @xmath89 of first and second kind respectively @xcite . we can now define the `` radius '' of the q - ball , @xmath100 , as the solution of the equation @xmath101 . in the limit @xmath102 we have @xmath103 , from which it follows that @xmath104 where @xmath105 represents the first zero of the bessel function of order @xmath106 of first kind , @xmath107 @xcite . is an increasing function of @xmath106 with @xmath108 and @xmath109 , @xmath110 , @xmath111 , @xmath112 , @xmath113 @xcite , etc . moreover , the asymptotic expansion of @xmath105 , as @xmath114 , is : @xmath115 , where @xmath116 is the first negative zero of the airy function @xmath117 @xcite . ] ( we will se in the following that the condition @xmath102 will correspond to have large values of the charge @xmath1 . ) if the thickness of the q - ball is much smaller than its radius ( we will see , below , that indeed large charges @xmath1 implies that @xmath118 ) , we can write @xmath119 inserting the above solution in eq . and minimizing with respect to @xmath9 , we find the parameter @xmath9 as a function of the charge @xmath1 : @xmath120 where , from now on , quantities with the subscript `` 0 '' refer to the case of spherically - symmetric supersymmetric q - balls analyzed in section ii . inserting eq . in eqs . and we find , respectively , the q - ball radius and energy as a function of the charge : @xmath121 and @xmath122 while , inserting eq . in eq . and taking into account eq . , we obtain @xmath123 where we have introduced the `` critical charge '' @xmath124 as @xmath125 ^ 4 \ , q^{\rm ( cr)}_0.\ ] ] therefore , the q - ball solution we found in the limit @xmath102 , corresponds indeed to the case of large charges compared to @xmath124 . also , from eq . and taking into account eqs . and , we find that for large charges , @xmath126 , it results @xmath127 , and this justifies our approximation to neglect the thickness of the q - ball in computing its profile . finally , using eq . , we find classical stability , @xmath128 , for @xmath129 , with @xmath130 taking into account that the general form of a supersymmetric q - ball solution [ given by eqs . , - , and ] is of the form with @xmath83 a real function , and taking into account the discussion at the beginning of this section , we conclude that the nonspherically - symmetric q - ball solutions we found describe q - balls with total angular momentum directed along the @xmath34-axis and equal to @xmath131 . moreover , observing that the angular part of the q - ball solution is proportional to the spherical harmonic @xmath132 , we deduce that @xmath89 determines its parity @xmath133 : @xmath134 however , not all values of @xmath89 are admitted since , at fixed charge @xmath1 , a q - ball with definite angular momentum @xmath135 ( or , which is the same , with definite value of @xmath79 ) is such that its energy is minimum . looking at eq . and taking into account that @xmath136 is an increasing function of @xmath89 , we deduce that at fixed @xmath1 and @xmath79 , two values of @xmath89 are allowed : for even ( odd ) @xmath137 , @xmath138 if the parity of the q - ball solution is positive ( negative ) and @xmath139 if the parity of the q - ball solution is negative ( positive ) . accordingly , the energy spectrum of allowed states of a spinning supersymmetric q - ball looks like that in fig . 1 . . also shown are the corresponding values of the radius @xmath100 , the angular momentum @xmath135 , and the parity @xmath133 of the state.,scaledwidth=45.0% ] in fig . 2 , we plot the spinning supersymmetric q - ball s profile , @xmath140 , as a function of @xmath141 and @xmath142 for different values of @xmath89 and @xmath79 , at fixed charge @xmath143 . we observe that , using well - known properties of spherical harmonics , the profiles for negative values of @xmath79 coincide with the corresponding positive ones if @xmath137 is even , while they get an extra minus sign if @xmath137 is odd . , for @xmath143 . from upper to lower panel : @xmath144 , @xmath145 , @xmath146.,scaledwidth=45.0% ] , for @xmath143 . from upper to lower panel : @xmath144 , @xmath145 , @xmath146.,scaledwidth=45.0% ] , for @xmath143 . from upper to lower panel : @xmath144 , @xmath145 , @xmath146.,scaledwidth=45.0% ] before concluding , we would like to show that spinning supersymmetric q - balls are stable against small perturbations about their classical configurations . we will closely follow an analysis performed in ref . @xcite ( see also references therein ) regarding the stability of q - balls arising in some theories with nonrenormalizable scalar potentials . writing @xmath147 and varying the functional @xmath18 [ defined by eq . ] with respect to @xmath2 , we find the equation of motion for the field @xmath148 : @xmath149 where @xmath150 is the linear differential operator @xmath151 since eq . is linear in @xmath24 [ due to the form of the potential @xmath152 , see eq . ] , the evolution of small perturbations @xmath153 about the background spinning supersymmetric q - ball configurations is described by a similar equation : @xmath154 the solutions @xmath155 of the above equation are easily found : @xmath156 where @xmath157 , @xmath158 and @xmath159 being the eigenvectors and eigenvalues of @xmath150 : @xmath160 from eq . , we get that the background solution is unstable [ i.e. @xmath161 grows unboundedly with time ] if there exists a @xmath159 such that @xmath162 < 0 $ ] . however , this is not the case since @xmath163 is real . in fact , writing eq . as @xmath164 and remembering that the eigenvalues of the operator @xmath165 are strictly positive real numbers , we obtain @xmath166 where in the last inequality we used eq . . the above equation shows that @xmath163 is a real quantity , as anticipated . we have succeeded in obtaining , analytically , nontopological solitonic solutions with nonvanishing angular momentum in ( 3 + 1)-dimensional minkowski spacetime in the theory of a self - interacting complex scalar field carrying a conserved global @xmath0 charge . this kind of solitons ( known as spinning q - balls ) naturally emerge in a particular class of supersymmetric extensions of the standard model of particle physics where supersymmetry is spontaneously broken at low energy . the scalar field is in this case a gauge - singlet combination of squarks and sleptons corresponding to some flat direction of the supersymmetric potential , while the conserved global charge is some combination of baryon and lepton numbers . in this class of models an effective potential for the flat directions arises due to the breaking of supersymmetry . we have shown that such a type of potential admits , as the nonperturbative ground state of the theory , axisymmetric q - balls whose angular momentum is directed along the axis of symmetry . working in the limit of large charges , we have found that the state of a _ spinning supersymmetric q - ball _ can be labeled by the triple @xmath167 , where @xmath1 is the conserved @xmath0 charge , @xmath89 is positive integer that can take the values @xmath168 and @xmath169 and defines the parity of the state , @xmath170 , while @xmath79 is a integer which gives the projection of the angular momentum on the axis of symmetry through @xmath171 . moreover , we have found the expressions for the energy and radius of spinning supersymmetric q - balls , which fully determine their astrophysical and cosmological properties . it turns out that they do not explicitly depend on @xmath79 and , at fixed charge , are increasing functions of @xmath89 . this indicates that spinning supersymmetric q - balls are indeed excitations of spherically - symmetric supersymmetric q - balls . they are classically stable due to conservation of angular momentum and parity and stable against small perturbations about their classical configurations , and could form during collisions between supersymmetric q - balls and/or during the process of fragmentation of the affleck - dine condensate at the end of inflation . a. g. cohen , _ et al . _ , nucl . phys . b * 272 * , 301 ( 1986 ) ; a. m. safian , s. r. coleman and m. axenides , _ ibid . _ * 297 * ( 1988 ) 498 ; k. m. lee , _ et al . _ , d * 39 * , 1665 ( 1989 ) ; a. kusenko , m. e. shaposhnikov and p. g. tinyakov , pisma zh . fiz . * 67 * , 229 ( 1998 ) ; [ jetp lett . * 67 * , 247 ( 1998 ) ] ; k. enqvist and j. mcdonald , phys . b * 425 * , 309 ( 1998 ) ; m. axenides , _ et al . _ , _ ibid . _ * 447 * ( 1999 ) 67 ; t. multamaki and i. vilja , nucl . b * 574 * , 130 ( 2000 ) ; r. battye and p. sutcliffe , _ ibid . _ * 590 * , 329 ( 2000 ) ; s. kasuya and m. kawasaki , phys . rev . lett . * 85 * , 2677 ( 2000 ) ; s. theodorakis , phys . rev . d * 61 * , 047701 ( 2000 ) ; m. axenides , _ et al . _ , _ ibid . _ * 61 * , 085006 ( 2000 ) ; s. kasuya and m. kawasaki , _ ibid . _ * 61 * , 041301 ( 2000 ) ; _ ibid . _ * 62 * , 023512 ( 2000 ) ; _ ibid . _ * 64 * , 123515 ( 2001 ) ; k. n. anagnostopoulos , _ et al . _ , _ ibid . _ * 64 * ( 2001 ) 125006 ; f. paccetti correia and m. g. schmidt , eur . j. c * 21 * , 181 ( 2001 ) ; a. kusenko and p. j. steinhardt , phys . rev . lett . * 87 * , 141301 ( 2001 ) ; n. graham , phys . b * 513 * , 112 ( 2001 ) ; t. multamaki and i. vilja , _ ibid . _ * 535 * , 170 ( 2002 ) ; m. fujii and k. hamaguchi , _ ibid . _ * 525 * , 143 ( 2002 ) ; k. enqvist , _ et al . _ , _ ibid . _ * 526 * , 9 ( 2002 ) ; m. postma , phys . rev . d * 65 * , 085035 ( 2002 ) ; m. kawasaki , f. takahashi and m. yamaguchi , _ ibid . _ * 66 * , 043516 ( 2002 ) ; a. kusenko , l. loveridge and m. shaposhnikov , _ ibid . _ * 72 * , 025015 ( 2005 ) ; b. kleihaus , j. kunz and m. list , _ ibid . _ * 72 * , 064002 ( 2005 ) ; t. a. ioannidou , a. kouiroukidis and n. d. vlachos , j. math . phys . * 46 * , 042306 ( 2005 ) ; s. kasuya and f. takahashi , jcap * 0711 * , 019 ( 2007 ) ; s. clark , arxiv:0706.1429 [ hep - 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22 sep 2006 , astro - ph/0701333 ; s. cecchini _ et al . _ [ slim collaboration ] , eur . phys . j. c * 57 * , 525 ( 2008 ) [ arxiv:0805.1797 [ hep - ex ] ] ; z. sahnoun , arxiv:0812.3248 [ hep - ex ] .

we construct nontopological solitonic solutions in ( 3 + 1)-dimensional minkowski spacetime carrying a conserved global @xmath0 charge and nonvanishing angular momentum in a supersymmetric extension of the standard model with low - energy , gauge - mediated symmetry breaking .

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Proceed to summarize the following text: multi - dimensional or multi - way data is prevalent nowadays , which can be represented by tensors . an @xmath0th - order tensor is a multi - way array of size @xmath1 , where the @xmath2th dimension or mode is of size @xmath3 . for example , a tensor can be induced by the discretization of a multivariate function @xcite . given a multivariate function @xmath4 defined on a domain @xmath5^n$ ] , we can get a tensor with entries containing the function values at grid points . for another example , we can obtain tensors based on observed data @xcite . we can collect and integrate measurements from different modalities by neuroimaging technologies such as functional magnetic resonance imaging ( fmri ) and electroencephalography ( eeg ) : subjects , time , frequency , electrodes , task conditions , trials , and so on . furthermore , high - order tensors can be created by a process called tensorization or quantization @xcite , by which a large - scale vectors and matrices are reshaped into higher - order tensors . however , it is impossible to store a high - order tensor because the number of entries , @xmath6 when @xmath7 , grows exponentially as the order @xmath0 increases . this is called the `` curse - of - dimensionality '' . even for @xmath8 , with @xmath9 we obtain @xmath10 entries . such a huge storage and computational costs required for high dimensional problems prohibit the use of standard numerical algorithms . to make high dimensional problems tractable , there were developed approximation methods including sparse grids @xcite and low - rank tensor approximations @xcite . in this paper , we focus on the latter approach , where computational operations are performed on tensor formats , i.e. , low - parametric representations of tensors . in this paper , we consider several tensor formats , especially the tensor train ( tt ) format , which is one of the simplest tensor networks developed with the aim of overcoming the curse - of - dimensionality . extensive overviews of the modern low - rank tensor approximation techniques are presented in @xcite . the tt format is equivalent to the matrix product states ( mps ) for open boundary conditions proposed in computational physics , and it has taken a key role in density matrix renormalization group ( dmrg ) methods for simulating quantum many - body systems @xcite . it was later re - discovered in numerical analysis community @xcite . the tt - based numerical algorithms can accomplish algorithmic stability and adaptive determination of ranks by employing the singular value decomposition ( svd ) @xcite . its scope of application is quickly expanding for addressing high - dimensional problems such as multi - dimensional integrals , stochastic and parametric pdes , computational finance , and machine learning @xcite . on the other hand , a comprehensive survey on traditional low - rank tensor approximation techniques for cp and tucker formats is presented in @xcite . despite the large interest in high - order tensors in tt format , mathematical representations of the tt tensors are usually limited to the representations based on scalar operations on matrices and vectors , which leads to complex and tedious index notation in the tensor calculus . for example , a tt tensor is defined by each entry represented as products of matrices @xcite . on the other hand , representations of traditional low - rank tensor formats have been developed based on multilinear operations such as the kronecker product , khatri - rao product , hadamard product , and mode-@xmath2 multilinear product @xcite , which enables coordinate - free notation . through the utilization of the multilinear operations , the traditional tensor formats expanded the area of application to chemometrics , signal processing , numerical linear algebra , computer vision , data mining , graph analysis , and neuroscience @xcite . in this work , we develop extended definitions of multilinear operations on tensors . based on the tensor operations , we provide a number of new and useful representations of the tt format . we also provide graphical representations of the tt format , motivated by @xcite , which are helpful in understanding the underlying principles and tt - based numerical algorithms . based on the tt representations of large - scale vectors and matrices , we show that the basic numerical operations such as the addition , contraction , matrix - vector product , and quadratic form are conveniently described by the suggested representations . we demonstrate the usefulness of the proposed tensor operations in tensor calculus by giving a proof of orthonormality of the so - called frame matrices . moreover , we derive explicit representations of localized linear maps in tt format that have been implicitly presented in matrix forms in the literature in the context of alternating linear scheme ( als ) for solving various optimization problems . the suggested mathematical operations and tt representations can be further applied to describing tt - based numerical methods such as the solutions to large - scale systems of linear equations and eigenvalue problems @xcite . this paper is organized as follows . in section 2 , we introduce notations for tensors and definitions for tensor operations . in section 3 , we provide the mathematical and graphical representations of the tt format . we also review mathematical properties the tt format as a low - rank approximation . in section 4 , we describe basic numerical operations on tensors in tt format such as the addition , hadamard product , matrix - vector multiplication , and quadratic form in terms of the multilinear operations and tt representations . discussion and conclusions are given in section 5 . the notations in this paper follow the convention provided by @xcite . table [ table : notation_ten ] summarizes the notations for tensors . scalars , vectors , and matrices are denoted by lowercase , lowercase bold , and uppercase bold letters @xmath11 , @xmath12 , and @xmath13 , respectively . tensors are denoted by underlined uppercase bold letters @xmath14 . the @xmath15th entry of @xmath14 of size @xmath16 is denoted by @xmath17 or @xmath18 . a subtensor of @xmath14 obtained by fixing the indices @xmath19 is denoted by @xmath20 or @xmath21 . we may omit ` : ' as @xmath22 if the rest of the indices are clear to readers . the mode-@xmath2 matricization of @xmath23 is denoted by @xmath24 . we denote the mode-@xmath25 matricization of @xmath14 by @xmath26)}\in{\mathbb{r}}^{i_1i_2\cdots i_n\times i_{n+1}\cdots i_n}$ ] in the sense that @xmath27\equiv \{1,2,\ldots , n\}$ ] is the set of integers from 1 to @xmath2 . in addition , we define the multi - index notation by @xmath28 for @xmath29 @xmath30 . by using this notation , we can write an entry of a kronecker product as @xmath31 . moreover , it is important to note that in this paper the vectorization and matricization are defined in accordance with the multi - index notation . that is , for @xmath23 , we have @xmath32 ) } \in{\mathbb{r}}^{i_1\cdots i_n \times i_{n+1}\cdots i_n } \quad & \leftrightarrow \quad { \mathbf{x}}(\overline{i_1\cdots i_n},\overline{i_{n+1}\cdots i_n } ) = { \underline{\mathbf{x}}}(i_1,i_2,\ldots , i_n ) , \end{split}\ ] ] for @xmath30 . .[table : notation_ten]notations for tensors [ cols="<,<",options="header " , ] let @xmath33 and @xmath34 be tt tensors . the sum @xmath35 can be expressed in the tt format @xmath36 that is , each tt - core of the sum is written as the direct sum of the tt - cores . alternatively , each entry of @xmath37 can be represented as products of matrices @xmath38 the tt - ranks for @xmath39 are the sums , @xmath40 . on the other hand , multiplication of @xmath14 with a scalar @xmath41 can be obtained by simply multiplying one core , e.g. , @xmath42 , with @xmath43 as @xmath44 . this does not increase the tt - ranks . we note that that the set of tensors with tt - ranks bounded by @xmath45 is not convex , since a linear combination @xmath46 generally increases the tt - ranks , which may exceed @xmath45 . the hadamard ( elementwise ) product @xmath47 of @xmath48 and @xmath49 can be written in the tt format as @xmath50 that is , each tt - core is written as the mode-@xmath51 kronecker product . as an alternative representation , each entry is written as products of matrices @xmath52 the tt - ranks for the hadamard product are the multiplications , @xmath53 . the contraction of two tensors @xmath54 and @xmath55 is defined by @xmath56 the contraction of a tt tensor @xmath33 with a rank - one tensor @xmath57 can be simiplified as @xmath58 the contraction of two tt tensors @xmath59 and @xmath60 can be calculated by combining the hadamard product and the contraction with the rank one tensor @xmath61 as @xmath62 where @xmath63 for @xmath30 . we remark that @xmath64 and @xmath65 are row and column vectors . we define a generalized contraction operator of two tt - cores as follows . the core contraction of two tt - cores @xmath66 and @xmath67 is defined by @xmath68 we can express the contraction of two tt - tensors @xmath33 and @xmath34 by @xmath69 the computational cost for calculating the contraction @xmath70 is @xmath71 , which is linear in @xmath0 . the matrix - vector product , or the linear mapping can also be efficiently represented by the tt format . the computational cost for computing a matrix - vector product in tt format is @xmath72 . 1 . suppose that a vector @xmath73 is in tt format and a matrix @xmath74 is in matrix tt format , and both are represented as kronecker products @xmath75 and @xmath76 then the matrix - vector product is represented in tt format @xmath77 where @xmath78 and @xmath79 + we can get the same expression in the case that @xmath80 and @xmath81 are represented as outer products . the linear mapping @xmath82 can also be represented by its entries written as products of matrices . suppose that @xmath80 and @xmath83 are in the tt formats @xmath84 and @xmath85 then the entries of the linear mapping @xmath86 is calculated by the contraction @xmath87 where the lateral slices @xmath88 of tt - cores @xmath89 are expressed as @xmath90 note that @xmath91 and @xmath92 are row and column vectors . 3 . we can further simplify the notation ( [ eqn : core_contraction_entry ] ) by considering the tt - core @xmath93 as an operator . let @xmath94 be a linear map defined by @xmath95 with each @xmath96th slice @xmath97 we can represent @xmath98 in a simplified form as follows . let @xmath99 be in matrix tt format and @xmath100 be in tt format . the linear mapping @xmath98 is represented by @xmath101 on the other hand , we recall that @xmath102 where @xmath103 is the frame matrix and @xmath104 . a large - scale matrix - vector multiplication reduces to a smaller matrix - vector multiplication as @xmath105 , where @xmath106 but we can not calculate @xmath107 by matrix - matrix multiplication for a large matrix @xmath108 . however , by using the representation ( [ eqn : linearmap_tensor_operation ] ) , we can show that @xmath107 can be calculated by recursive core contractions as follows . let @xmath109 be a linear map defined by @xmath110 for any @xmath111 . let @xmath107 be the matrix defined by . then @xmath112 for any @xmath111 . that is , the matrix - vector product @xmath113 , where @xmath114 , can be computed efficiently via the recursive core contractions in @xmath115 . figure [ fig : loc_lin_op ] illustrates the graph for the linear map @xmath116 . for each @xmath117 core tensor @xmath118 is connected to the core tensor @xmath119 , which is represented as @xmath120 in ( [ eqn : linearmap_local_operation ] ) . graph for the matrix - vector product @xmath121 represented by the linear mapping @xmath122 , where @xmath104 , width=264 ] the quadratic form @xmath123 for a symmetric and very large - scale matrix @xmath124 can be represented in tt format as follows . 1 . let @xmath108 and @xmath12 are represented as kronecker products @xmath75 and @xmath76 then the quadratic form is represented as the products @xmath125 where @xmath126 and @xmath127 for @xmath128 and @xmath129 for @xmath130 . the outer product representation leads to the same expression . recall that the @xmath2th core tensor of @xmath131 can be represented by the @xmath96th slices @xmath132 , @xmath133 . the quadratic form @xmath134 is computed by contraction of the hadamard product with the rank - one tensor as @xmath135 where @xmath136 we have @xmath137 and @xmath138 . 3 . instead , by using the core contraction defined in ( [ eqn : define_core_contraction ] ) , we can simplify the expression ( [ eqn : quadratic_mps_core ] ) by @xmath139 where @xmath140 is defined in the previous subsection . finally , we can express the quadratic form efficiently as @xmath141 on the other hand , from @xmath102 the quadratic form @xmath142 reduces to @xmath143 , where @xmath144 is a much smaller matrix than @xmath108 when tt - ranks @xmath145 and @xmath146 are moderate . since @xmath147 can not be calculated by matrix - matrix multiplication for a large matrix @xmath108 , we calculate it iteratively by recursive core contractions based on the distributed representation ( [ eqn : quadratic_tensor_represn ] ) as follows . let @xmath148 be a bilinear form defined by @xmath149 for any @xmath150 . let @xmath151 be the matrix defined by . then @xmath152 for any @xmath150 . that is , the bilinear form @xmath153 , where @xmath154 and @xmath114 , can be computed efficiently via the recursive core contractions in @xmath155 . figure [ fig : loc_quad_op ] illustrates the graph for the bilinear form @xmath156 . it is clear that each core tensor @xmath157 is connected to the core tensor @xmath119 , which is represented as @xmath158 in ( [ eqn : quadratic_local_represn ] ) . graph for the quadratic form @xmath159 represented by the bilinear form @xmath160 , where @xmath104 , width=283 ] in this paper , we proposed several new mathematical operations on tensors and developed novel representations of the tt formats . we generalized the standard matrix - based operations such as the kronecker product , hadamard product , and direct sum , and proposed tensor - based operations such as the self - contraction and core contraction . we have shown that the tensor - based operations are able to not only simplify traditional index notation for tt representations but also describe important basic operations which are very useful for computational algorithms . the self - contraction operator can be used for defining the tensor chain ( tc ) @xcite representations , and its properties should be more investigated in the future work . moreover , the definition of core contraction can also be generalized to any tensor network formats such as hierarchical tucker ( ht ) format @xcite . the partial contracted products of either the left or right core tensors are matricized and used as a building block of the frame matrices . we have shown that the suggested tensor operations can be used to prove the orthonormality of the frame matrices , which have been proved only by using index notation in the literature . the developed relationships may also play a key role in the alternating linear scheme ( als ) and modified alternating linear scheme ( mals ) algorithms @xcite for reducing the large - scale optimizations to iterative smaller scale problems . recent studies adjust the frame matrices in order to incorporate rank adaptivity and improve convergence for the als @xcite . in this work , we have derived the explicit representations of the localized linear map @xmath116 and bilinear form @xmath156 by the proposed tensor operations , which are important for tt - based iterative algorithms for breaking the curse - of - dimensionality @xcite . the global convergence of the iterative methods remains as a future work . in addition , it is important to keep the tt - ranks moderate for a feasible computational cost . a. cichocki , r. zdunek , a .- phan , and s. amari ( 2009 ) . _ nonnegative matrix and tensor factorizations : applications to exploratory multi - way data analysis and blind source separation _ , chichester : wiley . a. cichocki ( 2013 ) . era of big data processing : a new approach via tensor networks and tensor decompositions . invited talk at _ international workshop on sisa-2013 _ , nagoya , 1 oct . arxiv:1403.2048 . s. v. dolgov , b. n. khoromskij , i. v. oseledets , and d. v. savostyanov ( 2014 ) . computation of extreme eigenvalues in higher dimensions using block tensor train format . _ computer physics communications _ , 185(4 ) , 12071216 . s. v. dolgov and i. v. oseledets ( 2012 ) solution of linear systems and matrix inversion in the tt - format . _ siam j. sci . _ , 34 , a27182739 . s. v. dolgov and d. v. savostyanov ( 2013 ) . alternating minimal energy methods for linear systems in higher dimensions part i : spd systems . arxiv:1301.6068 . s. v. dolgov and d. v. savostyanov ( 2013 ) . alternating minimal energy methods for linear systems in higher dimensions . part ii : faster algorithm and application to nonsymmetric systems . arxiv:1304.1222 . a. falc and w. hackbusch ( 2012 ) . on minimal subspaces in tensor representations . _ , 12 , 765803 . l. grasedyck ( 2010 ) . hierarchical singular value decomposition of tensors . _ siam j. matrix anal . _ , 31(4 ) , 20292054 . s. holtz , t. rohwedder , and r. schneider ( 2012 ) . the alternating linear scheme for tensor optimization in the tensor train format . _ siam j. sci . _ , 34(2 ) , a683a713 . v. a. kazeev and b. n. khoromskij ( 2012 ) . low - rank explicit qtt representation of the laplace operator and its inverse . _ siam j. matrix analysis applications _ , 33(3 ) , 742758 . v. a. kazeev , b. n. khoromskij , and e. e. tyrtyshnikov ( 2013 ) . multilevel toepliz matrices generated by tensor - structured vectors and convolution with logarithmic complexity . _ siam j. sci . _ , 35(3 ) , a1511a1536 . t. g. kolda ( 2006 ) . multilinear operators for higher - order decompositions . technical report sand2006 - 2081 , sandia national laboratories . b. n. khoromskij and i. v. oseledets ( 2010 ) . dmrg+qtt approach to computation of the ground state for the molecular schrdinger operator . preprint 69 , mpi mis , leipzig . d. kressner , m. steinlechner , a. uschmajew ( 2013 ) . low - rank tensor methods with subspace correction for symmetric eigenvalue problems . mathicse technical report 40.2013 , epfl , lausanne . w. d. launey and j. seberry ( 1994 ) . the strong kronecker product . _ journal of combinatorial theory , series a _ , 66(2 ) , 192213 . doi:10.1016/0097 - 3165(94)90062 - 0 . i. v. oseledets and e. e. tyrtyshnikov ( 2009 ) . breaking the curse of dimensionality , or how to use svd in many dimensions . _ siam j. sci . _ , 31(5 ) , 37443759 .

we review and introduce new representations of tensor train decompositions for large - scale vectors , matrices , or low - order tensors . we provide extended definitions of mathematical multilinear operations such as kronecker , hadamard , and contracted products , with their properties for tensor calculus . then we introduce an effective low - rank tensor approximation technique called the tensor train ( tt ) format with a number of mathematical and graphical representations . we also provide a brief review of mathematical properties of the tt format as a low - rank approximation technique . with the aim of breaking the curse - of - dimensionality in large - scale numerical analysis , we describe basic operations on large - scale vectors and matrices in tt format . the suggested representations can be used for describing numerical methods based on the tt format for solving large - scale optimization problems such as the system of linear equations and eigenvalue problems .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently the validity of general relativity ( gr ) has been brought to question by yilmaz , et al . although such interpretations allow for gravitation to be mathematically consistent and singularity free . such revisions fail to describe the behavior of test particles as adequately as gr , elevating gr as the correct theory . today certain questions about gr remain relevant , such as how does it relate to vacuum energy and quantum mechanics in general . it has been shown in previous works that gr remains self consistent when including the quantum vacuum , or zero - point field . however , the search for a self consistent theory of quantum gravity , " remains a major theoretical challenge today . among the theoretical arguments against the standard interpretation of gr is the choice of mathematical coordinate systems . special relativity ( sr ) , is based upon the structure of a flat minkowski spacetime given in a four - dimensional coordinate system . recently attempts have been made describing coordinate systems with fractal spaces as opposed to natural ones . such an adaptation as the case with the yilmaz approach eliminates singularities within the field equations . recent observational and experimental data have also put into question the validity of gr . the national aeronautics and space administration ( nasa ) has reported an anomalous acceleration of @xmath3 , on spacecraft on the outer edge of the solar system . this data was obtained from information gathered by the jet propulsion laboratories ( jpl ) , and the deep space network ( dsn ) . thus far , no satisfactory conclusion has been given to explain the so called anomalous acceleration towards the sun . " not only have spacecraft provided some fundamental flaws with gravitation , but laboratory results as well . eugene podkletnov , has reported a gravitational shielding " effect with composite bulk @xmath4 ceramic plates . in light of all of these developments it is hard to consider gr as the correct theory . it is the opinion of the author that gr is a theory that works , " however it does nt necessarily make it the correct theory . the goal of this letter is to show that gr is not the correct theory of gravitation , but just works exceptionally well . just as previously newton s law of universal gravitation worked exceptionally well . this letter is not intended to be a replacement for gr , nor is it intended to present theoretical flaws of the that theory . this letter is only presented as an introductory work for an alternative theory of gravitation . the general theme of this letter is given by the following postulates : ( virtual gravitation ) . spacetime is not a null energy field , it consist of asymptotic vacuum fluctuations , and behaves as a virtual energy - sheet . " ( planckian invariance ) . the planck length is a gauge invariant function for all ( interacting ) brane observers . [ an adaptation to the postulate of new relativity . ] this letter is presented in the following format in section [ uni ] a brief introduction into unified field theories are given . in section [ qgh ] a few quantum gravity approaches are introduced . in section [ fg ] fractal geometry is introduced and its relations to a complex system are given . in section [ qedf ] the meaning of fractal geometry for qed is discussed . in section [ qcdf ] the meaning of a fractal geometry is discussed for qcd . in section [ card ] a new theoretical particle is introduced utilizing fractal geometry . in section [ flt ] a relationship between n - dimensional and two - dimensional systems are given . in section [ geo ] a philosophy of geometry is given . in section [ vac ] the effects of the quantum vacuum are discussed . in section [ stg ] a relationship between fractal geometries and the quantum vacuum are discussed . in section [ feystg ] the meaning of feynman diagrams are discussed . in section [ qm ] the validity of quantum mechanics is brought into question . in section [ bm ] an alternative description of gravity is given which may explain the epr paradox . in section [ cqg ] an overview of a canonical non riemannian gravitational field is given . in section [ can ] the planck length results as a function in canonical quantum gravity . in section [ aa ] a possible alternative for the anomalous acceleration " of spacecraft is given . in section [ pgeo ] pseudo geodesic equations are presented . in section [ dis ] a general discussion of this work is presented . in section [ pl ] a discussion of the meaning of the planck length the conclusions of this work is drawn in section [ con ] , which gives stronger definitions to equivalence principle in appendix [ ep ] . finally it is suggested that there may exist a detectable from of yang - mills gravity " in appendix [ ymg ] . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i am convinced that he [ god ] does not play dice . _ einstein _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ einstein , quit telling god what to do . _ " n . bohr _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the unification of gravitation with quantum mechanics began with einstein s objections to the newly developed quantum theory . although acknowledging the successes of the new theory he believed it to be incomplete . einstein was convinced that there was a deeper theory involved , one which would also include gr , a unified field theory was christened . soon came the work of kaluza and klien , giving a pseudo mathematical unification of electromagnetism and gravitation . the theory would soon die out and loose interest , until quantum mechanics came around . and asked the simple question , how does gravity behave at the quantum level ; the answer kalzua - klien gravity . research in this area soon exploded , extra dimensions were soon added to the field equations , superstring theory was born . particle physics began unifying fundamental forces as well , the weak force , the strong force , the electromagnetic force . but no gravitational force , cosmologists helped out , the big bang and nucelarsynthesis would help to explain the problem . soon physics became littered with grand unified theories ( guts ) and theories of everything ( toes ) , they all have the approach of a unified field theory . " however , they missed the simple point einstein was trying to make , how do quantum mechanics and relativity relate ? this is a hierarchical question , the relevant question is how do macroscopic and microscopic worlds communicate ? historically there have been two models formulated for the construction of a consistent quantum gravity theory . they are the canonical and hamiltonian approaches . these two approaches have had limited success , however more recently the theory of loop quantum gravity has been introduced into the sea . out of the three approaches presented loop quantum gravity is generally accepted as the correct approach . however , for a more accurate description of the historic developments of quantum gravity see . in this letter i will focus on the canonical approach as it relates to gauge invariance . _ if we re built from spirals while living in a giant spiral , then is it possible that everything we put our hands to is infused with the spiral ? _ " + max cohen in the motion picture @xmath5 the presence of matter within gr disturbs the field equations by the existence of singularities or point - particles . " how can one avoid this eye sore in the equations , simple fractal geometry . if matter is fractal it can not condense into points , however this allusion can still take place above the planck energy scale . it is hard to believe that this simple approach has only been attempted in recent times , fractal sets are more common in nature than simple polygons . first let us begin with a simple construction of a fractal set with the simple equation @xmath6 . one must also realize that a fractal is composed of a complex number system , i.e. a+ib . using this form one may wish to construct a complex averaging of the mean , which results from the golden mean @xmath7 . thus we have constructed a complex mean which has two possible solutions as seen below : c_m = + = \ { l ; a = c + a ; a c + . this complex mean thus has non communcating solutions . which from the stand point of imaginary numbers yields the general statements : = i = - i thus these statements would appear to be in agreement of the theory of quaternions . which is interesting enough in itself , a four - dimensional version of the complex number system . with this preliminary work set we can now construct a minkowski spacetime which that takes advantage of fractal dimensions . first one can construct the generalized three - dimensional manifold as a three - brane , and thus incorporating time as a fractal set . thus the fractal construction on spacetime is presented in the form @xmath0 a similar approach was made in ref . . it is here postulated the origin for this three - dimensional brane arises from planck scaling . the reason for this postulate is seen when the golden mean is applied to @xmath1-dimensions @xmath8 , the higher n the closer the mean is to 1 . thus only when @xmath9 approaches infinity will we see that @xmath0 will yield the standard minkowski space of 3 + 1 dimensions . this is of course in agreement with any system that we apply c=@xmath10=@xmath11=@xmath12=1 more evidently this corresponds to a time dilation effect in terms of special relativity . such that we have the following revision to the flat four - dimensional minkowski space : ( , z^2)=c^2-()z^2_1-()z^2_2-()z^2_3 [ k ] it is interesting to note that this pseudo metric appears to be an inverse of the standard minkowski spacetime , this importance is seen in section [ bm ] . so far this method has only left intriguing consequences , however it diverges from the point made earlier in this section . in string theory the atomism view of matter is replaced with vibrating strings , these vibrating strings correspond to a real geometry . however , fractal geometries are allowed to break of these strings into imaginary components . these imaginary components thus make the string a complex function , yielding a pseudo point - like string . to analyze this premise allow us to view the nambu - goto action @xmath13 . if strings can indeed be made to subside with fractal geometry , then they would break off into imaginary components by the empirical action : s=_0 ^2 where @xmath14 . this pseudo string forms many more string components in @xmath1-dimensions , that is the string fragment continues in infintium . however , in the physical sense the string exist a pseudo point - like particle , do to the scaling nature of the planck length . these fractal strings then interact within a field , known as gravitational or zero - point fields . when the fractal strings converge with other fractals , a self organization takes place , i.e. the production of virtual particles . this production is made possible through the non communacating mathematics associated with quantum mechanics . as the particles are produced they destroy one another , such that their world - sheets reverberate in a complex form . this complex reverberations in @xmath1-dimensions is responsible for the vibration of the string , which we navely associate with mass . electromagnetic waves are the result of four - dimensional interactions , and its real wave would correspond to the results found in classical mechanics . however , it would intern have a fractal complex field , which would cause the field to break periodically yielding a string fragment , a quanta . this quanta in non self regulating , i.e. it is the nature of the real wave which causes the string to reverberate . the above consideration may carry some controversy in the well known theory of quantum electrodynamics ( qed ) . if a quanta is just a fractal string , then what is the proper approach for the exchange of energy between the two systems ? well , the result appears as classical approximation , the quanta hits the string as a solid body , causing a change in geometry , which virtual particles oppose . this causes the string to bounce " back to its original form , emitting a real wave , but not necessarily a quanta , remember a quanta is given by a complex field . since quantum mechanics is sketched out onto a point particle - like environment , its consequences would agree with the qed model . in fact the fractal model yields a much physical picture for non communacating relationships than the quantum theory . the above result would agree with qed , however , its definitions are quite weak , in fact one may expand these definitions to quantum chromodynamics ( qcd ) . thus a nucleon may be made to reflect electromagnetic radiation as well , however , this dose not defy the documented experiments in any magnitude . it would be the interaction of the system , i.e. what string perimeters give way to the colored , and other gauged forces , that yields its particles " . each string fragment consist of its own local vibrations ( gauge invariance ) , which attributes its mass , i.e. differentiates between a higgs particle and a quark . these states then have their own local statistics , their real waves would then correspond differently than the electromagnetic field . which results in the production of the celebrated yang - mills field , and thus yields the production of colored particles such as the gluon . cantor pioneered the study of infinities with his new theory of cardinal numbers , however he faced opposition in his time for this new theory . cardinal numbers offer the best insight into to the study of fractal strings in n - dimensions , these interpretations in fact have direct physical consequences . most notably it can explain the situation unleashed by the infamous epr - bell paradox , where faster than light ( ftl ) communication appears possible ( depending on planckian scaling ) . for recent theoretical implications and interpretation of the epr - bell paradox see . in each scaling the laws of physics would be very different , and hence superluminal velocities would seem to appear in lower branes . thus the traditional interaction of strings should not be limited on a specified dimension , but behave as a set of cardinal numbers . thus a slight revision of the nambu - goto action should be given which yields s=_0 ^2 where @xmath15 $ ] . _ to divide a cube into two or other cubes , a fourth power , or , in general , any power whatever into two powers of the same denomination above the second is impossible _ " + fermat fermat s last theorem can be associated with fractal geometry in one respect , there is no general real solution to fractal geometry above dimension 2 . this may be a simple coincidence and may have no deeper meaning , however , this is contrary to the recently created holographic principle ( hp ) . the hp relates that the universe may exist in dimensions of infinitum status . however , the laws of physics are best projected onto a pseudo two - dimensional screen , and our three - dimensional world is only a pseudo manifestation of an @xmath1-dimensional continuum . in string theory we can view our universe as made up of two two - dimensional branes ( described by type iia d2 membranes ) . thus any other dimension outside the holographic conjecture yields no physical meaning and no solution . just as what is suggested by fermat s last theorem , therefore our four dimensional slice of the brane is a pseudo physical manifestation of the holographic screen . there is only one explanation for this result , there must exist a physical constant for specified energy scales , i.e. the planck scale . here another coincidence appears to arrive , the two dimensional wave equation for string theory @xmath16 . it seems that both mathematically and physically there is a special importance with dimension 2 . this discussion is largely philosophical , however it is interesting to note that _ cardinal strings are given by complex numbers . in fact a cardinal string in four - dimensions is remarkably similar to the two - dimensional form , and seems to correspond to a torus : s^2=_0 d^2 + _ 0 d^2 _ 0 d^4 ( x1 ^2_iiat^2_iiam ) from this complex structuring , and properties of cardinal numbers it can be seen why quaternions were alluded to in section [ fg ] . _ einstein s theory of gr transformed newton s theory of a gravitational force , to a direct consequence of geometry . however , although the idea of a force was replaced with a geometry , the geometry still yields a force when explained in riemannian geometry . an even more radical approach to gravitation as a geometry was produced by roger penrose in his theory of twistor spaces . the geometry itself is more important than physical masses , in fact masses only come important when one expands this theory . this is true for a fractal revision of string theory , it exists as a pure geometry , the interacting geometry in fact produces mass . this seems almost a radical stance from the point of view of gr , however geometry remains a key factor as the ideal of a force to gr . the vacuum exist from a state of virtual particles being produced via cardinal string fragments . since virtual particles are a pure construction of particles " in @xmath1-dimensions , they are not true strings ( i.e. they violate the hp ) . these particles thus carry no mass - energy equivalent within our universe , never the less they still posses a geometry . this situation only holds true when the system is localized , however , when interacting with non cardinal strings can induce an energy exchange . by the well known casimir effect the energy of the vacuum should be given by : @xmath17 and when interacting with an inertial mass system we have : @xmath18 this relates the fact that as mass is accelerated it pushes the quantum vacuum energy ( which is analogous to the assumption made be postulate one ) . in other words it reacts in the same fashion as air molecules do when inertial masses accelerate on earth ( producing pressure on the system ) . furthermore , it can be assumed that a material body increases its rest mass by absorbing this zero - point - energy ( this assumption must be given in order to satisfy conservation laws ) . moreover , since they are cardinal strings they are unified in a manner , thus the vacuum is a geometrical manifestation of string particles . that is the geometrical patterns formed through string interactions is what we call a gravitational field , i.e. a virtual gravitational field . since these string interactions are only virtual there is no reason to modify the einstein field equation , unless one wishes to discuss quantum string effects . furthermore @xmath1-dimensional spacetime metrics have shown to be very similar to the structure of four - dimensional spacetimes . therefore the classical gravitational field is removed from quantum mechanics as it exist in a virtual sense , thus quantum mechanics is a property of matter , i.e. interacting geometries . the vacuum however is not currently treated as the geometry of a fractal spacetime system , and hence is incompatible with other vacuum theories . however a fractal model for quantum mechanics appears to agree with at least one interpretation of the quantum vacuum . in fact this interpretation goes right along with loop quantum gravity , and string theory see ref . the hausdorff ( or fractal ) dimension suggest that dimensions maybe confined to a @xmath19 spacetime , with fractal string scaling . this principle was postulated earlier in this letter as the consequence of the planckian scaling , " and is the leading postulate in new relativity . here the importance now becomes what is the meaning of de - broglie phenomenon . the einstein de - broglie equation @xmath20 can be seen as a representation of the relativistic wave equation , i.e. mass is a quantifiable measure of energy . which can be applied directly to string theory , the vibrations of the string are given in a fractal frequency comparable to the compton wavelength . > from the above consideration it can be equally applied the the origin of a bodies mass intern comes from the gravitational field itself . this requires the use of feynman diagrams , and believe it or not this approach is indeed correct , if strings are represented by cardinal numbers ( and if quantum mechanics is considered to be correct ) . since this allows for ftl communication at the classical level it can be interpreted at least at the quantum level that mass originates from spacetime ( when measured at the planckian scale ) . in fact at the quantum level the production of virtual particles may be responsible for a light paths geodesic curvature , yielding a quantum gravity theory . more correctly it may be viewed that the quantum theory is in reality a classical approximation of string theory . ( brussels approximation ) . quantum mechanics exist as a classical approximation of string interactions which possesses an apparent time reversed symmetry . deriving this theorem let us consider the following thought experiment : if mass is composed of vibrating stings , and intern these strings produce gravitational fields in n - dimensions then space is vibrating . however , such an approach would imply that geometrically speaking the two systems are unaware of their own vibrations under gauge invariance . on the other hand , the interaction of fractal strings are given in a complex field , which itself is anti - communcating . thus spacetime , or string space is subjected to uncertainty principles as shown in ref . . since at the classical level , the fractal space can give way to real solutions i now make the assertion that the quantum particles are at flux , and not the space itself ( for argumentative purposes only ) . therefore when a quantum is observed , it is the space which becomes fuzzy , " not the particle , and when not observed the inverse follows . thus the theorem leads to two possible out comes during an observation sequence . i ) the fuzzy quantum particle becomes a point particle , when time symmetries are reversed . ii ) spacetime is fuzzy , however when collapsed by a point particle elucidates to a natural " state . thus it is seen that only when time symmetries are reversed does one obtain the laws commonly associate with quantum mechanics . making the only valid approximation of quantum mechanics the brussels interpretation , this may also explain the epr paradox . however our thought experiment does yield one solution which interpretation i and ii are consistent . when a particle is observed by a frame , it is in reality observed by the local states ( brane ) of the cardinal string , which may be in any number of states . as a point particle ( non local string ) enters the system it begins to collapse the wave function of the local state . this makes it appear that a fuzzy quantum particle has entered the system , much as a star appears to twinkle in the night sky . this collapsing of the state i will call the observational s frame present , " before the action the observational state was fuzzy . however , after the event a self organization took place , an event occurred , producing a present state . after the quanta is observed by the observational frame , its present state then becomes certain in the terminology of classical quantum mechanics . therefore local brane string interactions can not take place until a non local ( cardinal ) string collapses the wave function of the system . > from the feynman interpretation of the time reversed symmetries of the gravitational field , new conclusions about the nature of spacetime can be made . it is here postulated that classical mechanics is in reality a description of a quantum system given under an approximation of a time reversed symmetry . such that the following statements become true : * reversal of bohmian mechanics ( bm ) yields classical mechanics * reversal of brussels interpretation ( bi ) yields standard quantum mechanics ( sqm ) with the fractalization of spacetime given in section [ fg ] , we may conclude that ( with the use of bohmian mechanics ) that inertial mass yields an expansion of spacetime . thus as a body gains mass as it accelerates in classical spacetime it causes the fractalization of the bohmian system to increase ( which is analogous to a lorentz transformation ) . which thus gives the allusion that spacetime is contracting in the classical real frame . the gravitational force , thus is an inertial acceleration which radiates a pseudo center of gravity vector in terms of newtonian mechanics . however , this is how we interpret the events classically , in reality it is the expansion of the fractal bohmian space ( the @xmath21 term in section [ fg ] , e.g. the time dimension ) which yields inertial acceleration . therefore , it is the brussels interpretation of quantum mechanics which yields the epr paradoxes , the connection of the particles is created by the ( incorrect ) approximation of time reversed symmetries . that is to say the epr paradox only includes simple ( non fractal ) states , which by time reversal appears to yield ftl communication ( see figure [ fig ] ) . this appearance of ftl communication should nt be taken to seriously since recent experiments ( cfr . wang , et al ) appear to yield ftl communication . however , it is the string interactions which yield quantum mechanics , in fact it yields the same interpretations as bohmian mechanics . thus bohmian mechanics adequately describes the behavior of particles " while , the brussels interpretations yields standard quantum mechanics [ meaning that this system is only an approximation ] . since complex spaces have been presented as a solution to the singularity problem , it is natural for a formulation of a complex spacetime . to proceed in this manner one must neglect the cherished einstein - hilbert action @xmath22 and replace it with the tucker - wang action : @xmath23 therefore we can now discuss a complex gravitational field , without the traditional riemannian geometry . the classical einsteinian relativity gives the generic field for a spacetime geometry as @xmath24 . such that i now wish to make the generalized statement @xmath25 , or in canonical terms @xmath26 . as such a generalization of a purely idealistic spacetime governed by perfect fluid becomes : g^ab=16gt_ab where under ideal cases one can have the geometry @xmath27 . the reason the field takes on the term @xmath28 , as opposed to @xmath29 , can be seen with the use riemannian metrics . first , let us begin with a ricci symmetric tensor of the form : @xmath30 which within a constant field becomes @xmath31 . this field can thus transpose to @xmath32 , and couple to an opposing electromagnetic field by the connection @xmath33 which therefore leads to the following antisymmetric riemannian field @xmath34 whence therefore means that there must be an equivocal orthonormal action taking place , such that one has @xmath35 . in which the generic geometry for a perfect fluid becomes that of g^ab=16 gt^_ab which translates to the field equation : r^a_b -1 4g(e_(a ) ) , e_(b))r=-16 g c^4t^_ab this equation must be modified when given in an @xmath1-dimensional system such that on has : r^a_b -1 2(n)g(e_(a),e_(b))r=-8 ( n)g c^2(n)t^_ab [ wfe ] the above equation is in essence a canonical gravitational field equation , which appears to be a good candidate for a quantum gravity theory . where the geodesic equations become - _ [ ] ^ dx^ dx^ ( 1 + _ ab(n ) ) 0 however , this interpretation suggest that spacetime is quantitized by a canonical action however , quantum particles are given as classical particles . hence this interpretation would be an inverse of understood quantum mechanics . however , here a paradox opens up , when time is reversed particles remain in one quantum state , thus sqm is not retrieved . thus , it is seen that there exist no true quantum gravity " theory . in fact if one applies this formulation with the planck length it destroys the principle of _ planckian invariance and gives an allusion to the existence of an ther . _ therefore the planck length no longer remains a constant but becomes a dynamical function . first let us write the planck length in terms of @xmath1-dimensions and apply it to the above field equation such that we have : l_p = ( ^n /m^n _ p c^3n)^1/2n.momentum must be reevaluated from @xmath36 , such that @xmath37 . thereby the planck length , and mass are actually given by a particles rest momentum . such that the planck length is in reality given by the function : l_p = ( ^n /m_p_0 ^n c^3n)^1/2n . this thereby has major implications , that the planck length is not really a constant at all but a function of momentum . such that as an object increase in speed with respect to its rest momentum , its planck energy becomes larger . that is as a material body is subjected to length contraction , its planckian energy is modified to compensate for the effect . since the momentum is measured at rest m remains a constant , it is the velocity of the system which changes . thereby meaning that length contraction in special relativity is not given by lorentz transformations , but by the local rest momentum of the planck barrier . this results when we interpreted this action canonically however under bm it yields expected results . therefore it is seen that a canonical formulation fails to keep planckian invariance " which represents a failed attempt at a quantum gravity theory . finally i bring light to an alternative explanation for the acceleration of spacecraft . since the findings of the anomalous acceleration towards the sun , " there have been a number of possible explanations given . with the construction of a fractal @xmath1-dimensional spacetime , i view this as a quantum gravity effect . as an object accelerates its fractal geometry changes [ by means of bm ] , thus resulting in pressure on the system . pressures as the source of a gravitational field were pioneered long ago by einstein : @xmath38 this method is not ad hoc , gravitational pressures for atomic gases and radiation can be given by : p_gas = hqtp_rad=13at^4 which lends the general results @xmath39^{1/3}\\q^{4/3}=c(\beta)q^{4/3}\end{aligned}\ ] ] thus after a slight modification of eq.([wfe ] ) , one can obtain the following gravitational pressure : r^ca_b=-(t^ca_b-12g^c_abt)+2a^2^c_ab=(2-p ) [ rfe ] therefore the flat field equations can be given by : ^c _ ab(z)0,r^ca _ b(z)0 thus a line elements trajectory would be given by ( , z^2)=g_caz_cz_a therefore a metric in a complex fractal spacetime can be given by : _ ab=1,2^c = iz_a^c z_b^c0 since this quantum gravity effect originates from the planck length it is very unlikely that the yukawa interaction : @xmath40\ ] ] will take place ( unless special conditions arise ) . however , if such an effect does arise , it may yield peculiar motion for an obejects geodesic path . first let us begin with the two - dimensional lagrangian hamiltonian , so that we have an equation of motion from the simple action = h p_i;=h q_i in canonical terms motion is given by q_i - h p_i;p_i = h q_i lending a four - vector of the form @xmath41 . in such that a hamiltonian wave within a gravitational field would be in motion according to the geodesic path : -^ _ [ ] dx^ dsdx^ ds0 this geodesic unlike the prior for a classical particle , will not differentiate and thus its motion need not transverse through classical euclidean space . therefore it can be seen that complex spaces could impose unseen forces which would effect a geodesic path for a body ( or wave ) in motion . a proposal made in ref . , made a like was case in the relativistic sense so that one would have : f^=m_0w wdy ddx _ d. although there is no direct physical evidence of this , it is still however an intriguing explanation . after all the so called anomalous acceleration " is only experienced by small bodies , not massive ones such as planets . thus a quantum interpretation of this effect seems to fit the observed data better than any other approach . alternatively modanese has also predicted a macroscopic quantum gravity effect , however it is limited to the podkletnov experiment . the formulation of this theory was based on a desire for a reformulation of gr in order to describe a singularity free theory ; in which a fractal formulation of the field equations were derived . the second desire for this theory was the formulation of a quantum construction of gr , however the end result is a gravity theory which describes quantum mechanics . therefore the gravitational field and matter can be considered to be molded into the following form . matter exist as a pseudo point particle who s field of movement is restricted onto a two - dimensional ( complex ) frame . this two - dimensional frame s movement is governed by bm , and in part by the hp . matter , is thus in reality a fractal vibrating string fragment which continues on into @xmath1-dimensions . the fractalization of this cardinal string " produces virtual particles which posses a geometry , it is this ( virtual ) fractal geometry that is responsible for the gravitational field . in light of future studies it is likely that an adequate formulation for an alternative to gr be given in the following forms . one the acceptance of a fractal ( even if only quasi fractal ) structure of matter and space as an adequate formulation for the geometry of spacetime . two the acceptance of complex systems into the equations , e.g. quaternions , octonions , c * algebras , etc . and finally three , the acceptance of physical conditions which may not be popular , " but yield results that are not contradictory to known data . the mathematical conditions are the most intriguing to author because there seems to be a hidden mechanism in the mathematics . however , my advanced mathematics skills are mediocre at best so these avenues are still left open in this letter . several physical arguments against the existence of singularities have been given by loinger , as well as einstein s classic objections . thus one may inquire what happens at the planck length , i.e. what are the laws of physics ? here i now quote kip thorne , on our current understanding of singularites and ` quantum foam . ' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ how probable is that a black hole s singularity will give birth to ` new universes ? ' we do nt know . it might well never happen , or it might be quite common or we might be on completly the wrong track in believing that singularites are made of quantum foam . _ " thorne ( 1994 ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ this now leads a discussion to recent attempts to model gravity in terms of @xmath1-dimesnional spaces ( arkani - hamed , et al 62 - 69 ) , in which the planck length varies with the number of dimensions . of course the planck length could be infinitely small in an infinite system , clearly a challenge to the principle of planck invariance . however , we note that with mach s principle ( mp ) the planck length must be observed by an external mechanism to remain invariant . thus the planck length exist as a fractalization of bm , which becomes an observational frame in classical real mechanics . furthermore , from this it may be seen that the laws of physics as we understand them are in direct consequence of the planck length . we may also assume the chosen string field is quantitized ( i.e. given by bm ) , because its mass is attributed to a complex pseudo oscillation ( vibration ) . where i now quote david bohm ( cfr . bohm , 22 ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ we may conclude that all systems which oscillate are quantitized with @xmath42 whether these systems be mathematical oscillators , sound waves , or electromagnetic waves . _ " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ so what does physics look like beyond the planck length , remember the ( local ) laws of physics are given by two complex d2-branes . when we interpret these interactions we receive the traditional gr effects at the macroscopic level . however , at the planck scale singularities do nt exist such that the frame interacts via cardinal strings " and not classical gr . thus interactions on local branes cease , and supersymmetry takes over . only strings which are connected to a form of the d2-brane will have observable physical manifestations , this deals with planckian invariance " . in fact each dimension may have its own unique planck length which governs its own local laws ( explaining the limitation of classical string theory to a set number of dimensions ) . which leaves open several areas in @xmath1-dimensional black hole mechanics , and planck length physics . i have shown that there is enough evidence at present to challenge gr as the correct theory for gravitation . i have also introduced the study of a complex fractal spacetime system and its possible relationship to the planck length . the given formulation for a canonical gravitational field resulted in contradictory conclusions , thus ruling out a canonical approach to quantum gravity . " finally if my hypothesizes hold valid then sqm will begin to make invalid predictions for the behavior of particles near the planck length . thus a fractal correction for sqm will be needed under certain gravitational fields , which may be comparable to bm . a new weak equivalence principle ( wep ) for the gravitational field can be postulated utilizing a complex fractal minkowski spacetime ( cfms ) system ( see eq.([k ] ) ) . since there is no spatial acceleration for the gravitational field ( in respect to mp ) , it is the acceleration of the pseudo time dimension in the cfms which produces a gravitational curvature . therefore a material body would have the traditional minkowski spacetime , acting as a lorentz frame . since gravitational fields extend indefinitely , this should cause time to continually progress within an inertial acceleration frame . this therefore means that as an object enters a gravitational field it becomes less massive , in terms of a lorentz transformation . equivocally it can be stated that energy is lost in curved spacetime . a similar effect is all ready known , known as a gravitational time delay , " i.e. the _ shapiro effect . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ according to the general theory , the speed of a light wave depends on the strength of the gravitational potential along its path . _ " shapiro ( 1964 ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ this is however contradictory to sr , because it fails to describe inertial acceleration within a gravitational field correctly . however , the _ principle of relativity is still preserved , because the curvature of spacetime corrects for the cfms . therefore the reason the _ equivalence principle is fundamental in gr is because it is the only priori condition which satisfies the _ principle of relativity . thus without an equivalence principle , there would be no relativistic theory for the gravitational field . _ _ _ the logarithm gravitational time delay , may also be responsible for the apparent anomalous acceleration " of spacecraft . david crawford has offered a similar explanation , where the gravitational term arises from interplanetary dust . here i now hit upon a topic hinted upon in section [ qedf ] ; converting fractal geometry in the terminology of qcd . let us now rewrite eq.([rfe ] ) , so that we have an equation of the form : -(r_b ^ca-12g^c_abr^c ) = 8t^c_ab with this equation a yang - mills gravitational pressure can arise under the following field : s_e=14g^2 ^ 4 zf^_f^_+1_ok^ia _ b k^ib _ a ^2 which must be given in a conformal field , i.e. @xmath43 , thus we have : s=12_0+_0_p ^2 from this it is now seen that a cardinal string , " is in fact an @xmath1-dimensional world line . which can communacate with other world lines , where we have a self - organization of the system by v(s)=14 ^ 2 g_ab^jklm_a t_ab_jklm it is these interactions which generate a spinor space , which attributes mass to the geometry . therefore the yang - mills field is added to the gravitational field , by means of a gravitational pressure . here a less restrictive form of the strong equivalence principle ( sep ) can be applied : thus the interaction of two or more cardinal strings , " produces a twistor like action , represented by @xmath45 . this also means that certain gravitational anomalies may not only arise at the planck length , and may result in experimental verification . 10 yilmaz h. did the apple fall ? in m. barone and f. selleri , editors , frontiers of fundamental physics . 115 - 124 ( 1994 ) fackerell e. remarks on the yilmaz and alley papers . proceedings of the first australaian conference on general relativity and gravitation , ed . d. l. wiltshire , university of adelaide ( 1996 ) , 117 available url : http://www.physics.adelaide.edu.au/asgrc/acgrg1/fackerell.html castro c. on m theory , quantum paradoxes and the new relativity physics/0002019 castro c. why we live in 3 dimensions hep - th/0004152 ya.kobelev l. do gravitational and electromagnetic fields have rest masses in the fractal universe ? physics/0006043 ya.kobelev l. the theory of gravitation in the space - time with fractal dimensions and modified lorents transformations physics/0006029 ansoldi s. loop quantum mechanics and the fractal structure of quantum spacetime , chaos solitons fractals 10 ( 1999 ) 197 hep - th/9803229 anderson j , _ et al . _ indication from pioneer 10/11 , galileo , and ulysses data , of an apparent anomalous , weak , long - range acceleration , phys.rev.lett . 81 ( 1998 ) 2858 - 2861 gr - qc/9808081 podkletnov e. weak gravitation shielding properites of composite bulk @xmath46 superconductor below 70 k under e.m . field cond - mat/9701074 overduin j. and wesson p. kalzua - klien gravity . phys.rept . 283 ( 1997 ) 303 - 380 gr - qc/9805018 wallace d. the quantization of gravity an introduction gr - qc/0004005 rovelli c. loop quantum gravity . living reviews in relativity vol 1 ( 1998 ) max - planck - gesellschaft . /1998 - 1rovelli available url http://www.livingreviews.org/articles/volume1/1998-1rovelli rovelli c. notes for a brief history of quantum gravity gr - qc/0006061 motz l. and weaver j. the story of mathematics , 1993 ( new york : avon books ) szab l. removing the last obstacle to the einstein - fine resolution of the epr - bell problem quant - ph/0002030 t hooft g. dimensional reduction in quantum gravity gr - qc/9310026 casadio r. and harms b. black hole evaporation and large extra dimensions hep - th/0004004 krasnoholovets v. and ivanosky d. motion of a particle and the vacuum , physics essays , vol . 4 , pp . 554 - 563 ( 1993 ) quant - ph/9910023 jaekel m. and reynand s. movement and flutuations of the vacuum , rept.prog.phys . 60 ( 1997 ) 863 - 887 quant - ph/9706035 haisch b. on the relation between zero - point - field - induced inertial effect and the einstein - de broglie formula , phys.lett . a268 ( 2000 ) 224 - 227 gr - qc/9906084 ansoldi s. aurilia a. and spallucci e. loop quantum mechanics and the fractal structure of quantum spacetime , chaos solitons fractals 10 ( 1999 ) 197 hep - th/9803229 sasakura n. an uncertainty relation of space - time , prog.theor.phys . 102 ( 1999 ) 169 - 179 hep - th/9903146 bostm k. regaining time symmetry in the generalized quantum mechaincs of the brussels school quant - ph/0005024 berndl k , et al . a survey of bohmian mechanics , nuovo cim . b110 ( 1995 ) 737 - 750 quant - ph/9504010 loinger a. on the concept of mass point in general relativity gr - qc/0006033 loninger a. on the propagation speed of wavy metric tensors gr - qc/0007048 crawford d. a possible explanation for the anomalous acceleration of pioneer 10 astro - ph/9904150 tureyshev s , et al . the apparent anomalous long - rang acceleration of pioneer 10 and 11 gr - qc/9903024 stvang d. assumption of static gravitational field resulting in an apparently anomalous force gr - qc/9910054 einstein a. the meaning of relativity , 1921 ( new york : mjf books ) long j. experimental status of gravitational - strength forces in the sub - centimeter , regime nucl.phys . b539 ( 1999 ) 23 - 34 hep - ph/9805217 modanese g. theoretical analysis of a reported weak gravitational shielding effect . hep - th/9505094 wang l. kuxmich a. and dogarlu a. `` gain - assisted superluminal light propagation . 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in this letter recent developments are shown in experimental and theoretical physics which brings into question the validity of general relativity . this letter emphasizes the construction of a fractal @xmath0 spacetime , in @xmath1-dimensions in order to formalize a physical and consistent theory of ` quantum gravity . ' it is then shown that a ` quantum gravity ' effect could arise by means of the strong equivalence principle . which is made possible through a pressure of the form @xmath2 . where it is seen that nuclear pressures can be added to the gravitational field equations by means of twistor spaces . * keywords : * fractal geometry , new relativity , quantum vacuum , epr , zero - point field , quantum gravity , mach s principle , holographic principle , fermat s last theorem , alternative gravity , bohmian mechanics . pacs numbers . 4.50 , 4.60

You are an expert at summarizing long articles. Proceed to summarize the following text: massive stars have a strong impact on the evolution of galaxies . o type stars and their descendants , the wolf rayet stars , are the main source of uv photons , mass , energy and momentum to the interstellar medium . they play the main role in the ionization of the interstellar medium and dust heating . the milky way is the best place to access , simultaneously , massive stellar populations and their impact on the surrounding gas and dust . the sun s position in the galactic plane , however , produces a heavy obscuration in the optical window ( @xmath5 mag ) toward the inner galaxy , where massive star formation activity is the greatest . shifting to longer wavelengths , the perspective is much better , especially in the near infrared , because these wavelengths are long enough to lessen the effect of interstellar extinction ( @xmath6 mag ) and still short enough to probe the stellar photospheric features of massive stars @xcite . the study of giant h ii regions ( gh ii ) in the near infrared can address important astrophysical questions such as : 1 . characterizing the stellar content by deriving the initial mass function ( imf ) , star formation rate and age ; 2 . determining the physical processes involved in the formation of massive stars , through the identification of ob stars in very early evolutionary stages , such as embedded young stellar objects ( ysos ) and ultra compact h ii regions ( uch ii ) ; and 3 . tracing the spiral arms of the galaxy by measuring spectroscopic parallaxes of zero age main sequence ob stars . the exploration of the stellar content of obscured galactic gh ii regions has been studied recently by several groups : @xcite , @xcite , @xcite . in particular , blum and collaborators presented near infrared imaging and spectroscopic observations of three optically obscured gh ii regions : w43 , w42 and w31 . these observations revealed massive star clusters at the center of the h ii regions which had been previously discovered and studied only at longer wavelengths . in this work , we present results for ngc3576 ( g291.3 - 0.71 ) , located at a kinematic distance @xmath7 kpc , which we adopted from @xcite , after correcting for the standard galactic center distance ( @xmath8 kpc ) . ngc3576 appears in the visible passbands as a faint h ii region , but in the infrared it is among the most luminous in our galaxy @xcite . in fact , with 1.6 x 10@xmath9photons s@xmath10 inferred from the radio data , it can be classified as a gh ii ( following the suggestion of r. kennicutt for sources brighter than @xmath11 lyman continuum , @xmath12 @xmath13 , photons per second , private communication ) . a gh ii region has at least ten times the luminosity of the orion nebula and roughly the number emitted from the hottest single o3-type star , thus implying multiple hot stars . ngc3576 was observed in the radio continuum by @xcite . @xcite , @xcite , @xcite , and @xcite have detected radio recombination lines . maser sources have also been detected in the region : ch@xmath14oh @xcite and h@xmath15o @xcite . the detection of intense emission in the 10 @xmath2 m window , h@xmath15o masers , and the compact thermal emission in the radio are typical indications of the primitive stages of star formation and of a dense circumstellar environment . photometry from 1 to 2.5 @xmath2 m of the brightest sources was performed by moorwood & salinari ( 1981 ) and by @xcite showing that the spectral energy distributions of these objects suggests that they have excess emission . an intense co j=21 line at 230 ghz was observed by @xcite in the core region of ngc3576 . in the present paper , we present an investigation of the stellar content of ngc3576 through the @xmath16 , @xmath17 and @xmath18 imaging and @xmath1band spectroscopy ( described in 2 ) . in 3 we discuss our results , and our conclusions are summarized in 4 . @xmath16 ( @xmath19 @xmath20 1.3 @xmath2 m , @xmath21@xmath19 @xmath20 0.3 @xmath2 m ) , @xmath17 ( @xmath19 @xmath20 1.6 @xmath2 m , @xmath21@xmath19 @xmath20 0.3 @xmath2 m ) and @xmath18 ( @xmath19 @xmath20 2.1 @xmath2 m , @xmath21@xmath19 @xmath20 0.4 @xmath2 m ) images of ngc3576 were obtained on the nights of 1999 march 3 and 4 and 2000 may 19 and 20 with the f/14 tip tilt system on the cerro tololo interamerican observatory ( ctio ) 4-m blanco telescope using the facility imager osiris ) and on the nights of 1998 july 9 to 13 with the facility imager cirim ) mounted on the 1.5-m telescope . spectroscopic data were obtained with the blanco telescope and the facility near infrared spectrometer , irs , in 1998 may 17 and june 23 , and with osiris using the f/14 tip - tilt system in 1999 march 34 , 1999 may 23 , and 2001 july 7 and 12 . osiris delivers a plate scale of 0.16@xmath22/pixel , the irs 0.32@xmath22/pixel , and cirim 1.16@xmath22/pixel . all basic data reduction was accomplished using iraf . each image was flat - fielded using dome flats and then sky subtracted using a median - combined image of five to six frames . independent sky frames were obtained 510@xmath23 south of the ngc3576 cluster . the osiris 1999 march images were obtained under photometric conditions . total exposure times were 180s , 45s and 45s at @xmath16 , @xmath17 and @xmath18 , respectively . the individual @xmath16 , @xmath17 and @xmath18 frames were shifted and combined . these combined frames have point sources with fwhm of @xmath20 0.61@xmath24 , 0.88@xmath24 and 0.64@xmath24 at @xmath16 , @xmath17 and @xmath18 , respectively . dophot @xcite photometry was performed on the combined images . the flux calibration was accomplished using standard star gspc s427-d ( also known as [ pmk98 ] 9123 ) from @xcite which is on the las campanas observatory photometric system ( lco ) . the lco standards are essentially on the cit / ctio photometric system @xcite , though color transformations exist between the two systems for redder stars . no transformation exists between osiris and either cit / ctio or lco systems . the standard observations were made just after the ngc3576 data and within 0.24 airmass from the target . no corrections were applied for these small difference in airmass . aperture corrections measured inside 20 pixel radius circles were used to put the instrumental magnitudes on a flux scale . six uncrowded stars on the ngc3576 images were used for this purpose . since the brightest stars in the 1999 images were saturated , we have taken short exposure images in may 2000 . although the conditions in may 2000 were non - photometric , we used stars in common with the 1999 images to determine the zero point for the additional ( bright ) stars . uncertainties for the @xmath16 , @xmath17 and @xmath18 magnitudes in 1999 images include the formal dophot error added in quadrature to the error in the mean of the photometric standard and to the uncertainty of the aperture correction used in transforming from the dophot photometry to osiris magnitudes . the sum in quadrature of the aperture correction and standard star uncertainties are @xmath250.032 , @xmath250.034 and @xmath250.069 mag in @xmath16 , @xmath17 and @xmath18 , respectively . the scatter in the instrumental magnitudes in the set of stars from the 1999 images used to calibrate the may 2000 images are @xmath250.010 ( @xmath16 ) , @xmath250.012 ( @xmath17 ) and @xmath250.010 ( @xmath18 ) mag ; thus the errors in the bright star magnitudes are dominated by the uncertainty in the standard stars . the dophot errors were larger than @xmath25 0.01 mag , and we adopted a cutoff for errors larger than 0.05 mag , which corresponds to a limiting magnitude of @xmath26 @xmath27 15.0 . lower angular resolution images were obtained at @xmath16 , @xmath17 and @xmath18 using cirim at f/8 on the ctio 1.5-m telescope(1.16@xmath24 pixel@xmath10 ) . the individual frames in each filter were shifted and combined and have measured seeing of 2.2@xmath24 fwhm . although collected under photometric conditions , these images are not as deep as that of the 4-m telescope and were used only to transform to equatorial coordinates , since they encompassed a wider field than the osiris images taken at the 4-m telescope . the @xmath1band spectra of eight of brightest stars in the ngc3576 cluster were obtained : # 11 , # 48 , # 69 and # 160 with the irs and # 4 , # 48 , # 78 , # 95 and # 184 with osiris . the spectra were divided by the average continuum of sevaral a type stars to remove telluric absorption . the br@xmath28 photospheric feature was removed from the average a type star spectrum by eye by drawing a line between two continuum points . one dimensional spectra were obtained by extracting and summing the flux in @xmath25 2 pixel aperture . the extractions include background subtraction from apertures , 1 - 2@xmath22 on either side of the object . the wavelength calibration was accomplished by measuring the position of bright oh@xmath29 lines from the @xmath1band sky spectrum @xcite . the spectral resolution at 2.2 @xmath2 m is @xmath19/@xmath21@xmath19 @xmath20 3000 for osiris and @xmath19/@xmath21@xmath19 @xmath20 825 for the irs . the osiris @xmath16 , @xmath17 and @xmath1band images reveal an embedded star cluster . we detected 315 stars in the @xmath1band to a limiting magnitude of 16 ( see below ) in a region of 0.03 square degrees . figure [ finding ] shows a finding chart using the @xmath1band image . a false color image is presented in figure [ color ] , made by combining the three near infrared images and adopting the colors blue , green and red , for @xmath16 , @xmath17 , and @xmath18 , respectively . in this way , the bluest stars are likely foreground objects , and the reddest stars are probably @xmath1band excess objects , indicating the presence of hot dust for objects recently formed in the cluster ( background objects seen through a high column of interstellar dust would also appear red ) . the diffuse nebula is mainly due to br@xmath28 emission in the h ii region . the dark patches in the bottom right of figure [ color ] are zones of the giant molecular cloud from which ngc3576 is emerging . there is no doubt that this is a signature of a young cluster containing massive stars , now in the process of shredding the local molecular cloud . the cluster is asymmetric , with the majority of the stars encompassed in a semi circle ; and there is a definite appearance that the cluster is destroying the cloud from the ne to the sw . the ne is also the direction toward which the h ii region is seen in the visual . the @xmath30 _ versus _ @xmath18 color magnitude diagram ( cmd ) is displayed in figure [ cmd ] . the open circles indicate objects fainter than @xmath31 . a concentration of dots appear around @xmath32 , probably indicating the average color of cluster members . a number of stars display much redder colors , especially the brightest ones . the solid vertical line indicates the theoretical main sequence ( see below ) . the @xmath33 _ versus _ @xmath30 color color plot is displayed in figure [ ccd ] . open circles indicate objects fainter than @xmath34 . open triangles indicate stars fainter than @xmath31 and @xmath34 . the numbers labeling stars in both plots refer to the same objects . the inclined lines , from top to bottom , indicate interstellar reddening directions for main sequence m type @xcite , o type @xcite and t tauri @xcite stars . stars to the right of the solid line deviate from pure interstellar reddening , probably because of hot dust emission . the effect of this excess emission is stronger in the @xmath1band than at shorter wavelengths . although the open circles and triangles in the bottom right of figure [ ccd ] indicate only lower limits , these objects are also likely affected by thermal emission . they are bright in the @xmath1band and should be detected in @xmath16 and/or @xmath17 if affected only by interstellar reddening and if they were cluster members with typical extinction . the cluster characteristics indicate that the stars havent had enough time to evolve away from the main sequence . we expect that most of the high mass stars are close to the zero age main sequence ( zams ) . we can estimate the reddening toward the cluster from a simple approximation @xcite @xmath35 @xmath27 1.6@xmath36 and using the fact that the average intrinsic color of hot stars is almost zero @xcite . the stars brighter than @xmath37 have an average color of @xmath38 , corresponding to @xmath39 = 1.57 mag ( @xmath40 @xmath20 15.7 mag ) . the interstellar component of the reddening can be separated from that local to the cluster stars by using the star hd 97499 . this is a foreground star , since it is brighter and less reddened than stars in the cluster and is offset from the radio source line - of - sight by 2 arc - minutes . from its spectral type b1 - 2iv - v _ michigan spectral catalogue _ - @xcite and magnitude , a distance of 2.4 kpc was derived @xcite . our measurements of this star result in @xmath41 , indicating a@xmath42 . this gives a@xmath43mag kpc@xmath10 , which is close to the expected extinction for this position along the galactic plane . for 2.8 kpc distance of ngc3576 , the interstellar component is then a@xmath44 , leaving a local component of a@xmath45 . we were not able to independently check the distance derived by depree ( 1999 ) via spectroscopic parallax ( see below ) , but we expect that the radio kinematic distance is reliable for this galactic direction . in order to place the zams in the cmd , the corresponding bolometric magnitudes ( @xmath46 ) and effective temperatures ( @xmath47 ) must be transformed into @xmath30 colors and apparent @xmath18 magnitudes . @xcite models for the zams with solar abundances were adopted . the bolometric corrections ( bc ) applied to derive absolute visual magnitudes ( @xmath48 ) are from @xcite for spectral types @xmath49 to @xmath50 and from @xcite for later spectral types . the intrinsic colors ( @xmath51 , @xmath52 , and @xmath53 ) are from @xcite . using the distance modulus ( 12.24 ) and the apparent visual magnitude , we transform the @xmath46 into apparent @xmath54 magnitudes . the correspondence between spectral types and @xmath47 are from @xcite for @xmath49 to @xmath50 and from @xcite for later spectral types . since koornneef s colors are in the johnson system , which is nearly identical to saao system @xcite , we used the saao to cit / ctio relations to transform koornneef s @xmath30 color indices to the cit / ctio system . these corrections are about 1% and could be neglected when compared with the photometric errors and differential reddening . the zams is represented by a vertical solid line in figure [ cmd ] , shifted to @xmath55 kpc and reddened by a@xmath56 = 0.43 due to the interstellar component . when adding the average local reddening ( a@xmath57 , the zams line is displaced to the right and down , as indicated by the dashed lines . we can not fix the position of the zams , since there is a scatter in the reddening . the small group of relatively bright stars ( @xmath58 ) in between these two lines , suggests that some of them , the bluer ones , could mark the position of the zams . unfortunately , we do nt yet have spectra of these objects to check whether or not this is true . objects to the right of the o - type stars line ( figure [ ccd ] ) have colors deviating from pure interstellar reddening . this is frequently seen in young star clusters and is explained by hot dust in the circumstellar environment . we can estimate a lower limit to the excess emission in the @xmath1band by supposing that the excess at @xmath16 and @xmath17 are negligible , and that the intrinsic colors of the embedded stars are that of ob stars . indeed , assuming that our sample of stars is composed by young objects ( not contaminated by foreground or background stars ) , any object would have an intrinsic color in the range @xmath59 mag @xcite . let us adopt for all objects in our sample the intrinsic colors of a b2 v star : @xmath60 and @xmath61 @xcite . the error in the color index would be smaller than the uncertainty in the mathis law , we are using for the interstellar extinction . > from the difference between the observed @xmath33 and the adopted b 2 v star we obtain the color excess and by using the relation ( @xmath62 ) we derive the interstellar extinction in the @xmath63band and so the intrinsic apparent magnitudes @xmath64 , @xmath65 and @xmath54 . > > from another relation of mathis law ( @xmath66 ) , we derive the magnitude at @xmath1band corrected from the interstellar extinction . the difference between this number and @xmath54 gives the excess emission in the @xmath1band , due to dust thermal emission . for stars not detected in the @xmath63band we suppose they are affected by an interstellar extinction @xmath67 equal to the median value of those measured in the @xmath63band . in this way , we derive the extinction in the @xmath68band by using mathis law in the form : @xmath69 . from here the procedure to derive the @xmath1excess magnitude follows the same steps as we did before . for stars not detected in the @xmath68band , we still want to estimate the @xmath1excess , since potentially interesting objects are too red to be detected in the @xmath68band . a lower limit for this excess can be derived by using the same procedure as above , but assigning a limiting magnitude @xmath70 for objects not detected in the @xmath68band . our results are displayed in figure [ kexc ] . objects with very large excess in the upper right corner of figure [ kexc ] , can not be explained by errors in the de reddening procedure and could be real . they could represent the emission of accreting disks around the less massive objects of the cluster . in order to separate the cluster members from projected stars in the cluster direction , we imaged a region close to ngc3576 . the star counts were normalized by the relative areas projected on the sky and then binned in intervals of @xmath71 and @xmath72 . the stellar density in the field was then subtracted from that of the cluster in bins of magnitude and color intervals . this works well for foreground stars , since there were a few in the cluster field . regarding the background , the situation is more complex , however , since ngc3576 produces so high an obscuration , almost no background objects can be seen at this limiting magnitude . the completeness of dophot detections was determined through artificial star experiments . this was accomplished by inserting fake stars in random positions of the original frame , and then checking how many times dophot retrieved them . the psf of the fake star was determined from an average of real stars found in isolation and in areas of dark sky . in total , we inserted 2400 stars in the magnitude interval 8 @xmath73 k @xmath73 20 , which amounts to six times the number of real stars recovered in the original dophot run . for every @xmath74 we inserted simultaneously 5 stars , repeating the procedure for 20 times . the insertion of all the 100 stars at once would impact the stellar crowding and change the detection conditions . the incompleteness of the sample is defined as the percentage of times the fake star fails to be recovered . we performed these experiments in the whole frame and also in each of the three sub images that were cut out from it , displayed in figure [ finding ] . the upper right sub image is representative of detection limited by photon statistics , the one at the center , by high background and the lower left one , by stellar crowding . in figure [ completeness ] we present the photometric completeness . the _ dashed _ line ( without symbols ) refers to the whole image . the limit is different for different sub images , e.g. for an area with high nebular background ( _ circles _ ) , for a crowded area ( _ squares _ ) , or for an area with few stars and a dark sky ( _ triangles _ ) . the performance of the photometry is better than 90% for a 15th magnitude star found in isolation , as compared to stars in the nebular zone which need to be ten times brighter to be detected with the same efficiency since there is no objective way to define the sub - image limits we applied a single completeness correction to the whole frame ( dashed line ) . as seen in figure [ completeness ] such a correction is close to the curve limited by crowding . future work seeking to obtain deeper photometry in this cluster demands a substantial improvement of the spatial resolution and will require adaptative optics imaging from ground based telescopes . after correcting for non - cluster members , interstellar reddening , excess emission ( a lower limit ) and photometric completeness , the resulting @xmath1band luminosity function ( @xmath75 ) is presented in figure [ klf ] . a linear fit , excluding deviant measures by more than @xmath76 , has a slope @xmath77 . a similar klf slope was obtained for w42 ( @xmath78 ) by @xcite . we can evaluate the stellar masses by using @xcite models , assuming that the stars are on the zams and not the pre - main sequence ( but see below the discussion in 3.2 ) . this is a reasonable approximation for massive members of such a young cluster . the main errors in the stellar masses are due to the effects of circumstellar emission and stellar multiplicity . our correction to the excess emission is only a lower limit , since we assumed the excess was primarily in the @xmath18 band . @xcite have computed disk reprocessing models which show the excess in @xmath16 and @xmath17 can also be large for disks which reprocess the central star radiation . in general , we can expect the excess emission to result in an overestimate of the mass of any given star and the cluster as a whole . the slope of the mass function should be less effected . it is difficult to quantify the effect of binarity on the imf . if a given source is binary , for example , its combined mass would be larger than inferred from the luminosity of a `` single '' star and its combined ionizing flux would be smaller . the cluster total mass would be underestimated , the number of massive stars and the ionizing flux would be overestimated . the derived imf slope would be flatter than the actual one . with these limitations in mind we have transformed the klf into an imf . since other authors also do not typically correct for multiplicity , our results can be inter compared , as long as this parameter does nt change from cluster to cluster . the @xmath79 slope derived for ngc3576 is @xmath4 ( figure [ imf ] ) , which is consistent with salpeter s slope @xcite . a similar imf slope was obtained for the trapezium cluster ( @xmath80 ) by @xcite . flatter slopes have been reported only for a few clusters , most notably the arches and quintuplet clusters @xcite , both near the galactic center . flatter slopes may indicate that in the inner galaxy star forming regions , the relative number of high mass to the low mass stars is higher than elsewhere in galaxy . it is also possible that dynamical effects may be more important in the inner galaxy . @xcite have modeled the arches cluster data with a normal imf , but include the effects of dynamical evolution in the presence of the galactic center gravitational potential . they find the observed counts are consistent with an intial salpeter like imf . we derived an upper limit to the total mass of the cluster ( our imf is likely overestimated due to excess emission , see above ) by integrating the @xmath79 between @xmath81 - where the distribution is nearly continuous . the lower mass limit was adopted from @xcite taking into account the @xmath79 turnover measured by those authors in orion . the integrated cluster mass is m@xmath82 = 5.4 x 10@xmath83 m@xmath84 . as pointed out above , this is likely an upper limit . source # 48 is anomalously bright and needs to be treated separately . let us assume that it is affected by an interstellar reddening equal to the cluster average : a@xmath85 , which implies a derredned magnitude k@xmath86 . if we take at face value the @xmath87 color to represent only an excess at @xmath18 , the reprocessing disk would contribute with @xmath88 . this excess emission is clearly an underestimation , since the stellar flux is swamped by the disk emission to the point of veiling all the photospheric lines ( see below ) . moreover , if the excess emission was so small , the luminosity of # 48 would require a cluster of four 100 m@xmath84 stars , unresolved down to a limit of 0.6@xmath22 , which does nt seem to be the case . if object # 48 is a single o3 v star , it would contribute with nlyc @xmath12 1.17 @xmath25 0.05 x 10@xmath9 s@xmath10 . alternately , we can evaluate the excess emission by using @xcite models for reprocessing disks . by starting with the maximum excess emission @xmath89 valid for a o7-type star ( their table 4 ) we derive the m@xmath90 . using vacca et al . ( 1996 ) calibration , we obtain the corresponding stellar spectral type ( and mass ) that is much smaller than the o7-type we started with . the next step is reducing the excess emission ( adequate for a smaller stellar luminosity ) , deriving a larger final mass , and iterating until convergence . this was achieved for an excess emission @xmath91 , corresponding to a spectral type b1 v and mass m @xmath12 17 m@xmath84 . this is possibly a lower limit , since the intervening extinction toward # 48 is probably larger than the cluster average . we can say that # 48 is a late o / early b / early b yso , very similar to what has been found by @xcite in w31 for the brightest @xmath1band object in that cluster . object # 48 is buried in a large and dense disk , and its contribution to the cluster mass and ionizing photons are negligible . we have compared the locations of the @xmath1band sources in our images with the mid infrared sources of @xcite . source # 48 is very close to their irs1 and # 11 to their irs3 . these sources appear also in the iras small scale structure catalogue ( x1109 - 610 ) and the iras point source catalogue ( iras11097 - 6102 ) . in the case of irs1 , at least , recent high resolution mid - infrared images @xcite clearly indicate that it is associated with our object # 50 , 1@xmath22 to the south of the @xmath1band source # 48 . the number of lyman continuum photons derived from the imf , excluding object # 48 is nlyc @xmath12 0.42 @xmath25 0.22 x 10@xmath9 s@xmath10 . the contribution of this single object could be as large as nlyc @xmath12 1.17 x 10@xmath9 s@xmath10 , in the case it is an o3 v star . this is very close to the nlyc @xmath12 1.6 @xmath25 0.4 x 10@xmath9 s@xmath10 derived from radio observations @xcite scaled to a distance of 8 kpc to the galactic center . however , we have shown in the preceeding sub - section , object # 48 probably is a much less massive object , a b1 v star . it can be seen that the nlyc is highly sensitive to the particular procedure used to correct for the excess emission and extinction . very probably we have missed a handful of main sequence o - type stars responsible for the ionizing flux seen at radio wavelengths . moreover , the spectra of the eight brightest stars described in the next section , indicate that the ionizing stars must be apparently faint . the properties of the cluster are summarized in table 1 . the stellar cluster is located in the ne border of a molecular cloud . the stellar density increases toward the sw , ending abruptly , with a few sources embedded in the molecular cloud ( figure [ color ] ) . the spatial distribution of color indices ( figure [ cc_map ] ) also shows a similar gradient . this is due , in part , to the increasing extinction toward the inner molecular cloud . however , the red colors are intrinsic to many of the sources , since there are excess emission objects in this zone . this suggests that stars at the sw are younger than at ne , since recent models by @xcite predict formation times of the same order for stars of different masses . a similar scenario for this cluster , by which star formation is progressing toward the inner zones of the molecular cloud , has been suggested by @xcite and @xcite and our images dramatically confirm this to be the case ; see figure [ color ] . spectra of eight cluster members ( # 4 , # 11 , # 48 , # 69 , # 78 , # 95 , # 160 and # 184 ) , are shown in figure [ co ] and figure [ ftless ] . spectra of a foreground m - type and an a - type stars were added at the top of the figures for comparison . the a - type star also was divided by the average continuum slope of the other observed a - type stars . object labels are the same as in figure [ finding ] ; [ cmd ] ; and [ ccd ] . ordinates in figure [ co ] and figure [ ftless ] are normalized fluxes , as follows . we constructed templates for telluric absorption bands by observing a - type stars , close in time and airmass to the target stars . we removed the br@xmath28 line from those spectra by hand ( linear interpolation ) , since this region is free from telluric features . there are no other noticeable photospheric features in a - type stars in the @xmath1band . then we divided the spectrum of each target by the appropriate a - type spectrum and normalized the resulting spectrum at 2.19 @xmath2 m . the signal to noise ratio is s / n @xmath92 for # 4 , # 11 and # 95 and @xmath93 for the other objects . the spectra were placed on a flux scale by dividing by @xmath94 and multiplying by @xmath1band fluxes corresponding to the @xmath1magnitudes in table 2 . however , it is straight forward to compare the observered spectra ratioed only by the a type continuum . it can be seen that all the cluster member candidates display rising continuua to the red when compared to the a type and m type foreground stars . these eight stars are thus most likely cluster members , but we can not rule out that some may be background stars . we must take into account that these objects are projected toward the central part of the cluster and thus subject to high obscuration ( a@xmath85 ) due to the intra cluster gas and dust . objects # 48 , # 95 , and # 160 are undoubtedly cluster members , since , in addition , they have excess emission . object # 69 is bright in the @xmath17 and @xmath18 bands and judging from its @xmath30 color , it should be relatively bright in the @xmath63band , but it is nt detected at all in the @xmath63band osiris images , appearing above the reddening line in figure [ ccd ] . the spectrum of this object in figure [ ftless ] looks like its neighbor # 78 , that is 1.5 magnitude fainter and still is detected in the @xmath63band . this puzzling situation was clarified when we took a @xmath1band acquisition image ( feb/2002 ) with phoenix at gemini under very good seeing ( @xmath27 0.250.3 ) . in that image , object # 69 is shown as a small nebula , with no sign of a buried point - like source . object # 69 is a clump of dense material ionized by object # 48 or some other neighboring source . the h@xmath95 2.122 @xmath2 m emission could be either shock or ionization produced . the narrow features at 2.058 ( ) and 2.166 @xmath2 m ( br@xmath28 ) in figures [ co ] and [ ftless ] are due to contamination from the extended nebula . the spectra were extracted in such a way that the large scale nebular component is over subtracted . in some objects these lines are in emission , due to enhanced nebular emission close to the star . objects # 48 , # 160 , # 78 and # 69 show h@xmath95 2.122 @xmath2 m in emission . this feature appears is emission also in # 11 and # 95 , but since the he i and br@xmath28 components are not over subtracted , this feature may be due to contamination of extended nebular emission . the co bandhead at 2.2935 @xmath2 m is in absorption in # 4 , # 160 , # 184 and in emission in # 48 . none of those objects show photospheric lines indicating that they are still enshrouded in their birth cocoons . this is corroborated by the excess emission in the @xmath1band derived from photometry except for objects # 4 and # 184 ( table 2 ) . a variety of mechanisms and models have been proposed to explain the origin of co emission in ysos . these include circumstellar disks , stellar or disk winds , magnetic accretion mechanisms such as funnel flows , and inner disk instabilities similar to those which have been observed in fu orionis like objects and t tauri stars in a phase of disk accretion @xcite . @xcite shows that gas free - falling along the field lines yield the bandhead profiles , in agreement with those observed , with the shape of the profile determined mainly by inclination of the disk to the line of sight . hanson et al . ( 1997 ) reported the presence of co in emission in several masssive stars in m17 . the situation is less clear for objects # 4 , # 160 and # 184 displaying co in absorption . the absence of large color excess indicates that they could be cool pre - main sequence stars still in a contraction phase . while we can not rule out that they are evolved background m - type stars , they do appear projected on the core of the newly formed cluster . these objects deserve further study , and if they are indeed pre - mainsequence stars , then the imf determination above has a component whose masses have been overestimated . the h@xmath95 molecular emission is produced by shocks and may indicate the existence of gas outflow . since we subtracted the extended background close to the stars , the spectra show only the spatially unresolved component of h@xmath95 . we can thus be confident that h@xmath95 emission in # 48 and # 69 are emitted close to the stellar sources . however , since these two sources are only @xmath96 apart , the emission could be associated with either or both objects . it may be surprising that most of the stars with featureless spectra are close to the interstellar reddening line . however this is in accord with the hillenbrand et al . models ( 1992 ) for early type stars , which predict relatively small color excesses ( @xmath97h - k@xmath98 ) for objects with large excess emission ( @xmath99 ) . such models do not predict large departure from the reddening line , like displayed by objects # 160 and # 50 . those sources might be surrounded by local dusty clouds , in addition to the accreting disk . we have presented deep @xmath16 , @xmath17 and @xmath18 images of the newborn stellar cluster in ngc3576 ( figure [ color ] ) and @xmath1band spectra for eight cluster members . the @xmath1band excess emission displayed by objects # 4 , # 11 ( irs3 ? ) , # 48 , # 69 , # 78 , # 95 , # 160 and # 184 , in combination with their featureless continuum or co emission / absorption ( figure [ co ] ) , indicates that they are young , massive stars still in the process of accreting material from their birth cocoons . the lack of photospheric features and presence of disk signatures indicates that ngc3576 is one of youngest massive star clusters in the milky way . our data also confirm the scenario of star formation progressing from the ne toward the inner parts of the molecular cloud ( sw ) , but the very young age of the cluster may contradict the claim of enhanced he abundance in the ne part of the gh ii region by @xcite . there are no evolved stars in the cluster which could produce the enhancement and the time to diffuse nuclear processed material from neighboring regions into the nebular environment would be much larger than the cluster age . the fact that the nebular excitation increases toward the ne , producing stronger he lines , may have influenced the abundance calculations . since our data do not enable us to derive spectroscopic parallaxes ( no photospheric lines were detected in the luminous stars ) , we have adopted the radio distance obtained by @xcite revised to 2.8 kpc by using the most recent galactic center distance ( @xmath8 kpc , @xcite ) . the cluster parameters are not well constrained because of distance uncertainty and the difficulty in correcting the @xmath1band magnitude for the circumstellar dust emission and extinction . the overall picture summarized in table 1 of a massive and dense cluster remains valid . however , the ionizing flux derived from the imf is much smaller than that from radio observations . this is consistent with the fact that we havent found spectroscopically the massive main sequence stars that ionize the cluster . those stars probably remain behind heavily obscuring clouds . the fact that several of the brightest cluster members do not show revealed photospheres raises the question : where are the ionizing sources of ngc3576 ? it is plausible that some of the faint sources are in reality luminous objects seen through large extinction , and were not accurately dereddened because they escaped detection in the @xmath16 and @xmath68band . this may also explain why the number of lyman continuum photons derived from the imf is smaller than that measured at radio wavelengths . to tackle this question , we plan to obtain spectra from stars of @xmath100 and fainter using the gemini south 8m telescope . in order to understand better the circumstellar environment of the ysos we examined here , we have performed high resolution mid - infrared imaging with the gemini south telescope which will be analyzed in a future paper . our goal is to derive the characteristics of the circumstellar dust , in particular evaluating its contribution to the near infrared excess emission , extinction , and the possible presence of accreting disks . ll cluster distance & 2.8 @xmath25 0.3 kpc + cluster diameter & 1.5 pc + cluster mass & @xmath1015.4 x 10@xmath83 m@xmath84 + stellar density & @xmath1013.1 x 10@xmath83 m@xmath84 pc@xmath102 + imf slope & @xmath103 + nlyc phot . ( imf ) & 0.42 - 1.67 x 10@xmath9s@xmath10 + nlyc phot . ( radio ) & 1.6 @xmath25 0.4 x 10@xmath9s@xmath10 + lrcrrl # 4 & 1.51 & 0.93 & 9.95 & -0.01 & co abs + # 11 & @xmath1040.99 & 1.10 & 12.91 & -0.26 & + # 48 & 3.04 & 2.21 & 8.35 & -3.17 & h@xmath95 & co em + # 69 & @xmath1043.44 & 1.45&10.10 & -0.62 & h@xmath95 em + # 78 & 2.02 & 1.38&11.67 & -0.16 & h@xmath95 em + # 95 & 1.67 & 1.41 & 9.26 & -0.40 & + # 160 & @xmath1040.95 & 2.93 & 11.12 & -2.09 & co abs + # 184 & 1.23 & 0.81&10.40 & -0.07 & co abs +

we present deep , high angular resolution near infrared images of the obscured galactic giant h ii region ngc3576 . our images reach objects to @xmath0 . we collected high signal to noise @xmath1band spectra of eight of the brightest objects , some of which are affected by excess emission and some which follow a normal interstellar reddening law . none of them displayed photospheric features typical of massive ob type stars . this indicates that they are still enshrouded in their natal cocoons . the @xmath1band brightest source ( ngc3576#48 ) shows co 2.3 @xmath2 m bandhead emission , and three others have the same co feature in absorption . three sources display spatially unresolved @xmath3 emission , suggesting dense shocked regions close to the stars . we conclude that the remarkable object ngc3576#48 is an early b / late o star surrounded by a thick circumstellar disk / envelope . a number of other relatively bright cluster members also display excess emission in the @xmath1band , indicative of reprocessing disks around massive stars ( ysos ) . such emission appears common in other galactic giant h ii regions we have surveyed . the imf slope of the cluster , @xmath4 , is consistent with salpeter s distribution and similar to what has been observed in the magellanic cloud clusters and in the periphery of our galaxy .

You are an expert at summarizing long articles. Proceed to summarize the following text: the interaction between matter and radiation is one of the fundamental mechanisms shaping the distribution of the baryonic component in the universe , from stellar to cosmological scales . this process often couples wildly different scales and the large dynamical range makes accurate modelling difficult . a notable example is given by the reionization of cosmic hydrogen at redshift @xmath0 ( for a review , see , e.g. , meiksin 2009 ) . during this epoch , the hii regions generated by the first ionizing sources expand to intergalactic scales and overlap , leaving most of the universe highly ionized and re - heated by several thousand kelvin degrees . the brightest , high - redshift quasars , may have produced even larger hii regions before the end of reionization , with linear sizes extending up to a hundred comoving mpc . nonetheless , most of the evolution in the gas temperature and ionization state still happened on much smaller scales , i.e. within the ionization - fronts , whose sizes are comparable to the local photon mean - free - path just outside the hii regions , about three orders of magnitude smaller than the hii regions themselves for gas at mean cosmic density . even without considering such an extreme case in dynamical range , the numerical solution of the full radiative transfer equations still represents a big computational challenge . this is mostly due to two factors . the first is the high - dimensionality : seven variables are required for a full specification of the radiation field ( three spatial variables , two angular directions , photon frequency , and time ) . the second is the intrinsic `` non - locality '' of the problem : the radiation field at a given point is determined by the gas properties along the lines of sight towards all the sources of radiation ( including the diffuse medium itself ) . for these reasons , current numerical models rely on approximations aimed to decrease the problem dimensionality and , at the same time , the computational costs associated with the `` non - locality '' of the full radiation transfer ( e.g. , abel , norman & madau 1999 , gnedin & abel 2001 , maselli , ferrara & ciardi 2003 , razoumov & cardall 2005 , mellema et al . 2006 , rijkhorst et al . 2006 , ritzerveld & icke 2006 , whalen & norman 2006 ; for a review , see also iliev et al . 2006 and references therein ; more recent codes include , e.g. , trac & cen 2007 , semelin et al . 2007 , aubert & teyssier 2008 , altay et al . 2008 , pawlik & schaye 2008 , finlator et al . 2009 , petkova & springel 2009 ) . a widely used approximation is the so called `` method of characteristics '' which is particularly suited for cases where the radiation field is completely dominated by individual sources ( with no contribution from the radiative recombinations produced by the medium itself ) . in this scheme , the radiation field is determined simply from the column densities along a line of sight between the sources and a given point in the computational domain , discretized in cells ( or particles ) . in the `` long - characteristics '' flavour , the column densities are directly calculated with a single ray from the sources to the cell centre , summing up all the contributions cut through the cells in between . in the faster , but less accurate , `` short - characteristics '' version , the column densities are determined in an ordered way starting from the cells closer to the source and moving outwards after summing up ( using some interpolation scheme ) previous contributions . in both cases , sampling the radiation field with a single ray from the sources to the cell centre may be problematic in particular situations , i.e. when the cells are optically thick and in presence of strong density gradients . practically , the consequences may be the loss of photon conservation and numerical artifacts in the i - front shapes ( see , e.g. , mellema et al . 2006 ) . in a cosmological situation , e.g. for the reionization scenario described before , we would like to have an accurate method for the very typical case in which numerical resolution imposes optically thick cells . this may be obtained sampling the radiation field in a statistical , homogeneous way , using the widely used monte carlo techniques . in this case , a series of rays is randomly casted around the sources in order to obtain the right solid angle distribution , averaging the contributions to the radiation field within individual ( three - dimensional ) volume elements . the drawbacks of this method are the computational costs : many rays ( per cell ) are required to avoid statistical noise and the method becomes less and less efficient as one moves outwards from the sources ( but see , abel & wandelt 2002 ) . moreover , if the grid is composed by cells of different sizes , the required number of rays to be casted is determined by the size of the smallest cells , irrespectively of the volume fraction they actually occupy . in this work , we develop a novel approach aimed to obtain an accurate solution of the radiative transfer problem in cosmological situations in an efficient way : combining the monte carlo scheme with the `` cell - by - cell '' approach typical of the characteristic methods . as we mentioned at the beginning of this section , despite the large scales associated with , e.g. , the reionization process , the medium properties are very often determined within a small fraction of its volume , i.e. within the i - fronts . the basic idea is to concentrate the computational efforts onto this part of the computational volume , where they are actually needed . this is achieved thanks to a new algorithm which casts , with the correct solid angle distribution , a series of rays for each cell _ individually_. in this way , we are able to use an algorithm that adaptively determines : i ) the number of rays needed for a particular cell to achieve the convergence of the radiation field ( irrespectively of the cell size , or distance from the source ) , ii ) in which part of the volume to apply the ( expensive ) monte carlo ray - tracing . the result is that the computing time of the rt scales now with the number of cells to be evolved during a simulation - step , i.e. , typically the cells contained within the i - fronts , with a huge gain in computational speed ( and accuracy ) with respect to a classical monte carlo . with the new `` cell - by - cell '' approach we are also able to gain the full advantages of adaptive mesh refinement ( amr ) methods ( see , e.g. berger & oliger 1984 ) that increase the spatial resolution where needed . in particular , we implemented a scheme , based on the original clustering algorithm by berger & rigoutsos ( 1991 ) , to adaptively refine the mesh in correspondence of the i - fronts during the course of the simulation . this allows us to fully resolve the small scales associated with the i - fronts within the large , cosmological simulation boxes that follow the reionization process . resolving the i - fronts has a profound impacts for a large range of astrophysical applications , e.g. , to accurately predict the temperature state of the gas surrounding a bright quasar during reionization . in the validating test section ( section 4 ) , we will present an example of how resolving the i - front may be important in the determination of the gas temperature within the _ whole _ hii region . the paper is organized as follows . in section 2 , we review the basic radiative transfer equation . the computational method used by our new rt code , ` radamesh ` ( * * r**adiative - transfer on * * ada**ptive * mesh * ) , is presented in section 3 , in section 4 , we show the validating tests of the code . we conclude in section 5 . the cosmological radiative transfer equation in comoving coordinates ( e.g. , norman , paschos & abel 1998 ) is given by @xmath1 where @xmath2 is the monochromatic specific intensity of the radiation field , @xmath3 is a unit vector along the direction of propagation of the ray , @xmath4 is the ( time - dependent ) hubble constant , @xmath5 is the ratio of cosmic scale factors between photon emission at frequency @xmath6 and the present time t , @xmath7 denotes the opacity at frequency @xmath6 and @xmath8 is the source function of the medium . if the scale of interest @xmath9 is much smaller than the hubble radius , @xmath10 , ( as always in our case ) and the medium properties are changing on a time scale shorter than the light crossing time @xmath11 , equation ( [ generalrteq ] ) reduces to the classical , static radiative transfer equation : @xmath12 this equation admits the following solution , @xmath13 where @xmath14 is the optical depth along @xmath3 . this solution is in general adequate for cosmological simulations except on small distances from the sources ( or , equivalently , for very bright sources ) . in this case , the approximations break down allowing the ionization - fronts to expand faster than the speed of light . in section [ gammasec ] we discuss an approximate correction to solve this unphysical behaviour . it is often convenient to express @xmath15 as a sum of the attenuated , direct radiation from individual sources ( @xmath16 ) and the diffuse radiation ( @xmath17 ) generated within the medium : @xmath18 using equation ( [ ieq ] ) , the radiation field intensity can be specified at any point of the simulation box given the value of the optical depth to the individual sources and the source term of the medium . from the radiation field , it is then possible to derive for each species @xmath19 the photoionization rate ( per particle ) : @xmath20 and the gas photoheating rate ( per unit volume ) : @xmath21 where @xmath22 and @xmath23 are the frequency threshold and the cross - section for the ionization of species @xmath19 , respectively , and @xmath24 is the physical number density . analogously to eq . ( [ isplit ] ) , we can split @xmath25 and @xmath26 into the sum of direct ( @xmath27,@xmath28 ) and diffuse ( @xmath29,@xmath30 ) components . finally , @xmath31 and @xmath32 are used to compute the chemistry evolution of the neutral fraction of hydrogen ( @xmath33 ) , neutral helium ( @xmath34 ) , singly ionized helium ( @xmath35 ) , and the temperature ( expressed in terms of the total energy density @xmath36 ) according to the following rate equations : @xmath37 here @xmath38 , @xmath39 and @xmath40 are the temperature - dependent radiative recombination ( to all levels ) , collisional ionization , and dielectronic recombination coefficients , respectively ; @xmath41 is the number density of electrons , @xmath42 is the gas clumping factor for simplicity , we assume the same value of @xmath43 for each species in the current version of the code . , and @xmath44 is the total cooling rate ( including recombinations , collisional excitations , compton , brehmstrahlung and hubble cooling ) . we use the analytical fits of hui & gnedin ( 1998 ) for the ionization , recombination and cooling rates . one of the main challenges in the calculation of the photoionization / heating rate is represented by the presence of the diffuse term in eq . ( [ isplit ] ) , since it requires the transport of the diffuse photons generated from atomic recombinations in every point of the medium . a widely used solution to this problem is the so called `` on - the - spot '' approximation ( ots or _ case b _ , originally proposed by baker & menzel 1938 ) , which assumes that every diffuse , ionizing photon is absorbed exactly in the same point where it is generated ( i.e. , the photon mean - free - path is much smaller than the resolution scale ) . in this case , assuming for the moment a pure hydrogen medium , the diffuse part of the photoionization rate can be written as : @xmath45 where @xmath46 is the recombination coefficient to the hydrogen ground level , and @xmath47 is the analogous coefficient for the levels @xmath48 . substituting this relation in eq . ( [ fhieq ] ) , we obtain : @xmath49 i.e. , a relation that depends only on the photoionization rate from individual sources ( @xmath50 ) and , thus , greatly simplifies the radiative transfer problem . analogous relations can be obtained for hei and heii , assuming that every ionizing photon from recombinations is absorbed by the same species from which it originates ( obviously , this approximation breaks down if we consider that non - self - ionizing photons from heii recombinations can actually ionize both hei and hi , see section [ recradsec2 ] for a detailed discussion ) . as also noticed by ritzerveld ( 2005 ) , despite being widely used , the ots approximation is based on an incorrect argument : if the local mean - free - path is very small as assumed , i.e. , the local optical depth is very high for the diffuse photons generated locally , even higher the optical depth would be for the radiation coming from discrete , non - local sources ( unless their spectral energy distribution is much harder ) . for instance , in the classical strmgren sphere situation with a monochromatic source ( see test 1 below ) , the regime where the ots approximation may hold corresponds to regions where the directional flux from the central source is not able to penetrate , i.e. only at the extreme edge of the strmgren sphere . the opposite case ( called _ case a _ ) is represented by the situation in which the mean - free - path of the diffuse photons goes to infinity and , effectively , @xmath51 . despite the loss of photon conservation for a bounded region like a strmgren sphere , this approximation is actually more correct than the ots in most situations as we will show later . the computational volume in ` radamesh ` is discretized in a ( series of ) regular , block - structured grid(s ) where the physical properties of the medium are associated with a zone - centered grid element , i.e. with a _ cell_. the current possible choices for the computational domain in ` radamesh ` are : i ) single , uniform cartesian grid , ii ) _ static _ multi - mesh structure , iii ) _ evolving _ multi - mesh structure . in the case , the grid is spatially fixed to the initial , single or multi - mesh structure obtained , e.g. , from the output of an adaptive mesh refinement ( amr ) hydro - simulation . in the _ evolving _ case , this initial structure is adaptively refined ( unrefined ) each time the cells satisfy a chosen refinement ( unrefinement ) criterion . the refining procedure is described in details in section [ amrsec ] . the radiation field ( from individual sources and from diffuse radiation ) is discretized in a series of rays that are propagated through this single or multi - mesh structure using a ray - tracing algorithm . before discussing the ray - tracing method , it is necessary to describe more in detail the multi - mesh implementation in ` radamesh ` . multi - mesh domains in ` radamesh ` are composed of a nested hierarchy of rectangular grids of different sizes and levels of refinement , following the implementation called `` patch - based amr '' , originally described in berger & oliger ( 1984 ) . the position and aspect ratios of this patches are optimized in order to increase the spatial ( and temporal ) resolution where needed within the computational box . other possible multi - mesh implementations have been proposed in the past , like , e.g , the `` tree - based amr '' ( see khokhlov 1998 and references therein ) . in this case , each cell ( or @xmath52 group of cells in three - dimensions ; typically @xmath53 ) is refined into children cells , on a cell - by - cell basis , and a `` grid '' is the analogous of a single cell ( or a @xmath52 group of cells ) . these methods were originally developed for the numerical solution of hydrodynamical equation and they have advantages and disadvantages with respect to each other for this particular problem . in our case , patch - based amr presents a significant advantage with respect to other implementations : it reduces the total number of grids . one of the major problem presented by a ray - tracing algorithm is the fact that a significant part of the computational time may be spent in `` crossing '' the simulation box . this is in a sense unavoidable , because solving the radiative transfer equations is a highly non - local problem . optimizing the data and memory structure is thus fundamental in order to have an efficient algorithm . minimizing the number of `` crossing '' grids helps reducing the overhead associated with the ray - tracing in a multi - mesh structure . in order to increase the efficiency in `` crossing '' the computational volume , grids and cells in ` radamesh ` at different levels of refinement are connected with two separate hierarchical trees and linked lists . grids are strictly nested , i.e. , each grid at level @xmath54 is fully contained by a single grid at level @xmath55 ( the `` parent '' grid ) . note that , although this may increase the total number of grids , it allows a more efficient tree - search for both grids and cells . grids that share the same `` parent '' are connected via a linked list . thus , each grid @xmath56 at level @xmath54 may have a maximum of three associated pointers : i ) the `` parent '' grid at level @xmath55 , ii ) the `` next brother '' grid in the linked list at level @xmath54 , iii ) the `` son '' grid at level @xmath57 ( i.e. , the head of the linked list of grids fully contained within @xmath56 ) . grids without an associated `` son '' are called `` leaf '' grids . this hierarchical tree is a light structure that requires small amounts of memory ( if the number of grids is significantly smaller than the total number of cells , as it is always the case for `` patch - based amr '' ) and that can be reconstructed or saved easily given its strict nesting . if refined , a cell at level @xmath54 contains a group of @xmath58 subcells , where @xmath59 is the refinement factor . in principle , @xmath59 may vary with levels , although commonly is fixed to a constant value ( typically @xmath60 ) . analogously to the grid case , a cell without refinement is called a `` leaf '' cell . combined with the grid - tree , there is another simple hierarchical structure in ` radamesh ` associated directly with the cells : each refined cell at level @xmath54 points to the grid at @xmath57 that contains its @xmath58 subcells . this simple ( but more memory - consuming ) cell - tree is built efficiently when needed from the grid - tree and , for this reason , is not saved during output or back - up in order to save memory usage . the use of the cell - tree allows to recover very quickly the location of the relevant subgrid ( and thus the subcell ) at level @xmath57 when crossing the tree towards higher levels of refinement , without searching through the `` next brother '' linked list of the subgrids . when searching for cells at coarser ( or at the same ) level , the grid - tree is used instead . the combination of the grid - tree and the simple cell - tree represents a good balance between flexibility ( that translate into computational speed ) of the box crossing algorithm and memory consumption of the tree - structure . ray - tracing is a well - studied problem in computational geometry , with several efficient methods commonly used in computer graphics . one of the most efficient solution is the simple and fast grid - traversal algorithm of amanatides & woo ( 1987 ) , particular suited in the case of a single , uniform cartesian grid . basically , this simple algorithm determines the intersection points between the boundaries of the cells and a ray ( defined from its initial position and two direction angles ) as it traverses the computational grid . we employ the grid and cell hierarchical trees discussed above to extend this fast algorithm from the single , uniform grid to the multi - mesh structure . this is done simply adding to the original algorithm , after each cell boundary crossing , a ( recursive ) check for the need of changing the current traversing grid ( either at the same or at a different level ) . let us consider , for example , the case in which a ray is currently traversing a grid at level @xmath61 . there are two situations in which a change of grid is required : a ) the ray is still within the boundaries of the current grid but is entering a subgrid at level @xmath62 ; b ) the ray is exiting the boundaries of the current grid . we know that we are in the first situation simply checking the current cell tree : if the cell is associated with a subgrid , we immediately recover from the tree the memory location of the new grid at @xmath63 and , from the intersection coordinates , the corresponding cell within this grid . we recursively continue the search until we reach a `` leaf '' cell . in the second case , the search consists of two phases . first , we cross the grid tree towards the `` parent '' grids at level @xmath64 until we find a grid where the ray is fully contained ( and not at the grid boundaries ) or where the ray is _ _ entering _ _ the grid boundaries . note that usually this search is very fast given that , for a typical nested structure , we only need to go to the level @xmath65 . when the grid at @xmath64 is found , we check if the cell where the ray is currently located is a `` leaf '' . if this is not the case , we apply the same procedure as in case a ) . when traversing a `` leaf '' grid we can apply the original traversal algorithm without the need for the recursive check ( and change of grids ) described above . in this respect , the use of a `` patch - based amr '' , given its larger grids , result in a reduction of the computational costs of the traversing algorithm . ` radamesh ` is based on a ray - tracing algorithm that propagates the ( discretized ) radiation field through the computational volume . however , in the same spirit of the amr method , the key element in ` radamesh ` is not represented by the _ ray _ or photon package but , instead , by the computational _ cell_. in other words , the propagation of the radiation field ( and the solution of the radiative transfer equation ) is performed on a `` cell - by - cell '' basis rather than `` ray - by - ray '' . computationally , this translates into the change of the main loop in the algorithm , from rays to cells : the radiation field from the ( discrete and diffuse ) sources is propagated with a photon - conserving method separately for each cell in the computational volume . as we explain below , at the heart of the algorithm there is a new method that allows to draw a series of rays from a source , located at any position inside ( or outside ) the box , through a cubic cell with the _ correct _ solid angle distribution . this extends the `` monte carlo '' methods , where a series of rays is uniformly casted around a source to obtain the correct solid angle distribution , from the computational box to the cell level . this `` cell - by - cell monte carlo '' is a more natural approach when most of the physical evolution of the medium happens rapidly only in a small portion of the simulated box at a time . this is a typical case in cosmological simulation , either because of a difference in density ( e.g. , self - shielded dense clumps in a mostly ionized medium ) or in the ionized state ( e.g. , the expansion of an ionization - front in a mostly neutral igm during reionization ) with the cell - based approach , we can split the computational volume , either single or multi - mesh , grouping the cells with different evolving times or properties and focusing the computational efforts only where actually needed . this is obtained associating to each cell an individual time - step ( @xmath66 ) that is used to determine whether the cell can be considered _ active _ or not during the current simulation step . we discuss how we derive @xmath66 and how we define the _ active _ cell in section [ timestepsec ] . in detail , the computational algorithm consists of an iterative method divided into four main parts for each simulation step ( and for each _ active _ cell ) : i ) finding the ionization and photo - heating rates , ii ) choosing the time - step , iii ) solving the chemistry equation for the evolution of the medium properties , and , if needed , iv ) performing an adaptive refinement of the computational mesh . the first part of the algorithm is the most time - consuming ( and important ) part of the method and is described in detail in the following section . the photo - ionization ( @xmath31 ) and photo - heating rates ( @xmath26 ) are computed for each _ active _ , leaf cell ( during the current time - step ) with an iterative monte carlo procedure until convergence for both quantities is reached . the convergence level is typically set at 1% , but can be increased ( decreased ) depending on the requested level of accuracy . we compute separately , with two different methods , the contribution to the total photoionization / heating rate from discrete sources ( @xmath67 ) and from diffuse emission ( @xmath68 ) . most of the effort is dedicated to the calculation of ( @xmath67 ) , since in many ( cosmological ) cases the directional flux from discrete sources represents the major contribution to the total photoionization rate . the procedure to obtain the values of @xmath67 and @xmath69 is the following : for each iteration step , a packet of rays is propagated from selected points within the cell to the sources . these points are chosen with a monte carlo procedure based on a rejection algorithm that insures an uniform solid - angle distribution within the cell . the total flux is then re - scaled according to the fraction of the solid angle covered by the cell . we present in the appendix the analytical approximation that gives the solid angle of a ( cubic ) cell as seen by any point inside ( or outside ) the box . for each ray , we compute the hi , hei , and heii column densities both for the path length within the cell ( @xmath70 ) and for the path length connecting the cell edge to the source ( @xmath71 ) . the column densities are then used to calculate the frequency - dependent optical depths @xmath72 and @xmath73 . given the optical depths , we derive the probability that a photon with frequency @xmath6 is absorbed by the species @xmath19 within the cell : @xmath74 } { \sum\limits_{j=1}^{3}\{1-\mathrm{exp}[-\delta\tau_j(\nu)]\ } } \{1-\mathrm{exp}[-\delta\tau_i(\nu)]\}\ .\ ] ] the optical depths and the spectral energy distribution of the sources are sampled into @xmath75 ( logarithmically - spaced ) frequency bins . the probability distributions @xmath76 are used to compute the photoionization rate , including for the moment only the first term in eq . ( [ ieq ] ) , for each ray of the packet and for each source : @xmath77\ , \ ] ] where @xmath78 denotes the number of ionizing photons per unit time emitted by the source in the frequency bin @xmath79 . finally , the value of @xmath67 is obtained by averaging @xmath80 over the number of rays in the packet , the cell volume and the fraction of the solid angle covered by the cell ( see appendix ) . several packets of rays are generated until the value of @xmath67 converges to the required level . the procedure is repeated for each source and the single values of @xmath67 are added . the photoionization rates @xmath69 are obtained in a similar way , starting from @xmath81\ .\ ] ] in order to avoid redundant calculations of the column densities @xmath71 from the cell edges to the sources for each ray packet , we first evaluate @xmath71 on the cell vertexes `` visible '' from each source . if the difference between these values is less than a given ( small ) threshold , we do not use the full ray - casting algorithm . instead , we use the @xmath71 values at the vertexes to interpolate ( linearly ) the needed values on the cell faces . typically , the cells that need the full ray - casting algorithm are a few percent of the total volume and correspond to the ionization - front regions ( where the column - density varies rapidly ) . for the remaining cells , interpolating the column - densities is a good approximation and allows to substantially speed - up the computation . note that in this case , the algorithm is similar to a classical long - characteristic ray - tracing . to achieve a better performance it is also possible to choose a column - density threshold ( or , equivalently , an optical depth threshold at the ionization limit ) above which ` radamesh ` skips the calculation of the @xmath67 and @xmath69 values . this is computationally convenient , e.g. , in the early stages of the expansion of an ionization - front in a neutral and dense medium , when most of the volume is still optically thick to the source radiation . analogously , it is possible to choose a minimum value of @xmath70 and @xmath71 below which an optically - thin approximation is used instead of the full ray - tracing algorithm . this is typically the situation for the ionized region in proximity of the sources , where the optical depth is negligible . as mentioned in section 2 , eq . ( [ ieq ] ) permits i - fronts to propagate faster than light in near proximity of a bright source . an exact solution for this problem that does not require to fully solve the time - dependent radiative transfer equation and avoids the loss of photon conservation is only available in the case of an homogeneous medium ( e.g. , shapiro et al . 2006 ) as we have also discussed elsewhere ( cantalupo et al . 2008 ) . in the more general case of an inhomogeneous medium , we fix this unphysical behaviour by disregarding the ionizing flux of a given source at a distance @xmath82 , where @xmath83 is the source current lifetime . note that , the loss of photon conservation due to this approximation is typically restricted to the very early phases of the expansion of the i - front produced by very bright sources . once the value of the directional photoionization / heating rate have been obtained , we calculate ( with a simpler procedure ) the diffuse component rates for the same , _ active _ cells . also in this case we follow a `` cell - by - cell '' approach : given a particular ( _ active _ ) cell for which we want to compute @xmath84 and @xmath85 , we generate with a monte carlo procedure a set of rays that propagates outwards from the cell centre . for each cell encountered by the ray , we compute the medium opacity and the source term as a function of the @xmath86 , @xmath87 , @xmath88 , @xmath89 , and temperature . in particular , we include : i ) hi , hei and heii free - bound continuum ( from osterbrock 1989 ) , ii ) heii balmer continuum ( from ercolano & storey 2006 ) , iii ) heii two - photon continuum ( from nussbaumer & schmutz 1984 ) , and iv ) the heii ly@xmath90 line . we neglect the hei balmer continuum and hei emission lines given their ( relatively ) small contribution to the total emissivity . given the source function , we compute and add the respective @xmath84 and @xmath85 to the direct photoionization / heating rate , according to the equation presented in section 2 . in this method , each ray represents a fixed fraction of the solid angle ( @xmath91 ) of the sky as seen by the cell , determined by the chosen number of rays ( that we leave in the current version as a free parameter ) . note that this procedure is accurate if the three - dimensional diffuse field is homogeneous over a scale corresponding to @xmath92 , where @xmath93 is the distance from the cell . obviously , if the medium is clumpy , this method becomes more and more inaccurate at increasing distances from the cell , unless the number of rays is increased accordingly . note , however , that in most cosmological situations , the diffuse field represents a small component with respect to the direct radiation from sources . as mentioned above , we associate an individual time - step @xmath66 to each cell in the computational volume . we first estimate @xmath66 for the cells with the newly calculated @xmath25 by taking a fixed fraction @xmath94 of the minimum ionization timescale from the rate equations . the minimum value found for @xmath66 is stored ( as the new global time - step @xmath95 ) . at this point we check which cells can be considered _ active _ during the next simulation step comparing the individual @xmath66 with @xmath95 . if @xmath96 , where @xmath97 is a user - defined parameter , the cell is considered _ active _ and we assign @xmath98 . if a previously _ active _ cell does not meet this criterion we de - activate the cell until a simulation time @xmath99 ( or a number @xmath100 of simulation steps ) has elapsed . if @xmath101 all ( leaf ) cells are always considered active . since the cells within the ionization - front have the shortest @xmath66 , they are always selected as _ active_. in practice , the factor @xmath102 controls how broad is the region of _ active _ cells around the i - front . cells outside of this region are updated only when necessary , typically after several simulation time - steps , given their much longer ionization / recombination time - scales . even for very large values of @xmath102 ( e.g. , @xmath103 ) the gain in computational speed is huge , since most of the volume has evolution time - scales several orders of magnitude larger than the i - fronts . in this part of the computational algorithm , we solve the rate equation and we evolve the medium properties for the _ active _ cells with the newly calculated @xmath25 and @xmath26 . for the integration of the rate equations , we use the radau iia method ( hairer & wanner 1996 ) , an implicit runge - kutta scheme of variable , adaptive order . this allows this part of the code to be computationally stable and reasonably fast for the required accuracy . if activated , the adaptive mesh refinement is performed with the recursive clustering algorithm of berger & rigoutsos ( 1991 ) . before applying this algorithm , we flag the cells that satisfy a given refinement criterion . the currently implemented refinement criterion is based on a combination of three variables for each atomic species : i ) the cell neutral fraction , ii ) the cell ionization rate ( @xmath25 ) , and iii ) the ( frequency dependent ) cell optical depth @xmath104 . the ultimate goal is to flag and refine the fast evolving cells that are inside the i - front ( i.e. , the cells with the highest @xmath25 ) , until their @xmath105 is below the desired value . in practice , the refinement criterion should produce a multi - mesh grid where the front is well resolved , i.e. where the @xmath104 at the species energy threshold is well below unity . moreover , we would like also to have a pre - refined region just beyond the current position of the i - front , and a _ sharp _ cut corresponding to the regions where the i - front is just passed leaving the medium ( highly ) ionized . the size and shape of the refining region can be easily tuned with a criterion that includes the three variables discussed above , see test 6 for an example . the recursive amr algorithm starts from the base grid at the coarser level and it is composed by three main steps : i ) flagging , ii ) clustering , and iii ) refining / unrefining . once the cells to be refined at the coarser level ( @xmath106 ) are flagged as discussed above , they are clustered into new rectangular patches . these are splitted until the clustering efficiency ( i.e. , the ratio between the number of flagged and total number of cell in the new patch ) is greater than a chosen minimum threshold , typically 80% . in order to avoid the creation of grids that are too small ( and thus inefficient for the ray - tracing algorithm ) , it is also possible to chose a minimum size for the newly created patches . in most of the situations , we find that a minimum size of two ( parent ) cells gives the best balance between number of newly created cells and efficiency of the algorithm . we then _ propagate _ the physical properties from the coarser grid ( @xmath107 ) to the newly created patches at level @xmath108 ( refinement ) . currently , only straight injection refinement has been implemented . if a cell at level @xmath107 that was previously refined has not been flagged during the current time - step , the properties of its subcells at level @xmath108 are _ propagated _ back to the parent cell that becomes now a leaf cell ( un - refinement ) . temperature during un - refinement is weighted according to the subcell electron number densities , while the species neutral fractions ( @xmath109 ) are weighted according to the mass densities . this algorithm is recursively repeated , patch by patch , at finer levels , until there are newly flagged cells or the maximum number of refinement levels is reached . during clustering and refinement , the grid - tree and the cell - tree are updated , new memory is dynamically allocated for the newly created grids and , at the same time , the memory slots associated with the un - refined grids are released . by the algorithm construction , parent and son grids have memory slots allocated in close parts of the memory map . this ensures an efficient memory usage and reduces computational overhead during ray - tracing . in case where the amr is performed on an existing multi - mesh grid ( e.g. , the output of an hydrodynamical amr code ) , there is the very important option in ` radamesh ` to consider this initial grid as _ fixed _ : i.e. , the amr builds the new patches on the top of the existing multi - mesh grid without unrefining it below the original , initial level . this allows us to increase the resolution where needed for the radiative transfer ( on the i - front ) without losing the resolution achieved with the hydro code . the capacity of ` radamesh ` of adding further levels of refinement on an existing amr grid , without loosing previous information , is of great importance for several physical applications . for example for the study of the highly ionized ( but denser than the average ) regions in proximity of a qso that where already touched by its i - front and that would appear as ly@xmath90 forest in the quasar spectrum . myr in the plane @xmath110 ( top panel ) and @xmath111 ( bottom panel).,title="fig : " ] myr in the plane @xmath110 ( top panel ) and @xmath111 ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath112 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath112 myr ( bottom panel).,title="fig : " ] with respect to the strmgen radius @xmath113 ( bottom panel ) and the analytical solution ( top panel ) . see text for details . ] in this section we present the validating tests of the code based on the radiative transfer code comparison project ( iliev et al . 2006 ; i06 thereafter ) . these tests have been designed in order to compare all the important aspects of several radiative - transfer codes present in the literature . they include the correct tracking of both slow and fast ionization - fronts in homogeneous and inhomogeneous density fields , i - front trapping , spectrum hardening and the solution of the temperature state . the original tests in i06 ( test 1 to 4 ) are performed for a single , uniform grid , pure hydrogen medium and without recombination radiation ( using the ots approximation ) . in order to compare our results with the other rt codes , we use the same single grid ( pure - hydrogen ) configuration as used in i06 to reproduce the results of test 1 to 4 . in the second part of this section , we present a set of case studies aimed at substantiating the new characteristics of our code . in particular , we verify in test 5 and test 6 the multi - mesh , adaptive capability of ` radamesh ` . in test 7 and test 8 we show the effects of the diffuse radiation transfer for hydrogen only medium ( test 7 ) and including helium ( test 8) . and times @xmath114 myr ( top panel ) and @xmath115 myr ( bottom panel).,title="fig : " ] and times @xmath114 myr ( top panel ) and @xmath115 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath115 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , in particular : ` c^2-ray ` ( red , short - dashed line ) , ` crash ` ( cyan , dotted line ) , ` ftte ` ( blue , long - dashed line ) , and ` rsph ` ( green , dot - dashed line ) . , title="fig : " ] myr ( top panel ) and @xmath115 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , in particular : ` c^2-ray ` ( red , short - dashed line ) , ` crash ` ( cyan , dotted line ) , ` ftte ` ( blue , long - dashed line ) , and ` rsph ` ( green , dot - dashed line ) . , title="fig : " ] and times @xmath114 myr ( top panel ) and @xmath115 myr ( bottom panel).,title="fig : " ] and times @xmath114 myr ( top panel ) and @xmath115 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath115 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] myr ( top panel ) and @xmath115 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] this test represents the classical problem of the expansion of a hii region in an uniform ( pure - hydrogen ) medium around a single ionizing source . we assume that a steady , monochromatic ( @xmath116 ev ) source emitting @xmath117 ionizing photons per unit time turns on in an initially - neutral , uniform - density , static medium with hydrogen number density @xmath118 . likewise in i06 , we use for this test the ots approximation . the temperature is fixed at @xmath119 k. under these conditions , and assuming that the front is sharp ( i.e. that it is infinitely - thin , with the gas inside fully - ionized and the gas outside fully - neutral ) , there is a well - known analytical solution for the evolution of the i - front radius , @xmath120 , and velocity , @xmath121 , given by @xmath122^{1/3}\,,\\ \rm v_i&=&\frac{r_{\rm s}}{3t_{\rm rec}}\frac{\exp{(-t / t_{\rm rec } ) } } { \left[1-\exp(-t / t_{\rm rec})\right]^{2/3}}\ , , \label{strom0}\end{aligned}\ ] ] where @xmath123^{1/3}\,,\ ] ] is the strmgren radius , i.e. the radius at which recombinations balance the ionizations and the hii region expansion stops . here @xmath124 is the case b recombination coefficient and @xmath125^{-1}\,,\ ] ] is the recombination time . the hii region initially expands quickly and then slows down as the evolution time approaches the recombination time , @xmath126 . at a few recombination times , the i - front stops and in absence of gas motions remains static thereafter . the numerical parameters for this test are the followings : computational box dimension @xmath127 kpc ( the source is at one corner of the box ) , gas number density @xmath128 @xmath129 , initial ionization fraction ( given by collisional equilibrium ) @xmath130 , and ionization rate @xmath131 photonss@xmath132 . for these parameters the recombination time is @xmath133 myr . assuming a recombination rate @xmath134 at @xmath119 k , then @xmath135 kpc . note that the value of @xmath113 is actually independent from the use of the ots or case b approximation ( see test 7 ) . myr ( top panel ) and @xmath136 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel).,title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel).,title="fig : " ] in figure [ t1 ] , we show the images of the hi fraction in the plane y=0 and z=0 at time @xmath112 myr , when the equilibrium strmgen sphere is reached . the hii region is nicely spherical in both the planes , demonstrating that our algorithm produces an uniform coverage of the solid angle around the source . in figure [ t1f8 ] , we plot the spherically averaged radial profiles of the ionized ( @xmath137 ) and neutral fraction ( 1-@xmath137 ) at times @xmath138 and 500 myr . the thickness of the transition between the hii and hi regions is in agreement with both analytical and numerical expectations from most of the other codes in i06 . in particular , thanks to our new adaptive algorithm ( that ensures the convergence of the radiation field in any cell ) , the i - front thickness does not suffer from diffusive effects shown by other `` classical '' monte carlo methods ( cfr . , e.g. , the results of ` crash ` in i06 ) . finally , in figure [ t1f7 ] , we show the time evolution of the i - front position ( defined as the point of 50% ionization ) . the code tracks the i - front correctly , with the position never varying by more than few percent from the analytical solution . myr ( top panel ) and @xmath136 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] myr ( top panel ) and @xmath136 myr ( bottom panel ) . the results from ` radamesh ` are presented as solid black lines . the other lines represent the results from a sample of four different codes taken from the rt code comparison project ( i06 ) , see caption in figure [ t2f16 ] . , title="fig : " ] in this test we use the same parameters of test 1 , but now the ionizing source has a @xmath139 black - body spectrum and we allow the gas temperature to evolve . initially , the gas is fully neutral with a temperature t=100 k. there are no analytical solutions for this test , therefore we compare our results to what obtained by the other codes in i06 . in figure [ t2hi ] , we show the images of the neutral hydrogen fraction ( on the @xmath111 plane ) at times @xmath114 and @xmath115 myr . the spherically averaged hi profiles , for the same time - snapshots , are presented in figure [ t2f16 ] ( black , solid lines ) , together with a sample of the results obtained by the other codes in i06 ( see caption in the figure ) . the overall size of the hii region and the internal structure agree very well . the temperature images and the spherically averaged profiles at times @xmath114 and @xmath115 myr are presented in figures [ t2temp ] and [ t2f17 ] . also in this case , the resulting temperature structure agree well with most of the other codes in i06 . in particular , we are able to obtain a large pre - heated region without any sign of spatial anisotropy , since our algorithm does not suffer from the under - sampling the radiation field ( like in a `` classical '' monte carlo ) , even a large distances from the sources . myr obtained by ` radamesh ` ( top - left panel ) and by three of the four codes that performed the same tests in i06 , in particular : ` crash ` ( bottom - left panel ) , ` c^2-ray ` ( top - right panel ) , and ` ftte ` ( bottom - right panel ) . ] myr obtained by ` radamesh ` ( top - left panel ) and by three of the four codes that performed the same tests in i06 ( see caption in figure [ t4hi_005 ] ) . ] myr obtained by ` radamesh ` ( top - left panel ) and by three of the four codes that performed the same tests in i06 ( see caption in figure [ t4hi_005 ] ) . ] myr obtained by ` radamesh ` ( top - left panel ) and by three of the four codes that performed the same tests in i06 ( see caption in figure [ t4hi_005 ] ) . ] this test verifies that the propagation of an i - front within a dense clump is slowed down at the point of being stopped ( _ trapped _ ) and the production of a correct shadowing effect ( in both the ionization and the temperature state ) behind the clump . we use the same set - up as described by i06 : a spherical ( hydrogen - only ) uniform cloud of radius @xmath140 kpc , is located at the position @xmath141 kpc within a box of length @xmath127 kpc . the hydrogen number density and the initial temperature outside of the clump are , respectively , @xmath142 @xmath129 and @xmath143k , while inside the clump we have @xmath144 @xmath129 and @xmath145 k. the radiation has a black - body spectrum with @xmath146 k and a constant ionizing photon flux @xmath147 s@xmath132 @xmath148 , incident to the @xmath110 box side . given these parameters , we expect that the i - front should be trapped slightly beyond the clump centre , as discussed in i06 . in figure [ t3hi_ima ] , we show slices of the gas neutral fraction at the box mid - plane @xmath149 kpc ( passing through the centre of the clump ) at times @xmath150 myr ( top panel ) and @xmath136 myr ( bottom panel ) . the corresponding temperature slices are presented in figure [ t3temp_ima ] . at @xmath150 myr , the i - front is not yet trapped and it is still moving supersonically from the edge of the clump . the images show that the shadow is sharp and produced correctly behind the clump . as expected , the i - front is trapped slightly beyond the clump centre at @xmath136 myr . the position of the i - front and the hi profiles inside the clump , presented in figure [ t3hi_prof ] , agree well with the other codes in i06 , altough , especially at later times , the results are rather code dependent . the same is true for the temperature profiles , as shown in figure ( [ t3temp_prof ] ) . in this test , we follow the propagation of the ionization - fronts from multiple sources in a static cosmological density field . the initial conditions for the density and source spatial distribution / luminosities are provided by i06 . in particular , we use a time - slice at @xmath151 from a hydro - simulation with a box size of @xmath152 comoving mpc and @xmath153 cells . the initial temperature is fixed to t=100 k everywhere . the ionizing sources correspond to the 16 most massive halos in the box , with a luminosity proportional to the halo mass and a black - body spectrum with @xmath154 k ( see i06 for more details ) . in figures [ t4hi_005 ] and [ t4hi_02 ] , we present slices of neutral hydrogen fraction cut through the simulation box at coordinate @xmath155 , in box units , at time @xmath156 myr and @xmath157 myr , respectively . comparing our results with three of the four codes that performed this test in i06 ( ` c^2-ray ` , ` crash ` and ` ftte ` ) , we find a general agreement , although all the codes produce somewhat different morphologies . this general agreement is confirmed also examining the temperature slices in figures [ t4temp_005 ] and [ t4temp_02 ] . here the differences between the codes presented in i06 is higher and due to different assumption about spectral hardening and partially to the different algorithm used . the spectral hardening effect , namely the increased temperature in the regions that are still mostly neutral , is traced in detail by ` radamesh ` . myr and coordinate @xmath155 ( box units ) . the multi - mesh structure in the plane @xmath155 is overlaid . ] with this test , not present in the original set of i06 , we show and verify the multi - grid capability of ` radamesh ` in the case of a static multi - mesh . in particular we use the same initial condition of test 1 changing the original grid with three nested , concentric grids with @xmath158 cells , corresponding to two levels of refinement ( with a factor two and four increased resolution with respect to the base mesh ) . with this configuration , the centre of the box has a resolution equivalent to a @xmath159 cells grid . we limit the number of levels and the resolution of the base grid in this test for illustrative purposes . the source position is @xmath160 in box units . in figure ( [ t5 mg ] ) , we show the image of hi fraction and the computational meshes corresponding to the slice at coordinate @xmath155 and time @xmath161 myr . as we can see from the image , the front is tracked very well despite of the different grids ( and resolution ) , with no spurious effect introduced by the multi - grid structure . in the original set of tests in i06 , the initial conditions ( e.g. , box size , hydrogen density , source luminosity ) have been chosen in such a way to properly resolve the i - front with a single , uniform grid of @xmath153 cells . for example , in the final test of i06 , test 4 ( multiple sources in a `` cosmological '' density field ) , the box size is fixed to @xmath152 comoving mpc , corresponding to @xmath162 physical kpc at the simulation redshift ( @xmath151 ) for @xmath163 . however , in a more realistic cosmological situation , e.g. for the study of the reionization process or the expansion of the i - front around a bright , high - redshift qso , the box size must be at least 2 orders of magnitude larger than test 4 ( see , e.g. , meiksin 2009 ) . this test , not present in the original set of i06 , demonstrate the ability of ` radamesh ` to resolve the i - front of a bright quasar in a large cosmological box , with the help of an adaptively evolving multi - mesh . in particular we show that properly resolving the i - front is essential to accurately predict the temperature state of the gas , both in the i - front and in the qso hii region . for this purpose we run a series of simulation increasing the maximum level of refinement until convergence is reached for the temperature state of the gas . in the same spirit of test 2 , we use an uniform ( hydrogen - only ) medium with evolving temperature and we locate the source at one corner of the box . the box size is @xmath164 comoving mpc , the medium density is equal to the mean hydrogen density of the universe and it evolves with redshift . the initial redshift is @xmath165 , the total evolution time is @xmath166 yr ( corresponding to a final redshift @xmath167 ) . the source has a power - law spectrum with a ( frequency ) spectral slope of @xmath168 , sampled in 60 logarithmically spaced bins from 1 to 60 rydberg . the total ionization rate is @xmath169 ph s@xmath132 . these parameters are comparable to the respective quantities associated with the observed high - redshift quasars in sdss ( e.g. , fan et al . 2006 ) . for the assumed spectrum ( which has a mean ionizing photon energy of @xmath170 ryd ) the photon mean - free path is @xmath171 physical kpc at the initial redshift for a fully neutral patch of gas composed of hydrogen only . to resolve the i - front properly we would ideally need that @xmath172 is sampled by at least two cells , i.e. a spatial resolution of @xmath173 physical kpc . with an uniform grid this would correspond to a @xmath174 cells mesh , effectively out of the reach for current computational facilities . instead , we use a @xmath153 base grid and 5 additional levels of adaptive grid refinement that follow the i - front expansion , effectively achieving the required spatial resolution . in figure [ t6image ] , we show an image slice of the hydrogen neutral fraction at @xmath175 yr ( in the quasar rest - frame ) . the i - front scale is so small compared to the box size that it visually appears perfectly thin in the large image . zooming in by a large factor in a small region containing the i - front ( see inset in the same figure ) , and now the i - front thickness appears . in box units , the i - front size ( as measured from the points where the neutral fractions are 0.1 and 0.9 ) is roughly @xmath176 , i.e. @xmath177 physical kpc , corresponding to @xmath178 . overlaid , we show the adaptively refined multi - mesh structure that closely follows the i - front , with the highest level of refinement ( @xmath179 ) that encompasses the i - front itself . at this level of refinement , the i - front is fully resolved by at least 10 simulation cells . in figure [ t6plot ] , we show how resolving the i - front changes the predicted temperature profile inside the qso hii bubble and in the i - front region . from bottom to top , the lines represent the temperature profiles obtained varying the maximum level of refinement from @xmath180 ( i.e. , single uniform grid ) to @xmath181 . as expected , temperature convergence is reached for @xmath182 , i.e. at spatial resolution comparable or smaller than @xmath172 , indicating that we are effectively resolving the i - front . in fact , fully resolving the i - front improves the sampling of the spectral hardening ( within the i - front itself ) and a harder spectrum is able to produce a large _ increase _ in the gas temperature , as observed in figure [ t6plot ] note that the current temperature of a cell within the hii region at @xmath115 myr is mainly determined at an earlier epoch , i.e. , when the cell was located within the i - front . therefore , also the temperature increase _ within _ the hii region can be ascribed to the ( i - front ) spectral hardening effect . numerical effects on the gas temperature due to the different resolution are negligible , as we show in appendix b. . the prediction of the correct temperature state in the i - front and in the qso bubble is fundamental , e.g. , for the study of the reionization epoch with the near - zone ly@xmath90 forest ( see e.g. , bolton et al . 2010 ) or with the i - front ly@xmath90 emission ( cantalupo et al . 2008 ) . comoving mpc ) and i - front adaptive mesh refinement . the qso was turned on at @xmath165 and keep it at a constant ionizing rate of @xmath169 ph s@xmath132 for @xmath175 yr ( the final redshift is thus @xmath167 ) . from bottom to top , the lines correspond to increasing maximum level of refinement ( from @xmath180 to @xmath181 ) . the base grid is composed by @xmath153 cells . this plot clearly shows that the high - resolution achieved through amr is essential to correctly resolve the i - front and , thus , to properly recover the gas temperature profile around the qso in a cosmological context . ] we repeat test 1 dropping the ots approximation used in i06 and including now the full radiative transfer of diffuse photons produced by hydrogen recombinations . since all the cell in the strmgren sphere may be now source of ionizing photons , we must change the original test configuration , placing the ( discrete ) source at the centre of the box and increasing the box size by a factor of two . apart from these modifications , we use the same parameter set as in test 1 . for a monochromatic spectrum , homogeneous medium and for a fixed temperature , we know that the radius of the strmgren sphere ( @xmath183 ) obtained with the ots ( case b ) approximation must be equal to the one obtained with the full radiative transfer of the diffuse radiation . this is a fundamental constraint deriving from photon conservation . moreover , once the ionization equilibrium has been reached , there is an analytical approximation for the ratio between the diffuse and directional radiation field needed to compensate , at a given radius @xmath93 , the recombinations , as found by ritzerveld ( 2005 ) : @xmath184^{1-\alpha/\alpha_{{\mathrm}{b}}-1 } \ , \ ] ] where the first equality is only valid for a monochromatic spectrum . note that eq . ( [ ritz ] ) is exact for an `` outward - only '' system , i.e. when the contribution to @xmath185 is given by the recombinations at @xmath186 ( indeed , @xmath187 in ritzerveld s solution ) . therefore , we expect that this approximation should slightly underestimate @xmath188 for @xmath189 ( note , however that within this region the total radiation field will be dominated by @xmath190 ) . in figure [ gammadiff ] , we present the result obtained by ` radamesh ` including the full radiative transfer of diffuse photons ( black solid line ) . in very good agreement with the analytical expectations ( red dashed line ) , we found that the diffuse radiation field strength equals the directional field from the central source at @xmath191 , confirming that diffuse radiation is dominant in the outer parts of the strmgren sphere . the full rt results produce , as expected , a slightly higher value of diffuse radiation in the central parts of the hii regions with respect to the `` outward - only '' solution . because of photon conservation , this is correctly balanced by a lower diffuse field at the edge of the strmgren sphere when compared to the analytical approximation . in figure [ hifracdiff ] , we show how the hydrogen neutral fraction profile ( @xmath192 ) at `` equilibrium '' ( @xmath112 myr ) is modified by the full radiative transfer of diffuse photons ( red solid line ) with respect to the ots ( case b ) approximation ( blue , long - dashed line ) and the other extreme represented by the case a approximation ( black , short - dashed line ) . as expected , in the interior of the hii regions , where the gas is highly ionized , the mean - free - path of the diffusion photons is very large and thus the profile obtained by the full rt closely follows the one obtained assuming case a. however , as the local opacity increases moving outward , diffuse photons start to be absorbed and the hi fraction tends to become closer to the ots case . in the outer parts of the hii region ( @xmath193 ) , diffusion radiation becomes the dominant component and the gas is more ionized than in the ots case b approximation . however , in accordance with photon conservation , the final size of the strmgren sphere is very close to the one obtained with case b : the recombination radiation escaped from the inner region has been absorbed ( creating an `` excess '' ) closer to the strmgren radius , but the final photon balance is the same . as a remark , it is interesting to note that the ots approximation is actually never valid inside the strmgren sphere for this very simple , standard test ( apart , obviously , at the strmgren radius ) . actually , case a is a much better approximation if one is interested in the inner part of the hii region , or , analogously , in the early phases of the ionization - front expansion . ( see text for details ) . in this test , the medium temperature has been fixed to @xmath119k ] myr for case a ( short dashed black line ) , `` on - the - spot '' case b ( long dashed blue line ) approximations and including the full radiative transfer of diffuse emission from hii recombinations ( solid red line ) . in this test , the medium temperature has been fixed to @xmath119k . ] the previous test has been limited to the very simple case of a monochromatic spectrum ( for both diffuse and direct radiation ) and a hydrogen only , fixed temperature medium . in this way , we were able to compare our results to the analytical prediction from photon conservation arguments . however , the real effect induced by the full rt of recombination radiation is much more complex since its characteristic spectral energy distribution ( sed ) is in general different from the sed of the discrete sources . moreover , the presence of helium may have a profound impact also on the neutral hydrogen distribution , given that both the bound - free and the balmer continuum of hei and heii ( together with two - photon and line emissions ) are able to ionize hi . in this test , we verify these effects using the same configuration of test 2 , including now the full rt of recombination radiation from hydrogen and helium . the only parameters , apart the inclusion of the diffuse component rt , that have been modified with respect to test 2 are the following : i ) the discrete source has been placed in the centre of the box and the box size has been increased by a factor 2 , ii ) the medium include hydrogen and helium , iii ) we evolve the simulation to @xmath112 myr . note that we do not expect in this case a full equilibrium to be reached , given the complex interplay between hydrogen and helium recombination emission and their ionization state . in figure [ hifracdiff_t2b ] , we present the hi profile obtained by the full rt of diffuse radiation ( red solid line ) in comparison with the case a approximation ( black , short - dashed line ) and the ots case b ( blue , long - dashed line ) at @xmath112 myr . in figure [ hefracdiff_t2b ] , we show the analogous profiles for the hei , heii and heiii fractions ( see labels in the figure ) . as evident from figure [ hifracdiff_t2b ] , there is now an excess of hi photoionizations with respect to case b , that leads to a larger strmgren sphere size . this excess is due to diffuse photons produced by helium recombinations , in particular the balmer continuum ( plus ly@xmath90 and two - photon continuum ) produced in the heiii region , that alters the total photon budget available for the photoionization of hi . in effect , a fraction of the high - energy , heii - ionizing photons from the central source are `` converted '' via heiii recombinations to lower energy , hi - ionizing photons ( and also hei - ionizing photons ; indeed , a similar but less conspicuous effect is also visible for the hei fractions in figure [ hefracdiff_t2b ] ) . therefore , assuming case b for hydrogen and helium actually results in the loss of this important component of the photon budget . this enlightens the importance of including heii and heiii recombination emission also for studies that are only interested in the properties of the hydrogen component . myr for case a ( short dashed black line ) , case b ( long dashed blue line ) approximations and including the full radiative transfer of ionizing diffuse emission from hii , heii and heiii recombinations ( solid red line ) . ] myr for case a ( short dashed black lines ) , `` on - the - spot '' case b ( long dashed blue lines ) approximations and including the full radiative transfer of ionizing diffuse emission from hii , heii and heiii recombinations ( solid red lines ) . ] we have presented a new , three - dimensional radiative transfer code , called ` radamesh ` , based on an adaptive monte carlo ray - tracing scheme . the algorithm has been specifically developed to efficiently resolve the small scales associated with the ionization - fronts within large , cosmological simulations , with the help of an adaptive mesh refinement ( amr ) scheme combined with a new `` cell - by - cell monte carlo '' approach : rays are casted separately from selected cells within the computational box to the sources with a method that ensures their correct solid angle distribution . this method has several advantages with respect to a classical monte carlo ray - tracing scheme where a series of rays is uniformly casted around a source to obtain the correct solid angle distribution . , in particular for multi - mesh domains . indeed , in a classical monte carlo , the number of rays to cast is determined by the smallest , most distant cells to the source , i.e. from the ( few ) cells at the highest refinement level , oversampling the rest of the box . in ` radamesh ` , the number of rays is proportional to the number of cells to be evolved during the current time - step , i.e. the fast evolving cells typically located within the i - front . this translates into a huge gain in computational speed and efficiency , given the small fraction of the computational volume occupied by the i - front . moreover , we are now able to choose a local criterion that adaptively determines the number of rays needed in a particular cell , e.g. set by the convergence of the radiation field . this ensures to obtain the required accuracy in the prediction of the temperature or ionization state of the gas in a very efficient way : concentrating the computational efforts only where actually needed . in the same spirit of hydrodynamical amr codes , ` radamesh ` is also able to increase the spatial resolution where required , adaptively refining the mesh in correspondence of the ionization - fronts . the implemented algorithm is based on the original patch - based amr method of berger & rigoutsos ( 1991 ) . ` radamesh ` is able to trace the ionization - fronts from multiple sources as well as the diffuse ionizing radiation produced by hydrogen and helium recombinations with a multi - frequency approach . the time - evolution of six different species ( hi , hii , hei , heii , heiii , e ) and the gas temperature is followed with a time - dependent , non - equilibrium chemistry solver based on an implicit runge - kutta scheme of variable , adaptive order . we performed all the four tests present in the radiative transfer code comparison project of iliev et al . 2006 , plus four additional , new tests aimed to substantiate and show the new characteristic of ` radamesh ` . the first four tests include the correct tracking of the i - front in homogeneous ( test 1 ) and in - homogeneous density fields with multiple sources ( test 4 ) , i - front trapping behind a dense clump ( test 3 ) , and the effect of photo - heating on the gas temperature state ( test 2 ) . these tests , like in the original i06 work , have been performed on a single , static mesh , without emission from atomic recombinations in order to better compare our results with the other codes . ` radamesh ` results are in very good agreement with the majority of the other codes present in i06 . in particular , thanks to the new adaptive algorithm , we have shown ( test 1 ) that the recovered i - front thickness and structure does not suffer from any diffusive effects and numerical broadening typical of other monte carlo codes ( given their poor sampling at large distances from the source ) . the same is true when the gas temperature is examined ( test 2 ) . the second set of tests ( test 5 to test 8) shows the ability of ` radamesh ` to deal with both static ( test 5 ) and adaptively evolving ( test 6 ) multi - mesh structures , and with diffuse radiation produced by hydrogen ( test 7 ) and helium ( test 8) recombinations . in test 5 , we reproduce the same results of the classical strmgren sphere situation ( test 1 ) with a nested hierarchy of meshes at different refinement levels , showing that no spurious effects are introduced by the multi - mesh structure . most importantly , we have shown in test 6 that ` radamesh ` is able to fully resolve an expanding i - front from a bright source ( e.g. , a quasar ) within a large , cosmological box ( 100 comoving mpc size ) , thanks to an adaptively evolving mesh with several levels of refinement . the ability of resolving the i - front on cosmological scales is fundamental for a large range of applications . for instance , recovering the correct gas temperature within the i - front and the qso bubble is important , e.g. , for the study of the reionization epoch with the near - zone ly@xmath90 forest ( see e.g. , bolton et al . 2010 ) or with the ly@xmath90 emission generated within the i - front ( cantalupo et al . 2008 ) . in test 6 , we have demonstrated that a proper treatment of the spectral hardening inside a resolved i - front may result in a substantial increase of the gas temperature ( @xmath194 k ) within the _ whole _ hii region surrounding the bright , central source . the effect of diffuse radiation generated by hydrogen recombinations on the classic strmgren sphere case ( with monochromatic radiation ) has been presented in test 7 . here , we have verified that our treatment of diffuse radiation produces the same strmgren sphere size with respect to the case b approximation , in agreement with photon conservation . moreover , in accordance with the analytical prediction by ritzerveld ( 2005 ) , we have verified that the diffuse field becomes the dominant component at a distance from the source corresponding to 87% of the strmgren radius . the neutral fraction profile in the inner part of the hii region follows the same profile obtained with case a approximation , while in the outer part the gas is more ionized and the i - front narrower than the result obtained with the widely used , case b approximation . in test 8 , we have also considered the effect of hei and heii recombination radiation on the hydrogen and helium ionization state . we have found that heii diffuse radiation , especially the hi - ionizing , heii balmer continuum ( together with heii two - photon continuum and ly@xmath90 emission ) may have an important effect also on the hydrogen ionization state , substantially increasing the strmgren sphere size . at present , ` radamesh ` is not yet coupled with hydro - dynamics but is able to post - process the output of three different hydrodynamical codes , both grid - based , e.g. , ` ramses ` ( teyssier 2002 ) , ` charm ` ( miniati & colella 2007 ) and particle - based , e.g. ` gadget ` ( springel 2005 ) . these outputs are efficiently converted to ` radamesh ` patch - based structures with a fast algorithm based on the clustering method of berger & rigoustous ( 1991 ) , also used into the amr module . the default format is very similar to the widely used ` chombo hdf5 ` structure https://seesar.lbl.gov/anag/chombo , allowing ` radamesh ` outputs to be directly visualized with the most recent , high - performance visualization packages ( e.g. , ` visit ` https://wci.llnl.gov/codes/visit ) . currently , ` radamesh ` is efficiently parallelized with openmp . all the results presented in this paper have been obtained with a few hours of computational time on a 8-core intel - xeon workstation . sc thanks francesco miniati for useful discussions and martin haehnelt for comments on an earlier version of this manuscript . cp acknowledges the role of piero madau who nearly 10 years ago sensitised him to the cosmological radiative - transfer problem and followed the development of some unpublished algorithms . part of the figures presented in this paper have been realized with the visualization software visit , we are grateful to llnl to have made public this powerful tool . abel , t. , norman , m. l. , & madau , p. 1999 , apj , 523 , 66 abel , t. , & wandelt , b. d. 2002 , mnras , 330 , l53 altay , g. , croft , r. a. c. , & pelupessy , i. 2008 , mnras , 386 , 1931 amanatides , j. , & woo , a. , `` a fast voxel traversal algorithm for ray tracing '' , proc . eurographics 87 , amsterdam , the netherlands , august 1987 , pp 1 - 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dimensional levi - civita symbol ( note the implicit sum over the indices @xmath197 and @xmath198 ) , @xmath199 is the @xmath200 matrix containing the cell vertexes coordinates ( translated to the cartesian system with the source at the origin ) , @xmath201 and @xmath202 are , respectively , the cell centre and the source coordinates , @xmath54 is the linear cell size , @xmath203 is the @xmath204 unit matrix ( i.e. , the matrix of ones ) , and @xmath205 is the following @xmath200 matrix : @xmath206 . all lengths and coordinates are in box units . note that the presence of the factors containing @xmath207 and @xmath196 reduce the total number of terms in the overall sum to four per cell face . the function @xmath208 removes from the sum the solid angle of a face _ invisible _ to the source , since this has a negative sign . in figure [ satest ] , we show the comparison between the recovered cell solid angle obtained by a full monte carlo simulation with @xmath153 cubic cells and @xmath166 rays ( black circles ) , and the above analytical approximation ( red solid line ) . the solid angle is shown as a function of the distance from the source , placed at @xmath209 , and for two different directions : parallel to the @xmath137-axis ( i.e. , normal to two cell faces ; lower line ) and along the line of sight that connects the source to the box vertex @xmath210 ( upper line ) . note that these two directions correspond to the minimum and maximum possible solid angle of a cubic cell at a given distance from the source . the analytical approximation is in very good agreement with the monte carlo results . grid . the solid angle is shown as a function of the distance from the source , as seen by two different angular directions ( see labels ) . ] in test 6 , we have shown that the spectral hardening inside a resolved i - front may result in a substantial increase of the gas temperature ( @xmath194 k ) within the _ whole _ hii region surrounding a bright quasar . to better demonstrate that this result is not affected by numerical effects due to the different grid resolutions , we have repeated test 6 assuming a source with similar ionizing rate but with a soft spectrum ( a black - body with @xmath211 k ) . in this case , spectral hardening is minimal and therefore we do not expect that increasing the resolution on the i - front should substantially change the gas temperature as in test 6 . as we show in figure [ t6b ] ( line colors and style have the same meaning as in figure 18 ) , this is indeed the case : different maximum levels of refinement produce now very similar temperatures inside the hii region showing that numerical effects are not important . note that the ( larger ) shift in the i - front position in figure [ t6b ] ( with respect to figure 18 ) is due to the fact that , at the lowest resolution , cells are very optically thick to radiation with frequencies right above the ionization threshold ( the vast majority , given the soft spectrum assumed here ) . this slightly alters photon - conservation , which holds to high accuracy when the front is resolved into less optically thick elements , as shown in test 1 . k ) . in this case , spectral hardening is minimal and different maximum levels of refinement produce now very similar temperatures inside the hii region . this demonstrates that numerical effects are not substantially affecting the results of test 6 . ]

we present a new three - dimensional radiative transfer ( rt ) code , ` radamesh ` , based on a ray - tracing , photon - conserving and adaptive ( in space and time ) scheme . ` radamesh ` uses a novel monte carlo approach to sample the radiation field within the computational domain on a `` cell - by - cell '' basis . thanks to this algorithm , the computational efforts are now focused where actually needed , i.e. within the ionization - fronts ( i - fronts ) . this results in an increased accuracy level and , at the same time , a huge gain in computational speed with respect to a `` classical '' monte carlo rt , especially when combined with an adaptive mesh refinement ( amr ) scheme . among several new features , ` radamesh ` is able to adaptively refine the computational mesh in correspondence of the i - fronts , allowing to fully resolve them within large , cosmological boxes . we follow the propagation of ionizing radiation from an arbitrary number of sources and from the recombination radiation produced by h and he . the chemical state of six species ( hi , hii , hei , heii , heiii , e ) and gas temperatures are computed with a time - dependent , non - equilibrium chemistry solver . we present several validating tests of the code , including the standard tests from the rt code comparison project and a new set of tests aimed at substantiating the new characteristics of ` radamesh ` . using our amr scheme , we show that properly resolving the i - front of a bright quasar during reionization produces a large increase of the predicted gas temperature within the whole hii region . also , we discuss how h and he recombination radiation is able to substantially change the ionization state of both species ( for the classical strmgren sphere test ) with respect to the widely used `` on - the - spot '' approximation . [ firstpage ] radiative transfer - methods : numerical - hii regions - intergalactic medium - diffuse radiation - cosmology : theory

You are an expert at summarizing long articles. Proceed to summarize the following text: as in our previous analysis of run 1a data @xcite , we conduct a general search for new particles with a narrow natural width that decay to dijets . in addition , we search for the following particles summarized in fig . [ fig_particles ] : axigluons @xcite from chiral qcd ( @xmath7 ) , excited states @xcite of composite quarks ( @xmath8 ) , color octet technirhos @xcite ( @xmath9 ) , new gauge bosons ( @xmath10,@xmath11 ) , and scalar @xmath12 diquarks @xcite ( @xmath13 and @xmath14 ) . using four triggers from run 1a and 1b , we combine dijet mass spectra above a mass of 150 gev / c@xmath1 , 241 gev / c@xmath1 , 292 gev / c@xmath1 , and 388 gev / c@xmath1 with integrated luminosities of = 7.5 in = 3.3 in = 3.3 in .089 pb@xmath0 , 1.92 pb@xmath0 , 9.52 pb@xmath0 , and 69.8 pb@xmath0 respectively . jets are defined with a fixed cone clustering algorithm ( r=0.7 ) and then corrected for detector response , energy lost outside the cone , and underlying event . we take the two highest @xmath15 jets and require that they have pseudorapidity @xmath16 and a cms scattering angle @xmath17| < 2/3 $ ] . the @xmath18 cut provides uniform acceptance as a function of mass and reduces the qcd background which peaks at @xmath19 . in fig . [ fig_dijet ] the dijet mass distribution is presented as a differential cross section in bins of the mass resolution ( @xmath20% ) . at high mass the data is systematically higher than a prediction from pythia plus a cdf detector simulation , similar to the inclusive jet @xmath21 spectrum @xcite . to search for new particles we determine the qcd background by fitting the data to a smooth function of three parameters @xcite ; fig . [ fig_dijet ] shows the fractional difference between the data and the fit ( @xmath22 ) . we note upward fluctuations near 200 gev / c@xmath1 ( @xmath23 ) , 550 gev / c@xmath1 ( @xmath24 ) and 850 gev / c@xmath1 ( @xmath25 ) . for narrow resonances it is sufficient to determine the mass resolution for only one type of new particle because the detector resolution dominates the width . in fig . [ fig_resonance ] we show the mass resolution for excited quarks ( q * ) from pythia plus a cdf detector simulation ; the long tail at low mass comes from gluon radiation . for each value of new particle mass in 50 gev / c@xmath1 steps , we perform a binned maximum likelihood fit of the data to the background parameterization and the mass resonance shape . in fig . [ fig_resonance ] we display the best fit and 95% confidence level upper limit for a 550 gev / c@xmath1 resonance . for the mass region @xmath26 gev / c@xmath1 , there are 2947 events in the data , @xmath27 events ( @xmath28 ) in the background for the fit without a resonance , @xmath29 events ( @xmath30 ) in the background for the fit that includes the resonance , and the value of the resonance cross section from the fit is @xmath31 pb ( statistical ) . in fig . [ fig_cos ] we study the angular distribution of the fluctuation in the mass region @xmath26 gev / c@xmath1 . the angular distribution is compatible with both qcd alone , and with = 3.3 in = 3.3 in qcd + 5% excited quark ( best fit ) . this amount of excited quark is coincidentally the same as found in the mass fit . although the fluctuation is interesting , we conclude it is not yet = 3.3 in = 3.3 in statistically significant , and proceed to set limits on new particle production . = 3.3 in = 3.3 in from the likelihood distribution including experimental systematic uncertainties @xcite we obtain the 95% cl upper limit on the cross section for new particles shown in fig . [ fig_dijet_limit ] . we compare this to the cross section for axigluons ( excluding @xmath32 gev / c@xmath1 ) , excited quarks ( excluding @xmath33 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath35 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . the calculations are lowest order @xcite using cteq2l parton distributions @xcite and one - loop @xmath39 and require @xmath16 and @xmath40 . the large mass of the top quark suggests that the third generation may be special . topcolor @xcite assumes that the top mass is large mainly because of a dynamical @xmath5 condensate generated by a new strong dynamics coupling to the third generation . here the @xmath41 of qcd is a low energy symmetry arising from the breaking of an @xmath42 coupling to the third generation and an @xmath43 coupling to the first two generations only . there are then massive color octet bosons , topgluons @xmath44 , which couple largely to @xmath4 and @xmath5 . the topgluon is strongly produced and decays mainly to the third generation ( @xmath45 ) with a relatively large natural width ( @xmath46 ) . here we search for the topgluon in the @xmath4 channel . an additional @xmath47 symmetry is introduced @xcite to keep the @xmath48 quark light while the top quark is heavy ; this leads to a topcolor @xmath6 , which again couples largely to @xmath4 and @xmath5 . the topcolor @xmath6 is electroweakly produced and decays mainly to the third generation ( @xmath49 ) with a narrow natural width ( @xmath50 ) . here we search for the topcolor @xmath6 in both the the @xmath4 and @xmath5 channel ; the @xmath5 channel is the most sensitive because the coupling to @xmath5 is larger . we start with the dijet search in 19 pb@xmath0 of run 1a data @xcite and additionally require at least one of the two leading jets be tagged as a bottom quark . the b - tag requires a displaced vertex in the the secondary vertex detector @xcite . the @xmath4 event efficiency is @xmath51% independent of dijet mass . from fits to the @xmath52 distribution , we estimate that the sample is roughly 50% bottom , 30% charm , and 20% mistags of plain jets . pythia predicts that 1/5 of these bottom quarks are direct @xmath4 , and the rest are from gluon splitting and flavor excitation . consequentially , only about @xmath53% @xmath54 10% of our sample is direct @xmath4 . we expect both the purity and efficiency to increase when we use the run 1b dataset and a new tagging algorithm @xcite . with higher tagging efficiency we should be able to make better use of double b - tagged events like the one in fig . [ fig_btag_event ] . = 2.6 in = 2.6 in in fig . [ fig_btag ] we show the b - tagged dijet mass distribution corrected for the @xmath4 efficiency . also shown is the untagged dijet mass distribution from run 1a , and both are well fit with our standard parameterization @xcite . the b - tagged dijet data has an upward fluctuation near 600 gev / c@xmath1 . we model the shape of a narrow resonance using pythia z@xmath3 production and a cdf detector simulation . in fig . [ fig_btag ] we fit the b - tagged data to a 600 gev / c@xmath1 narrow resonance , and find a cross section of @xmath55 pb ( statistical ) . note that this is comparable to the dijet fluctuation in both mass and rate . however , there are only 8 events in the last two data bins of fig . [ fig_btag ] , and the fluctuation is only a @xmath56 effect , so we proceed to set limits on new particle production . we perform two kinds of fits for the limits . first , narrow resonances are modelled as described above , and the mass resolution in the cdf detector is shown in fig . [ fig_b_bbar_res ] . second , wide resonances characteristic of topgluons @xcite , including interference with normal gluons , was incorporated into pythia and a cdf detector simulation . the mass resolution in fig . [ fig_b_bbar_res ] displays destructive interference to the left of the resonance ; models with destructive interference on the right side of the resonance will be considered in the future . = 3.2 in = 3.2 in limits on new particle production are shown in fig . [ fig_btag_limit ] . the theoretical cross sections are lowest order and use cteq2l parton distributions . for narrow resonances the production cross sections are nt large enough for us to set mass limits at this time . for topgluons the production cross sections @xcite are larger , and we are able to exclude at 95% cl topgluons of width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . = 3.0 in = 3.0 in = 7.7 in to search for new particles decaying to @xmath5 we start with the data sample from the top mass measurement @xcite . there we used top decays to w + four jets with at least one b - tag , and found 19 events on a background of @xmath63 , resulting in a top mass of @xmath64(stat)@xmath65(sys ) gev / c@xmath1 . that analysis fit the entire event for the top hypothesis , discarding events with @xmath66 ( poor fit ) . here we add the additional constraint that the top mass is 176 gev / c@xmath1 , which significantly enhances our resolution of the @xmath5 mass . two of the 19 events fail the @xmath66 cut when the top mass constraint is added to the fit , leaving us with 17 events . the @xmath5 mass distribution expected from a narrow resonance , normalized to the topcolor @xmath6 predicted rate @xcite , is shown in fig . [ fig_ttbar ] . here we used pythia @xmath67 . also in fig . [ fig_ttbar ] is the monte carlo distribution of the background , on the left standard model top production from herwig , and on the right qcd w + jets background from vecbos with parton showers from herwig . all monte carlos include a cdf detector simulation . on the left in fig . [ fig_ttbar ] , the comparison of the topcolor z@xmath3 to sm @xmath5 simulations illustrates that in this data sample we are sensitive to topcolor @xmath6 up to a mass of roughly 600 gev / c@xmath1 . finally , on the right in fig . [ fig_ttbar ] , we present the @xmath5 candidate mass distribution from cdf compared to the total standard model prediction . given the statistics the agreement is quite good overall . the small shoulder of 6 events on a background of @xmath68 in the region @xmath69 gev / c@xmath1 is in an interesting mass region , given the dijet and @xmath4 search results , but is not statistically significant . upper limits on the @xmath5 cross section as a function of @xmath5 mass , and on a topcolor @xmath6 , are currently in progress . = 3.1 in = 3.1 in we have searched for new particles decaying to dijets , @xmath4 , and @xmath5 . in the dijet channel we set the most significant direct mass exclusions to date on the hadronic decays of axigluons ( excluding @xmath70 gev / c@xmath1 ) , excited quarks ( excluding @xmath71 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath3 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and for the first time e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . in the @xmath4 channel we set the first limits on topcolor , excluding a model of topgluons for width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . the search for topcolor in the @xmath5 channel has just begun and limits are in progress . limits are only a consolation prize ; the main emphasis of our search is to explore the possibility of a signal . although we do not have significant evidence for new particle production , the @xmath72 gev / c@xmath1 region shows upward fluctuations in all three channels . we can not ignore the exciting possibility that these apparently separate fluctuations may be the first signs of a new physics beyond the standard model . the remaining integrated luminosity for run 1b , currently being accumulated and analyzed , has the potential to either kill the fluctuations or reveal what may be the most interesting new physics in a generation . + f. abe et al . ( cdf ) , phys . lett . * 74 * , 3538 ( 1995 ) . p. frampton and s. glashow , phys . b190 * , 157 ( 1987 ) . u. baur et al . , int . j. mod . phys a2 , 1285(1987 ) & pr*d42 * , 815(1990 ) . k. lane et al . , pr*d44 * , 2768(1991 ) & phys . lett . * b327 * , 129(1994 ) . j. hewett and t. rizzo , phys . rep . * 183 * , 193 ( 1989 ) . talk by anwar bhatti in these proceedings . parameterization @xmath73 with parameters a , n and p. for new gauge bosons we use a k - factor to account for higher order terms . j. botts et al . ( cteq collaboration ) phys . lett . * b304 * , 159 ( 1993 ) . c. hill and s. parke , phys . rev . * d49 * , 4454 ( 1994 ) . c. hill , phys . b345 * , 483 ( 1994 ) . f. abe et al . ( cdf ) , phys . rev . * d50*,2966 ( 1994 ) . f. abe et al . ( cdf ) , phys . lett . * 74 * , 2626 ( 1995 ) . we model the interference between normal gluons and topgluons using hybrid model c in phys . rev . * d49 * , 4454 ( 1994 ) . g. burdman , c. hill , and s. parke private communication .

we present three searches for new particles at cdf . first , using 70 pb@xmath0 of data we search the dijet mass spectrum for resonances . there is an upward fluctuation near 550 gev / c@xmath1 ( 2.6@xmath2 ) with an angular distribution that is adequately described by either qcd alone or qcd plus 5% signal . there is insufficient evidence to claim a signal , but we set the most stringent mass limits on the hadronic decays of axigluons , excited quarks , technirhos , w@xmath3 , z@xmath3 , and e6 diquarks . second , using 19 pb@xmath0 of data we search the b - tagged dijet mass spectrum for @xmath4 resonances . again , an upward fluctuation near 600 gev / c@xmath1 ( 2 @xmath2 ) is not significant enough to claim a signal , so we set the first mass limits on topcolor bosons . finally , using 67 pb@xmath0 of data we search the top quark sample for @xmath5 resonances like a topcolor @xmath6 . other than an insignificant shoulder of 6 events on a background of 2.4 in the mass region 475 - 550 gev / c@xmath1 , there is no evidence for new particle production . mass limits , currently in progress , should be sensitive to a topcolor z@xmath3 near 600 gev / c@xmath1 . in all three searches there is insufficient evidence to claim new particle production , yet there is an exciting possibility that the upward fluctuations are the first signs of new physics beyond the standard model . fermilab - conf-95/152-e + cdf / pub / exotic / public/3192 + + * search for new particles decaying to dijets , + @xmath4 , and @xmath5 at cdf + * _ fermilab ms 318 + batavia , il 60510 + _

You are an expert at summarizing long articles. Proceed to summarize the following text: in the standard big bang model of nucleosynthesis ( sbbn ) , the light isotopes d , @xmath7he , @xmath3he and @xmath8li were produced by nuclear reactions a few minutes after the birth of the universe . given the number of light neutrino species @xmath9 = 3 and the neutron lifetime , the abundances of these light elements depend on one cosmological parameter only , the baryon - to - photon ratio @xmath10 , which in turn is directly related to the density of ordinary baryonic matter @xmath11 . the ideal objects for determination of the primordial helium abundance are blue compact dwarf ( bcd ) galaxies . these dwarf systems are the least chemically evolved galaxies known , so they contain very little helium manufactured by stars after the big bang . because the big bang production of @xmath3he is relatively insensitive to the density of matter , the primordial abundance of @xmath3he must be determined to very high precision ( better than a few percent relative accuracy ) in order to put useful constraints on @xmath12 . this precision can be achieved with very high signal - to - noise ratio optical spectra of bcds . these bcds are undergoing intense bursts of star formation , giving birth to high excitation supergiant h ii regions , which allow an accurate determination of the helium abundance in the ionized gas through the bcd s emission - line spectrum . the primordial helium mass fraction @xmath13 of @xmath3he is usually derived by extrapolating the @xmath4 o / h and @xmath4 n / h correlations to o / h = n / h = 0 , as proposed originally by peimbert & torres - peimbert ( 1974 , 1976 ) and pagel , terlevich & melnick ( 1986 ) . many attempts at determining @xmath13 have been made , using these correlations on various samples of dwarf irregulars and bcds ( e.g. , pagel et al . 1992 ; izotov , thuan & lipovetsky 1994 , 1997 , hereafter itl94 and itl97 ; olive , steigman & skillman 1997 ; izotov & thuan 1998 ; pagel 2000 ) . another way to infer the primordial @xmath3he abundance is to measure the helium abundance in the most metal - deficient bcds , which is very close to the primordial value ( e.g. , izotov et al . 1999 ) . recent @xmath13 determinations made by izotov & thuan ( 1998 ) and izotov et al . ( 1999 ) resulted in a very narrow range of @xmath13 @xmath5 0.244 0.245 . they used high signal - to - noise spectroscopic observations of bcds reduced in a homogeneous way . a self - consistent method was applied to correct he i emission line intensities for the collisional and fluorescent enhancement mechanisms which lead to deviations from the values predicted by recombination theory . the use of several he i lines allows one to discriminate between collisional and fluorescent enhancements which change the line intensities in different ways . it also allows one to estimate the importance of underlying stellar absorption in each galaxy and to improve the precision of the @xmath13 determination through the use of several lines . the details of this approach are discussed by itl97 and izotov & thuan ( 1998 ) . our present study is the continuation of helium abundance determinations in the lowest - metallicity bcds based on spectroscopic observations with the 10 m keck telescope . earlier such studies have been done for the bcds sbs 0335052 ( @xmath1/40 , izotov et al . 1999 ) and sbs 0940 + 544 ( @xmath1/27 , guseva et al . we focus here on two southern bcds , tol 1214277 and tol 65 @xmath14 tol 1223359 , the very low metallicity of which ( @xmath5 @xmath1/25 ) has been established by earlier spectroscopic work ( kunth & sargent 1983 ; campbell et al . 1986 ; pagel et al . 1992 ; masegosa , moles & campos - aguilar 1994 ; fricke et al . the motivation of our work is as follows . first , these galaxies are relatively bright targets for a large telescope which allows the derivation of the @xmath3he abundance with great precision . second , tol 1214277 and tol 65 are the second and third lowest - metallicity bcds in the pagel et al . ( 1992 ) sample ( after i zw 18 ) , for which those authors derived very low @xmath3he mass fractions @xmath4 = 0.233 and 0.231 respectively . those values are in disagreement with the significantly larger @xmath4 s derived in later studies of other very low metallicity bcds . third , recently several papers have appeared ( e.g. , ballantyne , ferland & martin 2000 ; viegas , gruenwald & viegas 2000 ; sauer & jedamzik 2001 ; peimbert , peimbert & luridiana 2001 ; stasiska & izotov 2001 ) where systematic effects on the @xmath3he abundance determination are discussed . here we estimate the systematic uncertainties for several of the best - observed low - metallicity bcds , including tol 1214277 and tol 65 . in sect . 2 we describe the observations and data reduction . heavy element abundances are derived in sect . the results of the @xmath3he abundance determination are discussed in sect . we discuss the systematic uncertainties in @xmath3he abundance determinations in sect . the summary is in sect . the keck ii spectroscopic observations of tol 1214277 and tol 65 were carried out on january 9 , 2000 , with the low - resolution imaging spectrograph ( lris ) ( oke et al . 1995 ) , using the 300 groove mm@xmath15 grating , which provides a dispersion 2.52 pixel@xmath15 and a spectral resolution of about 8 in first order . the slit was 1@xmath16180 , centered on the brightest central regions and oriented with a position angle p.a . = 19@xmath17 for tol 1214277 and 51@xmath17 for tol 65 ( fig . [ fig1 ] ) . no binning along the spatial axis has been done , yielding a spatial sampling of 02 pixel@xmath15 . the total exposure time was 45 min for each galaxy , broken into three 15-min exposures . all exposures were taken at airmasses of 1.7 and 1.8 for tol 1214277 and tol 65 , respectively . the seeing was 09 . no blocking filter was used . therefore some second - order contamination is present in the red parts of spectra at wavelengths @xmath18 7000 . the spectrophotometric standard stars feige 34 and hz 44 were observed for flux calibration . spectra of a hg - ne - ar comparison lamp were obtained before and after each observation to provide the wavelength calibration . data reduction of the observations was carried out at the main astronomical observatory of ukraine using the iraf software package . this reduction included bias subtraction , cosmic - ray removal and flat - field correction using exposures of a quartz incandescent lamp . after wavelength calibration , night - sky background subtraction , and correction for atmospheric extinction , each frame was calibrated to absolute fluxes . because both bcds were observed at large airmasses , the atmospheric differential refraction can be important ( filippenko 1982 ) . this effect is smaller for tol 1214277 because of its higher declination and slit position angle close to the parallactic one . to minimize the effect of the atmospheric differential refraction we extracted one - dimensional spectra from large apertures 1 @xmath16 68 for tol 1214277 and 1 @xmath16 84 for tol 65 . the spectra are shown in figs . [ fig2 ] and [ fig3 ] . they are dominated by very strong emission lines . remarkable spectral features in tol 1214277 are the strong nebular he ii @xmath24686 and [ fe v ] @xmath24227 emission lines suggesting a very hard stellar radiation field in the bcd ( fig . [ fig2 ] ) . the latter line was detected first by fricke et al . ( 2001 ) in the vlt spectrum of tol 1214277 . this line is absent in the spectrum of tol 65 where he ii @xmath24686 is also weaker , implying that the radiation in this galaxy is softer . the fluxes of the nebular lines have been measured by fitting gaussians to the line profiles . the errors in the line fluxes include the errors in placement of the continuum and those in the gaussian fitting . we also take into account the errors introduced by uncertainties in the spectral energy distributions of the standard stars . standard star flux deviations for both feige 34 and hz 44 are taken to be 1% ( oke 1990 ; bohlin 1996 ) . these 1@xmath6 errors have been propagated in calculations of the electron temperature , electron number density and elemental abundances . the observed and extinction - corrected emission line fluxes and observed equivalent widths are shown in table [ tab1 ] together with the extinction coefficient @xmath19(h@xmath20 ) , derived from the decrement of hydrogen emission lines which includes both milky way and target galaxy extinction , the absolute flux @xmath21(h@xmath20 ) of the h@xmath20 emission line and average equivalent width @xmath22(abs ) of the hydrogen absorption lines . the errors of the fluxes and equivalent widths of the emission lines introduced by second - order contamination are relatively small . we estimate this effect by measuring the fluxes of the [ o ii ] @xmath23727 and [ ne iii ] @xmath23868 emission lines in the second order spectra of both galaxies and find that they are respectively @xmath5 1.5% and @xmath5 3 4% of those in the first order spectra . this implies that the effect of the second - order contamination is less than 1% at wavelengths shorter 7500 , smaller than the flux errors of the weak lines seen at 7000 7300 . therefore , we do not take into account this effect . to derive heavy element abundances , we have followed the procedure detailed in itl94 and itl97 . we adopted a two - zone photoionized h ii region model ( stasiska 1990 ) including a high - ionization zone with temperature @xmath23(o iii ) , and a low - ionization zone with temperature @xmath23(o ii ) . we have determined @xmath23(o iii ) from the [ o iii]@xmath24363/(@xmath24959+@xmath25007 ) ratio using a five - level atom model . that temperature is used for the derivation of the o@xmath24 , ne@xmath24 and ar@xmath25 ionic abundances . to derive @xmath23(o ii ) , we have utilized the relation between @xmath23(o ii ) and @xmath23(o iii ) ( itl94 ) , based on a fit to the photoionization models of stasiska ( 1990 ) . the temperature @xmath23(o ii ) is used to derive the o@xmath26 , n@xmath26 , s@xmath26 and fe@xmath26 ionic abundances . for ar@xmath24 and s@xmath24 we have adopted an electron temperature intermediate between @xmath23(o iii ) and @xmath23(o ii ) following the prescriptions of garnett ( 1992 ) . the electron number density @xmath27(s ii ) ( table [ tab2 ] ) is derived from the [ s ii ] @xmath26717/@xmath26731 ratio . we point out that the flux of the [ s ii ] @xmath26717 emission line in tol 1214277 spectrum is significantly reduced due to coincidence with a night sky absorption line . this artificially low intensity results in a significant overestimate of @xmath27(s ii ) . however , the heavy element abundances in low - density h ii regions do not depend on @xmath27 . therefore , the uncertainties in @xmath27(s ii ) ( table [ tab2 ] ) do not contribute significantly to the error budget of the heavy element abundances . the oxygen abundance is derived as @xmath28 where the o@xmath25 abundance is derived from the relation @xmath29 total abundances of other heavy elements were computed after correction for unseen stages of ionization as described in itl94 and thuan , izotov & lipovetsky ( 1995 ) . the abundances of oxygen and other heavy elements obtained in this study for both galaxies are in general agreement with previous studies . our value for the oxygen abundance in tol 1214277 , 12 + log ( o / h ) = 7.54 @xmath0 0.01 , compares with values of 7.54@xmath00.04 ( campbell et al . 1986 ) , 7.59@xmath00.05 ( pagel et al . 1992 ) , 7.57@xmath00.01 ( masegosa et al . 1994 ) , 7.58@xmath30 ( kobulnicky & skillman 1996 ) , 7.52@xmath00.01 ( fricke et al . in tol 65 we derive an oxygen abundance 12 + log ( o / h ) = 7.54 @xmath0 0.01 which is in agreement with 7.53@xmath00.05 ( kunth & sargent 1983 ) , 7.42@xmath00.07 ( campbell et al . 1986 ) , 7.59@xmath00.05 ( pagel et al . 1992 ) , 7.40 7.54 ( masegosa et al . 1994 ) , 7.56@xmath31 ( kobulnicky & skillman 1996 ) . the spectral resolution of the tol 1214277 spectrum is not enough to measure the flux of the weak [ n ii ] @xmath26583 which is contaminated by the h@xmath32 emission line . therefore , to derive the nitrogen abundance we use the [ n ii]/h@xmath32 flux ratio obtained by pagel et al . ( 1992 ) from a higher resolution spectrum . we obtain log n / o = 1.64@xmath00.02 . this value is lower than previous values of 1.46@xmath00.06 ( pagel et al . 1992 ) and 1.45@xmath33 ( kobulnicky & skillman 1996 ) . the difference in n / o comes mainly from the differing fluxes of [ o ii ] @xmath23727 emission line . this line is used for determination of the ionization correction factor for nitrogen and it varies from @xmath34(@xmath23727)/@xmath34(h@xmath20 ) = 0.23 ( pagel et al . 1992 ) to 0.34 in this paper , the difference being 0.17 dex . fricke et al . ( 2001 ) derived log n / o = 1.50@xmath00.02 from vlt observations . however , no flux calibration was available for those observations and hence fricke et al . ( 2001 ) used earlier spectroscopic observations of tol 1214277 with the kpno 2.1 m telescope to calibrate their vlt spectrum . the [ n ii ] @xmath26583 emission line is stronger in the spectrum of tol 65 and we derive log n / o = 1.64 @xmath0 0.02 . all other log n / o values for tol 65 are derived from a single observation by kunth & sargent ( 1983 ) who obtained log n / o = 1.75 @xmath0 0.07 , while pagel et al . ( 1992 ) quote 1.81@xmath00.15 , and kobulnicky & skillman ( 1996 ) quote 1.79@xmath35 . our value of log n / o in tol 65 obtained from the high signal - to - noise ratio spectrum is the same as that derived for tol 1214277 and is very close to the mean value of 1.60 , derived for the most metal - deficient galaxies with 12 + log o / h @xmath36 7.6 ( thuan et al . 1995 ; izotov & thuan 1999 ) , further supporting the very low dispersion of the n / o ratio in those galaxies . such a constant n / o abundance ratio in the lowest - metallicity bcds favors primary production of nitrogen in massive stars ( izotov & thuan 1999 ) and may have important implications for analysis of the abundance patterns in high - redshift damped ly@xmath32 systems ( izotov , schaerer & charbonnel 2001 ) . the element - to - oxygen abundance ratios in tol 1214277 and tol 65 for @xmath32-product elements are the same within the errors ( table [ tab2 ] ) and they are close to the mean values derived for low - metallicity bcds ( izotov & thuan 1999 ) . the exception is iron . while the fe / o abundance ratio in tol 65 , though slightly lower , is consistent with the mean value for bcds ( izotov & thuan 1999 ) , the fe / o abundance ratio in tol 1214277 is 2@xmath6 larger . a similar fe / o abundance ratio was found by izotov et al . ( 1997 , 1999 ) for the bcd sbs 0335052 and was interpreted to be a result of the contamination of the [ fe iii ] @xmath24658 emission line by stellar or nebular c iv @xmath24658 line emission . this interpretation seems to be likely , because both sbs 0335052 and tol 1214277 are galaxies with the strongest nebular he ii @xmath24686 emission lines , implying very hard ionizing radiation . only in those two bcds was the [ fe v ] @xmath24227 emission line detected , again supporting the presence of hard radiation ( fricke et al . our derived abundance of o@xmath25 in tol 1214277 is 5.5% of the total oxygen abundance ( table [ tab2 ] ) . hence we expect that a significant amount of carbon in the h ii region is present in the form of c@xmath37 , implying the presence of c iv @xmath24658 line emission . he i emission - line fluxes are converted to singly ionized helium abundances @xmath38 @xmath14 he@xmath26/h@xmath26 using theoretical he i recombination line emissivities by smits ( 1996 ) . however , collisional and fluorescent enhancements can cause the observed he i fluxes to deviate from recombination values . in order to correct for these effects , we have adopted the following procedure , discussed in more detail in itl94 and itl97 . we evaluate the electron number density @xmath27(he ii ) and the optical depth @xmath39(@xmath23889 ) in the he i @xmath23889 line in a self - consistent way , so that the he i @xmath2@xmath23889/5876 , 4471/5876 , 6678/5876 and 7065/5876 line ratios have their recombination values , after correction for collisional and fluorescent enhancement . corrections are determined using the formulae by kingdon & ferland ( 1995 ) for collisional enhancement and the izotov & thuan ( 1998 ) fits to robbins ( 1968 ) calculations for fluorescent enhancement . the he i @xmath23889 and 7065 lines play an important role because they are particularly sensitive to both optical depth and electron number density . since the he i @xmath23889 line is blended with the h8 @xmath23889 line , we have subtracted the latter , assuming its intensity to be equal to 0.106 @xmath34(h@xmath20 ) ( aller 1984 ) , after correction for interstellar extinction and underlying stellar absorption in hydrogen lines . the singly ionized helium abundance @xmath38 and @xmath3he mass fraction @xmath4 is obtained for each of the three he i lines @xmath2@xmath2 4471 , 5876 and 6678 . we then derive the weighted mean @xmath38 of these three determinations , the weight of each line being determined by its intensity . however , this weighted mean value may be underestimated due to the lower value of @xmath38(4471 ) resulting from underlying stellar absorption . therefore , in subsequent discussions we also use the weighted mean values of @xmath4 derived from the intensities of only two lines , he i @xmath25876 and @xmath26678 . additionally , we have added to @xmath38 the abundance of doubly ionized helium @xmath40 which is derived from the he ii @xmath24686 emission line flux . finally the helium mass fraction is calculated as @xmath41}{1 + 4y } , \label{eq : y}\ ] ] where @xmath42 = @xmath38 + @xmath40 is the number density of helium relative to hydrogen ( pagel et al . 1992 ) . the results of the @xmath3he abundance determination are presented in table [ tab3 ] , where we show the adopted electron temperature @xmath23 and the derived electron number density @xmath27 in the he@xmath26 zone , the optical depth in the he i @xmath23889 emission line , ionic abundances @xmath38 and @xmath40 , total abundances @xmath42 and helium mass fractions @xmath4 derived for each line and the two weighted means . the errors in @xmath23(o iii ) and @xmath27 ( he ii ) are propagated in calculations of the helium mass fractions @xmath4 . the helium mass fraction in tol 1214277 is slightly lower for the he i @xmath24471 emission line , which is most subject to underlying stellar absorption . similarly , systematically lower @xmath4 from the he i @xmath24471 emission line was derived earlier in two bcds , sbs 0335052 ( @xmath1/40 , izotov et al . 1999 ) and sbs 0940 + 544 ( @xmath1/27 , guseva et al . 2001 ) , observed with keck . the contribution of the doubly ionized he in tol 1214277 is significant and amounts to 6% of the total he abundance . earlier , fricke et al . ( 2001 ) arrived at the same conclusion for this bcd . the helium mass fraction in tol 1214277 is @xmath4 = 0.2458 @xmath0 0.0039 when all three lines are used , and 0.2466 @xmath0 0.0043 when the he i @xmath24471 emission line is excluded . the effect of the underlying stellar absorption is more significant in tol 65 . the helium mass fraction @xmath4 derived from the he i @xmath24471 emission line is @xmath5 10% lower than that derived from other lines . the contribution of doubly ionized helium in tol 65 is smaller than that in tol 1214277 and amounts to @xmath5 1.4% of the total helium abundance . the helium mass fraction in tol 65 is @xmath4 = 0.2410 @xmath0 0.0050 when all three lines are used , and 0.2463 @xmath0 0.0057 when the he i @xmath24471 emission line is excluded . because of the effect of underlying absorption in the he i @xmath24471 emission line , we finally adopt for tol 1214277 and tol 65 respectively @xmath4 = 0.2466 @xmath0 0.0043 and 0.2463 @xmath0 0.0057 . these values are very similar to 0.2463 @xmath0 0.0015 in sbs 0335052 ( izotov et al . 1999 ) and 0.2468 @xmath0 0.0034 in sbs 0940 + 544 ( guseva et al . 2001 ) derived from the analysis of high signal - to - noise ratio keck spectra . the similarity of @xmath4 in the lowest - metallicity bcds suggests that statistical errors in the helium abundance determination are small . however , some systematic effects may change the value of @xmath4 . we already pointed out one such effect , the underlying stellar absorption , which if not accounted for , results in the underestimation of @xmath4 . this effect is more pronounced for the he i @xmath24471 emission line . recently , gonzlez delgado , leitherer & heckman ( 1999 ) have produced synthetic spectra of h balmer and he i absorption lines in starburst and poststarburst galaxies . they predict that the equivalent width of the he i @xmath24471 absorption line for young starbursts with an age @xmath43 @xmath44 5 myr , which is the case for tol 1214277 and tol 65 , can be in the range 0.4 0.6 . comparing with equivalent widths of the he i @xmath24471 emission line 7.6 and 4.8 in tol 1214277 and tol 65 , we conclude that the underestimate of the he abundance can be as high as @xmath5 10% in the case of the he i @xmath24471 emission line . for the other two he i lines @xmath25876 and @xmath26678 lines the underestimate seems to be significantly smaller because of larger emission line equivalent widths . unfortunately , gonzlez delgado et al . ( 1999 ) did not calculate equivalent widths for the he i @xmath25876 and @xmath26678 absorption lines . the weighted mean helium mass fraction in our calculations is mainly defined by the strongest he i @xmath25876 emission line with the highest weight . the upward correction of @xmath4 for this line due to the underlying stellar absorption is not larger than @xmath5 1% for tol 1214277 and @xmath5 1% 2% for tol 65 . here we assume that the equivalent width of the he i @xmath25876 absorption line is 0.4 , which is @xmath5 100 times smaller than the equivalent width of the emission line in tol 1214277 and @xmath5 80 times smaller in tol 65 ( table [ tab1 ] ) . another source of systematic uncertainties comes from the assumption that the h@xmath26 and he@xmath26 zones in the h ii region are coincident . however , depending on the hardness of the ionizing radiation , the radius of the he@xmath26 zone can be smaller than the radius of the h@xmath26 zone in the case of soft ionizing radiation and larger in the case of hard radiation . in the former case , a correction for unseen neutral helium should be made , resulting in an ionization correction factor @xmath45(he ) @xmath18 1 and hence a higher helium abundance . in the latter case , the situation is opposite and @xmath45(he ) @xmath36 1 . furthermore , the electron temperature @xmath23 in the o@xmath24 zone derived from the collisionally excited [ o iii ] lines was assumed in our calculations to be constant and is the same as that in the h@xmath46 and he@xmath46 zones . however , @xmath23(o iii ) tends to be larger than the temperature for the recombination lines of h i and he ii and , if applied , results in an overestimate of the helium abundance . both these effects have been discussed in several studies ( e.g. , pagel et al . 1992 ; itl97 ; steigman , viegas & gruenwald 1997 ; olive et al . 1997 ; viegas et al . 2000 ; peimbert , peimbert & ruiz 2000 ; ballantyne et al . 2000 ; sauer & jedamzik 2001 ) . it was shown that the correction of the helium abundance can be as high as several percent in either downward or upward directions depending on the hardness of the radiation . the hardness is characterized by the `` radiation softness parameter '' @xmath10 defined by vlchez & pagel ( 1988 ) as @xmath47 besides @xmath10 some other parameters have been used to derive @xmath45(he ) , in particular the [ o iii ] @xmath25007/h@xmath20 and [ o iii ] @xmath25007/[o i ] @xmath26300 emission line flux ratios ( ballantyne et al . 2000 ) , the ionization parameter @xmath48 , which is the ratio of ionizing photon density to gas density , and the combination of all preceeding parameters ( sauer & jedamzik 2001 ) . sauer & jedamzik ( 2001 ) calculated an extensive grid of photoionized h ii region models aiming to derive the correction factors as functions of @xmath10 and @xmath48 . their conclusion was that a downward correction of @xmath4 as much as 6% and 2% is required respectively for ionization parameters log @xmath48 = 3.0 and 2.5 ( see their fig . 18 ) . however , the downward correction is @xmath44 1% if log @xmath48 @xmath49 2.0 . for tol 1214277 and tol 65 we find respectively log @xmath10 = 0.29 and 0.08 . the ratios ( [ o iii ] @xmath24959 + 5007)/[o ii ] @xmath23727 of 19.9 and 7.2 in those galaxies ( table [ tab1 ] ) at an oxygen abundance 12 + log o / h = 7.54 correspond to an ionization parameter log @xmath48 @xmath49 2.0 ( mcgaugh 1991 ) . in particular , campbell ( 1988 ) derived log @xmath48 = 1.61@xmath50 for tol 1214277 at the 70% confidence level . with these @xmath10 and @xmath48 values , the downward correction of @xmath4 due to ionization effects and variations of the electron temperature in tol 1214277 and tol 65 is unlikely to be greater than @xmath5 1% . taking into account the fact that the upward correction of @xmath4 due to underlying stellar absorption can be as high as 1% 2% , we conclude that both effects seem to offset each other and the combined systematic uncertainty is @xmath44 1% in tol 1214277 and tol 65 . similar conclusions can be drawn for sbs 0335052 ( izotov et al . 1999 ) and sbs 0940 + 544 ( guseva et al . 2001 ) . another approach has been developed by peimbert ( 1967 ) to take into account the difference in the electron temperature in the o@xmath51 zone as compared to the h@xmath46 and he@xmath46 zones . he developed a formalism introducing an average temperature @xmath52 and a mean square temperature variation @xmath53 in an h ii region . then the temperatures in the o@xmath51 and h@xmath46 and he@xmath46 zones are expressed as different functions of @xmath52 and @xmath53 , and in hot h ii regions @xmath54(o iii ) @xmath55 @xmath54(h ii ) , @xmath54(he ii ) . this approach has been applied by peimbert et al . ( 2001 ) for the determination of the he abundance in some low - metallicity dwarf galaxies , including the two most - metal deficient bcds , i zw 18 and sbs 0335052 . they use the observations by izotov et al . ( 1999 ) and find that while the ionization correction factors @xmath45(he ) in both galaxies are very close to unity , the difference in @xmath54(o iii ) and @xmath54(he ii ) results in the reduction of the he mass fraction by 2 3 percent compared to the case with @xmath54(o iii ) = @xmath54(he ii ) . additionally , peimbert et al . ( 2001 ) considered the effect of collisional excitation of hydrogen emission lines , first noted by davidson & kinman ( 1985 ) . neglecting this effect results in an artificially large extinction and hence overcorrection of the he i @xmath25876 and @xmath26678 emission lines . peimbert et al . ( 2001 ) find that this effect in i zw 18 and sbs 0335052 leads to an the upward correction of @xmath4 by @xmath5 2% . hence , after correction for all the systematic effects considered , they obtain @xmath4 = 0.241 and 0.245 , respectively for i zw 18 and sbs 0335052 . these values are similar to @xmath4 = 0.243 and 0.246 derived earlier by izotov et al . ( 1999 ) for those galaxies . the importance of the correction for collisional excitation of the hydrogen emission lines has been pointed out also by stasiska & izotov ( 2001 ) who concluded that this effect can result in an upward @xmath4 correction of up to 5% , assuming that the excess of the h@xmath32/h@xmath20 flux ratio above the theoretical recombination value is due only to collisional excitation . however , in practice , some part of the h@xmath32/h@xmath20 flux ratio excess is due to interstellar extinction and the correction of @xmath4 for collisional excitation of the hydrogen lines is likely smaller , only @xmath5 2 3 percent , similar to the value obtained by peimbert et al . ( 2001 ) . because the physical conditions in the h ii regions of sbs 0940 + 544 ( guseva et al . 2001 ) , tol 1214277 and tol 65 ( this paper ) are similar to those in i zw 18 and sbs 0335052 , we expect that the estimates above for the systematic errors are valid for all the very metal - deficient high - excitation h ii regions considered in this paper . besides the above effects , the uncertainties in the he i recombination coefficients of @xmath5 1.5% also may play a role ( benjamin , skillman & smits 1999 ) . although some systematic effects are still difficult to estimate and are poorly studied , when taken into account together , it seems that they largely offset each other . because of the uncertainties of these effects , we conservatively assume that combined systematic error in the he abundance determination may be tentatively set to 2% ( the error is 2@xmath6 ) . our studies show that the helium mass fraction in the lowest - metallicity bcds observed with the keck telescope lies in the range 0.246 0.247 . correction for the small contribution of @xmath3he produced in stars @xmath56@xmath4 = 0.0010 0.0017 ( izotov et al . 1999 ) , results in a mean primordial @xmath3he mass fraction @xmath13 of 0.245 @xmath0 0.003 ( rms ) @xmath0 0.005(sys ) ( 2@xmath6 ) obtained from the keck observations of the four galaxies , in agreement with the previous studies of itl97 , izotov & thuan ( 1998 ) and izotov et al . this @xmath13 predicts a baryon mass fraction @xmath57 = 0.017@xmath00.005(rms)@xmath58(sys ) ( 2@xmath6 ) , consistent with 0.020@xmath00.002 ( 2@xmath6 ) derived from the primordial deuterium abundance ( burles & tytler 1998a , 1998b ; burles et al . it is also consistent with the estimation of high-@xmath59 peaks in the angular power spectrum of the cosmic microwave background ( cmb ) ( netterfield et al . 2001 ) . this overall consistency gives uniform support to the standard big bang nucleosynthesis model . in particular , if the baryon mass fraction @xmath57 = 0.022@xmath00.003 ( 1@xmath6 ) inferred from the cmb power spectrum is adopted , then , in the frame of the sbbn , the predicted primordial @xmath3he mass fraction is @xmath13 = 0.248@xmath00.001 ( lopez & turner 1999 ; burles , nollett & turner 2001 ) . our @xmath57 is also consistent with @xmath57 = 0.025@xmath00.001 ( 1@xmath6 ) derived by pettini & bowen ( 2001 ) from the deuterium abundance measurements in the @xmath60 = 2.0762 damped lyman @xmath32 system toward the qso 2206199 if 2@xmath6 systematic error of 2% in @xmath13 is assumed . the main conclusions drawn from our keck spectroscopic analysis of the extremely metal - deficient bcds tol 1214277 and tol 65 may be summarized as follows : \1 . the oxygen abundances in tol 1214277 and tol 65 are 12 + log o / h = 7.54 @xmath0 0.01 , or 1/24 solar . we find that the nitrogen - to - oxygen abundance ratio in both galaxies is log n / o = 1.64 @xmath0 0.02 , close to the mean value of 1.60 found for the other most - metal deficient bcds with @xmath61 @xmath36 @xmath1/20 ( thuan et al . 1995 ; izotov & thuan 1999 ) . alpha - product element - to - oxygen abundance ratios are in the same range as those found for bcds . the exception is the apparently higher fe / o abundance ratio in tol 1214277 which we argue is due to the contamination of [ fe iii ] @xmath24658 emission line by c iv @xmath24658 emission . the @xmath3he mass fractions in tol 1214277 and tol 65 are respectively @xmath4 = 0.2466@xmath00.0043 and 0.2463@xmath00.0057 . these values , after small corrections for the helium produced in stars , correspond to a primordial @xmath3he mass fraction of 0.245 , in excellent agreement with previous studies of itl97 , izotov & thuan ( 1998 ) and izotov et al . 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, , 278 , 683 stasiska , g. 1990 , , 83 , 501 stasiska , g. , & izotov , y. i. 2001 , , in press steigman , g. , viegas , s. m. , & gruenwald , r. 1997 , , 490 , 187 thuan , t. x. , izotov , y. i. , & lipovetsky , v. a. 1995 , , 445 , 108 viegas , s. , gruenwald , r. , & steigman , g. 2000 , , 531 , 813 vlchez , j. m. , & pagel , b. e. j. 1988 , , 231 , 257 lccrcccr 3727 [ o ii ] & 0.332@xmath00.006&0.341@xmath00.006 & 42.3&&0.634@xmath00.010&0.674@xmath00.011 & 67.1 + 3750 h12 & 0.031@xmath00.002&0.033@xmath00.003 & 4.1&&0.012@xmath00.001&0.046@xmath00.007 & 1.3 + 3770 h11 & 0.034@xmath00.002&0.035@xmath00.003 & 4.5&&0.024@xmath00.002&0.059@xmath00.005 & 2.7 + 3798 h10 & 0.050@xmath00.002&0.052@xmath00.003 & 6.8&&0.033@xmath00.002&0.068@xmath00.004 & 3.6 + 3835 h9 & 0.052@xmath00.002&0.054@xmath00.003 & 7.1&&0.046@xmath00.002&0.082@xmath00.004 & 4.9 + 3868 [ ne iii ] & 0.343@xmath00.006&0.351@xmath00.006 & 47.3&&0.246@xmath00.004&0.259@xmath00.005 & 26.1 + 3889 he i + h8 & 0.203@xmath00.004&0.208@xmath00.004 & 28.2&&0.154@xmath00.003&0.197@xmath00.004 & 16.2 + 3968 [ ne iii ] + h7 & 0.291@xmath00.005&0.298@xmath00.005 & 43.3&&0.208@xmath00.004&0.251@xmath00.005 & 21.9 + 4026 he i & 0.019@xmath00.001&0.019@xmath00.001 & 3.0&&0.012@xmath00.002&0.013@xmath00.002 & 1.3 + 4069 [ s ii ] & 0.008@xmath00.001&0.008@xmath00.001 & 1.2&&0.011@xmath00.002&0.011@xmath00.002 & 1.2 + 4101 h@xmath62 & 0.268@xmath00.005&0.274@xmath00.005 & 45.6&&0.222@xmath00.004&0.260@xmath00.005 & 25.8 + 4227 [ fe v ] & 0.007@xmath00.001&0.007@xmath00.001 & 1.3 & & & & + 4340 h@xmath63 & 0.481@xmath00.008&0.488@xmath00.008 & 96.9&&0.443@xmath00.007&0.477@xmath00.008 & 58.0 + 4363 [ o iii ] & 0.167@xmath00.003&0.169@xmath00.003 & 34.0&&0.093@xmath00.002&0.095@xmath00.002 & 12.3 + 4471 he i & 0.035@xmath00.001&0.035@xmath00.001 & 7.6&&0.034@xmath00.002&0.034@xmath00.002 & 4.8 + 4658 [ fe iii ] & 0.005@xmath00.001&0.005@xmath00.001 & 1.2&&0.006@xmath00.001&0.006@xmath00.001 & 1.0 + 4686 he ii & 0.049@xmath00.002&0.049@xmath00.002 & 12.1&&0.012@xmath00.001&0.012@xmath00.001 & 1.9 + 4711 [ ar iv ] + he i&0.027@xmath00.001&0.027@xmath00.001 & 6.7&&0.011@xmath00.001&0.011@xmath00.001 & 1.8 + 4740 [ ar iv ] & 0.016@xmath00.001&0.016@xmath00.001 & 4.0&&0.006@xmath00.001&0.006@xmath00.001 & 0.9 + 4861 h@xmath20 & 1.000@xmath00.015&1.000@xmath00.015 & 267.5&&1.000@xmath00.015&1.000@xmath00.015 & 174.1 + 4922 he i & 0.012@xmath00.001&0.012@xmath00.001 & 3.2&&0.011@xmath00.001&0.010@xmath00.001 & 1.9 + 4959 [ o iii ] & 1.707@xmath00.025&1.703@xmath00.025 & 478.3&&1.243@xmath00.019&1.212@xmath00.018 & 225.7 + 5007 [ o iii ] & 5.100@xmath00.075&5.082@xmath00.075 & 1467.2&&3.733@xmath00.055&3.628@xmath00.055 & 693.4 + 5200 [ n i ] & & & & & 0.007@xmath00.001&0.007@xmath00.001 & 1.4 + 5876 he i & 0.093@xmath00.002&0.091@xmath00.002 & 41.4&&0.111@xmath00.002&0.103@xmath00.002 & 31.4 + 6300 [ o i ] & 0.012@xmath00.001&0.011@xmath00.001 & 6.1&&0.020@xmath00.001&0.019@xmath00.001 & 6.7 + 6312 [ s iii ] & 0.008@xmath00.001&0.008@xmath00.001 & 4.2&&0.011@xmath00.001&0.010@xmath00.001 & 3.8 + 6363 [ o i ] & 0.003@xmath00.001&0.003@xmath00.001 & 1.5&&0.008@xmath00.001&0.007@xmath00.001 & 2.6 + 6563 h@xmath32 & 2.822@xmath00.042&2.737@xmath00.044 & 1571.1&&3.074@xmath00.045&2.769@xmath00.045 & 1079.6 + 6583 [ n ii ] & & & & & 0.020@xmath00.001&0.018@xmath00.001 & 4.7 + 6678 he i & 0.026@xmath00.001&0.025@xmath00.001 & 15.4&&0.031@xmath00.001&0.027@xmath00.001 & 11.2 + 6717 [ s ii ] & 0.024@xmath00.001&0.023@xmath00.001 & 14.1&&0.071@xmath00.002&0.064@xmath00.002 & 26.7 + 6731 [ s ii ] & 0.021@xmath00.001&0.020@xmath00.001 & 12.7&&0.052@xmath00.001&0.046@xmath00.001 & 19.7 + 7065 he i & 0.025@xmath00.001&0.024@xmath00.001 & 17.1&&0.031@xmath00.001&0.028@xmath00.001 & 13.6 + 7135 [ ar iii ] & 0.022@xmath00.001&0.022@xmath00.001 & 15.6&&0.035@xmath00.001&0.031@xmath00.001 & 15.7 + 7281 he i & 0.006@xmath00.001&0.005@xmath00.001 & 4.0&&0.006@xmath00.001&0.005@xmath00.001 & 2.9 + 7320 [ o ii ] & 0.005@xmath00.001&0.005@xmath00.001 & 3.8&&0.016@xmath00.001&0.014@xmath00.001 & 7.6 + 7330 [ o ii ] & 0.004@xmath00.001&0.004@xmath00.001 & 3.0&&0.011@xmath00.001&0.010@xmath00.001 & 5.4 + + @xmath19(h@xmath20 ) dex & & & + @xmath21(h@xmath20 ) & & & + @xmath22(abs ) & & & + lccc @xmath23(o iii)(k ) & 19790@xmath0260 & & 17320@xmath0240 + @xmath23(o ii)(k ) & 15630@xmath0190 & & 14770@xmath0200 + @xmath23(s iii)(k ) & 18130@xmath0210 & & 16080@xmath0200 + @xmath27(s ii)(@xmath64 ) & 400@xmath0120 & & 50@xmath050 + + o@xmath26/h@xmath26(@xmath1610@xmath65 ) & 0.273@xmath00.010&&0.614@xmath00.024 + o@xmath24/h@xmath26(@xmath1610@xmath65 ) & 2.982@xmath00.095&&2.816@xmath00.101 + o@xmath25/h@xmath26(@xmath1610@xmath65 ) & 0.191@xmath00.011&&0.046@xmath00.005 + o / h(@xmath1610@xmath65 ) & 3.447@xmath00.096&&3.477@xmath00.104 + 12 + log(o / h ) & 7.538@xmath00.012&&7.541@xmath00.013 + + n@xmath46/h@xmath26(@xmath1610@xmath8 ) & & & 1.402@xmath00.065 + icf(n ) & & & 5.66 + log(n / o ) & & & 1.642@xmath00.024 + + ne@xmath24/h@xmath26(@xmath1610@xmath65 ) & 0.418@xmath00.014&&0.425@xmath00.016 + icf(ne ) & 1.16 & & 1.23 + log(ne / o ) & 0.854@xmath00.019 & & 0.821@xmath00.021 + + s@xmath26/h@xmath26(@xmath1610@xmath8 ) & 0.414@xmath00.014&&1.118@xmath00.031 + s@xmath24/h@xmath26(@xmath1610@xmath8 ) & 2.304@xmath00.228&&4.272@xmath00.353 + icf(s ) & 2.90 & & 1.69 + log(s / o ) & 1.640@xmath00.039 & & 1.581@xmath00.031 + + ar@xmath24/h@xmath26(@xmath1610@xmath8 ) & 0.576@xmath00.024&&0.981@xmath00.036 + ar@xmath25/h@xmath26(@xmath1610@xmath8 ) & 1.252@xmath00.087&&0.596@xmath00.129 + icf(ar ) & 1.01 & & 1.03 + log(ar / o ) & 2.273@xmath00.025 & & 2.332@xmath00.039 + + fe@xmath24/h@xmath26(@xmath1610@xmath8 ) & 0.864@xmath00.222&&1.187@xmath00.274 + icf(fe ) & 15.8 & & 7.07 + log(fe / o ) & 1.404@xmath00.112 & & 1.617@xmath00.101 + @xmath66o / fe@xmath67 $ ] & 0.017@xmath00.112 & & 0.197@xmath00.101 + lccc @xmath23(o iii)(k ) & 19790@xmath0260 & & 17320@xmath0240 + @xmath27(he ii)(@xmath64 ) & 25@xmath01 & & 150@xmath050 + @xmath39(@xmath23889 ) & 0.01 & & 0.01 + + @xmath38(@xmath24471 ) & 0.0755@xmath00.0029&&0.0706@xmath00.0033 + @xmath38(@xmath25876 ) & 0.0773@xmath00.0016&&0.0812@xmath00.0022 + @xmath38(@xmath26678 ) & 0.0773@xmath00.0028&&0.0796@xmath00.0031 + @xmath38(weighted mean ) & 0.0770@xmath00.0013&&0.0784@xmath00.0016 + @xmath38(@xmath25876 + @xmath26678)&0.0773@xmath00.0014&&0.0807@xmath00.0018 + @xmath40(@xmath24686 ) & 0.0046@xmath00.0001&&0.0011@xmath00.0001 + + @xmath42(@xmath24471 ) & 0.0801@xmath00.0029&&0.0717@xmath00.0033 + @xmath42(@xmath25876 ) & 0.0819@xmath00.0016&&0.0823@xmath00.0022 + @xmath42(@xmath26678 ) & 0.0819@xmath00.0028&&0.0796@xmath00.0031 + @xmath42(weighted mean ) & 0.0816@xmath00.0013&&0.0795@xmath00.0016 + @xmath42(@xmath25876 + @xmath26678 ) & 0.0819@xmath00.0014&&0.0818@xmath00.0018 + + @xmath4(@xmath24471 ) & 0.2424@xmath00.0090&&0.2226@xmath00.0107 + @xmath4(@xmath25876 ) & 0.2466@xmath00.0050&&0.2475@xmath00.0069 + @xmath4(@xmath26678 ) & 0.2466@xmath00.0087&&0.2439@xmath00.0098 + @xmath4 ( weighted mean ) & 0.2458@xmath00.0039&&0.2410@xmath00.0050 + @xmath4 ( @xmath25876 + @xmath26678 ) & 0.2466@xmath00.0043&&0.2463@xmath00.0057 +

we present high - quality keck telescope spectroscopic observations of the two metal - deficient blue compact dwarf ( bcd ) galaxies tol 1214277 and tol 65 . these data are used to derive the heavy - element and helium abundances . we find that the oxygen abundances in tol 1214277 and tol 65 are the same , 12 + log o / h = 7.54 @xmath0 0.01 , or @xmath1/24 , despite the different ionization conditions in these galaxies . the nitrogen - to - oxygen abundance ratio in both galaxies is log n / o = 1.64@xmath00.02 and lies in the narrow range found for the other most metal - deficient bcds . we use the five strongest he i emission lines @xmath2@xmath23889 , 4471 , 5876 , 6678 and 7065 , to correct self - consistently their intensities for collisional and fluorescent enhancement mechanisms and to derive the @xmath3he abundance . underlying stellar absorption is found to be important for the he i @xmath24471 emission line in both galaxies , being larger in tol 65 . the weighted @xmath3he mass fractions in tol 1214277 and tol 65 are respectively @xmath4 = 0.2458 @xmath0 0.0039 and 0.2410 @xmath0 0.0050 when the three he i emission lines , @xmath2@xmath24471 , 5876 and 6678 , are used , and are , respectively , 0.2466 @xmath0 0.0043 and 0.2463 @xmath0 0.0057 when the he i 4471 emission line is excluded . these values are in very good agreement with recent measurements of the @xmath3he mass fraction in others of the most metal - deficient bcds by izotov and coworkers . we find that the combined effect of the systematic uncertainties due to the underlying he i stellar absorption lines , ionization and temperature structure of the h ii region and collisional excitation of the hydrogen emission lines is likely small , not exceeding @xmath5 2% ( the error is 2@xmath6 ) . our results support the validity of the standard big bang model of nucleosynthesis .

You are an expert at summarizing long articles. Proceed to summarize the following text: hole - doped manganites r@xmath4a@xmath5mno@xmath1 ( r = la , nd , pr , ... and a = sr , ca , ... ) , exhibit colossal magnetoresistive@xcite ( cmr ) properties , associated with the mixed manganese valence mn@xmath6 ( @xmath7)/mn@xmath8(@xmath9 ) resulting from the substitution of trivalent r ions by @xmath10 divalent a ions . the @xmath10 @xmath11 0.5 doping is particularly interesting , since the magnetic interaction is affected by i ) the ordering of the mn@xmath6 and mn@xmath8 charges@xcite , commonly referred as charge - ordering ( co ) , ii ) the ordering of the @xmath12 electron orbitals on the mn@xmath6 sites and iii ) the coupling between orbital degree of freedom and the lattice . for example , nd@xmath0ca@xmath0mno@xmath1 ( ncmo ) undergoes a co transition@xcite at @xmath2=245k with partial orbital ordering ( oo ) and magnetic correlations of short range . at lower temperatures , the oo increases and a long range ce - type@xcite antiferromagnetic ( afm ) state is established at @xmath3=145k . an insulator - to - metal transition occurs around this temperature in intermediate ( @xmath115 t or higher ) magnetic fields@xcite . the ce - type phase involves both fm and afm interactions . it consists of ferromagnetic ( fm ) chains in the ( a , b ) plane , afm coupled between each other within this plane , and along the @xmath13-axis . neutron@xcite and brillouin@xcite scattering experiments detect additional fm correlations in the low temperature ce - type afm phase . also resistivity noise measurements reveal two - level fluctuations@xcite related to the phase separation scenario@xcite , including mixed - phase states of different magnetic and electrical properties . in this context , the observed magnetic field induced metal - insulator transition@xcite reflects the percolative growth of conducting fm clusters embedded in an afm matrix@xcite . + in the present article , we investigate the stability of the low temperature afm phase of a nd@xmath0ca@xmath0mno@xmath1 single crystal using ac and dc magnetization measurements . the zero field cooled ( zfc ) and field cooled ( fc ) magnetizations are recorded vs. temperature and time after specific cooling protocols . similar measurements are performed on a gd@xmath0ca@xmath0mno@xmath1 ( gcmo ) single crystal for comparison . gcmo also shows charge - ordering at @xmath2 = 260k , but no long range afm , and it remains insulating at all temperatures , even in large magnetic fields@xcite . in the case of ncmo , the magnetization is cooling rate dependent below @xmath3 , and a weak spontaneous moment appears in the afm state . the corresponding excess magnetization appearing along the zig - zag chains of the ce - type structure is related to the presence of domain walls in the ( a , b ) plane breaking the orbital coherency . single crystals of ncmo@xcite and gcmo were grown using a floating zone furnace ( nec , japan ) . temperature and time dependent zero field cooled and field cooled magnetization measurements were performed using a quantum design mpms5 superconducting quantum interference device ( squid ) magnetometer . in the zfc and fc case , the magnetization @xmath14 was collected on re - heating in a small magnetic field @xmath15=20 oe after slow ( 3k / min ) and fast ( 60k / min ) cooling from room temperature down to 5k . @xmath14 vs. @xmath15 measurements were performed at @xmath16=35k after similar cooling protocols . additional ac - susceptibility @xmath17 measurements ( @xmath18=125hz , @xmath19=20 oe ) were recorded using the same cooling protocol on a lakeshore 7225 susceptometer for comparison . the volume susceptibility , @xmath20 in dc and @xmath17 in ac , is in the following plotted in dimensionless ( si ) units . the correspondence between @xmath20 in ( si ) units and @xmath14 in bohr magnetons per formula unit ( @xmath21/f.u ) is indicated in the text and figures . figure [ fig1 ] shows the cooling rate dependence of the zfc and fc magnetization for ncmo . ( a ) shows the temperature dependence of the zfc ( markers ) and fc ( simple line ) magnetization , measured on re - heating after fast ( continuous lines ) and slow ( dotted ) cooling . in the case of fast cooling or quench to low temperatures , an excess magnetization @xmath22 appears below @xmath3 , both in the zfc and fc curves . difference plots of the fc and zfc curves ( same symbols as in the main frame ) are added in the insert , showing that the excess magnetization @xmath22 [ = @xmath23 @xmath24 @xmath25 appears slightly below @xmath3 , around @xmath16=130k . this excess magnetization relaxes with time , as illustrated in fig . [ fig1](b ) which shows the temperature dependence of the zfc magnetization @xmath26(@xmath16 ) : as in fig . [ fig1](a ) , @xmath26 is recorded on reheating from the lowest temperature @xmath27=5k up to a temperature @xmath28=80k below @xmath3 after a fast cooling . the @xmath26(@xmath29 ) curve is marked using circles . the sample is cooled back ( 3k / min ) to @xmath27 and @xmath14 is recorded on re - heating up to @xmath30=300k , well above @xmath31 ; the @xmath26(@xmath32 ) curve is marked using crosses . the sample is cooled down ( 3k / min ) to @xmath27 , and @xmath14 is once again recorded up to @xmath30 ; the @xmath26(@xmath33 ) curve is plotted using a simple line . the @xmath26(@xmath16 ) curves obtained in fig . [ fig1](a ) are added as dotted lines for comparison . as expected , both the first ( from @xmath27 to @xmath28 after a fast cooling ) and the last ( from @xmath27 to @xmath30 after a slow cooling ) measurements coincide with the earlier results shown in fig . [ fig1](a ) . the second measurement , recorded from @xmath27 to @xmath30 after a temperature cycle to @xmath28 , yields instead a lower @xmath22 , showing that the excess magnetization is relaxing with time . it should be stressed that only a small dc - magnetic field , acting essentially as a non perturbing probe of the system , is used to record the magnetization ; i.e. the observed effects are not driven by the magnetic field employed in the experiments@xcite . to evidence this , we show in insert of fig . [ fig1](b ) the temperature dependence of the in - phase component of the ac - susceptibility recorded under the same conditions as the zfc and fc magnetization , after fast ( simple line ) and slow ( dotted line ) cooling ; a similar cooling rate dependence is observed in the ac - susceptibility . + in our weak probing field , @xmath22 amounts to 2.9@xmath3410@xmath35 @xmath21/f.u at @xmath16=35k ( c.f . fig . [ fig1](a ) ) . @xmath14 vs. @xmath15 measurements up to higher fields ( 4000 ka / m ) recorded after fast ( continuous line ) and slow ( dotted line ) cooling to @xmath16=35k are shown in figure [ fig2 ] . in both cases , a closely linear field dependence of the magnetization is observed , reflecting the afm order . the difference plot of the two curves ( see insert ) reveals a weak spontaneous moment of 4.4@xmath3410@xmath36 @xmath21/f.u . , superposed on a small excess susceptibility . the small moment reflects the presence of uncompensated spins in the afm state . the origin of these local defects of the magnetic structure will be discussed below . + in the above described measurements , @xmath14 was recorded using a magnetic field directed along the @xmath13-axis of ncmo . figure [ fig3](a ) shows the corresponding results on the zfc magnetization measured in the ( a , b ) plane . a cooling rate dependence is again observed , but the excess magnetization has a smaller magnitude , as seen in the main frame and also in the insert where the difference plots of the zfc curves for the two different orientations of @xmath15 are shown . most of the excess magnetization thus lies along the @xmath13 direction . the single crystals show no sign of twinning , but of course some magnetic anisotropy in the ( a , b ) plane would affect the magnitude of @xmath22 . + in the case of gcmo , a cooling rate dependence is not observed , and the magnetizations curves recorded along @xmath13 after slow and fast cooling virtually coincide , as shown in fig . [ fig3](b ) . on the other hand , the zfc and fc curves deviate from each other below @xmath37100k , indicating irreversibility below this temperature , and thus the development of magnetic correlation . the insert of fig . [ fig3](b ) shows the normalized difference between the zfc and fc curves for both gcmo and ncmo , which is a measure of the irreversibility . in the case of ncmo , the irreversibility arises sharply at @xmath3 , while it appears more gradually in gcmo . the magnetic correlation developing below @xmath37 100k in gcmo seems to remain of short range , as in the very similar y@xmath0ca@xmath0mno@xmath1@xcite manganite . the excess magnetization , and the observed cooling rate dependence of the magnetization at low temperatures in ncmo are instead related to the establishment of the long range afm state , possibly via the large increase in orthorhombic distortion@xcite and @xmath13-axis contraction@xcite occurring between @xmath2 and @xmath3 upon cooling . the spontaneous moment could then be related to the presence of defects in the low temperature antiferromagnetic arrangement , or an antiferromagnetic domain state . in this case , uncompensated spins at domain walls would give rise to an excess magnetization . the curie - like increase of the magnetization at very low temperatures ( below @xmath16 @xmath11 25k ) both in ncmo and gcmo is attributed to paramagnetic nd and gd ions respectively.@xcite + to further elucidate the dynamic nature of the magnetization of ncmo , we have performed fc - relaxation experiments , in which the fc magnetization @xmath38 is recorded versus time ( @xmath39 ) during @xmath40=10000s at @xmath41=35k after different thermal protocols ( * a * , * b * , * c * and * d * ) . the obtained @xmath38(@xmath16 ) curves are plotted in fig . [ fig4](a ) , and the corresponding relaxation curves @xmath38(@xmath39 ) are shown in fig . [ fig4](b ) ; all curves are labelled on the figure according to the thermal protocol ( * a*-*d * ) employed . * a * : the sample is rapidly cooled ( fast cooling rate ) to the measurement temperature @xmath41 @xmath42 @xmath3 in @xmath15=20 oe , and after achieving temperature stability ( @xmath1120s ) , the magnetization is recorded vs. time during @xmath40 . * b * : the sample is rapidly cooled to the lowest temperature , and the fc - relaxation collected after re - heating to @xmath41 . the temperature dependence of the magnetization is recorded during the re - heating to @xmath41 and above . * c * : the sample is rapidly cooled to @xmath41 , from where the cooling proceeds with a slower rate and the magnetization is recorded on cooling and re - heating to @xmath41 , where the fc relaxation is collected during @xmath40 . as in * b * , the magnetization is further recorded during the re - heating to room temperature . * d * is similar to * c * , but using a slower cooling rate when initially cooling to @xmath41 . as seen in fig . [ fig4](b ) , the relaxation curves obtained for experiments * a * and * b * are nearly identical , showing again that the cooling rate at all temperatures below @xmath3 is the key parameter of our effect . as observed earlier in fig . [ fig1](b ) , the relaxation diminishes when the effective cooling slows down , from experiments * a*,*b * to experiment * d*. the relaxation at @xmath41 reflects the evolution of the domain configuration of the afm state . + neutron powder diffraction studies on a similar charge and orbital ordered ce - type afm manganite@xcite indicate magnetic disorder in the mn@xmath6 sublattice , associated to domain boundaries breaking the long range orbital ordering . recent x - ray scattering results@xcite also indicate a partial orbital ordering of the low temperature phase , leading to an orbital domain state . the here observed cooling rate dependence of @xmath14 below @xmath3 could thus be related to the intrinsic inhomogeneities of the ce - type structure , and the nucleation or rearrangement of orbital domains and domain walls to accommodate the large contraction of the structure occurring upon cooling . the cooling rate determines the time allowed to the system to accomodate the structural modifications governed by the temperature . in a similar way , one can also perturb the ce - type state by introducing impurities in the structure , for example by replacing some of the mn cations by cr or ru@xcite . the orbital ordering is again affected , and fm - like correlations induced . + the cooling rate dependence of the fc magnetization depicted in fig . [ fig4](a ) is unusual and deserves some additional comments . in experiment b * , the magnetization curve always lies above the curve obtained for the slow cooling case , closely following the curve obtained for a fast cooling . in experiments * a * and * c * instead , @xmath43 at @xmath41=35k amounts to 0.0185 [ si ] immediatly after a rapid cooling from room temperature and the magnetization curve thus surprisingly remains below the fc curve obtained for a slow cooling . during a fast cooling to @xmath41 , the system can not accomodate the excess moments appearing with the domain walls , and its magnetization remains lower than in the slow cooling case . in experiment * b * , the sample is rapidly cooled down to the lowest temperature , and reaches a high magnetization value . albeit having a different initial magnetization level , the relaxation curves corresponding to experiments * a * and * b * ( shown in fig.[fig4](b ) ) appear nearly identical , which indicates that a similar magnetic configuration is probed in both experiments . the magnetization of a single crystal of the charge ordered manganite nd@xmath0ca@xmath0mno@xmath1 is cooling rate dependent below @xmath3 . the results reveal the presence of a weak spontaneous moment , related to uncompensated spins in the ce - type afm structure . the moment and its associated excess magnetization are connected to domain walls separating fully orbital ordered parts of the zig - zag chains of the ce - type structure . the cooling rate is then used to probe the orbital state , and study how it accommodates the large structural transformation occurring upon cooling . + the presence of magnetic inhomogeneities in the low temperature ce - type structure of nd@xmath0ca@xmath0mno@xmath1 is likely to weaken the afm state , so that the application of a large magnetic field could induce the observed insulator - metal transition near @xmath3 . in the case of gd@xmath0ca@xmath0mno@xmath1 , no long range antiferomagnetism is established at any temperature , and neither a cooling rate dependence nor an insulator - metal transition is observed . j. p. hill , c. s. nelson , m. v. zimmermann , y .- kim , d. gibbs , d. casa , b. keimer , y. murakami , c. venkataraman , t. gog , y. tomioka , y. tokura , v. kiryukhin , t. y. koo , and s .- w . cheong , _ unpublished , cond - mat/0105064_. = 20 oe . ( a ) the magnetization is recorded along the c - axis on re - heating after fast ( continuous line ) and slow ( dotted line ) cooling down to 5k . the corresponding value of @xmath38(@xmath16=35k ) in @xmath21/f.u is added for comparison . the insert shows the difference plots of the fc and zfc magnetization curves ( same symbols as in the main frame ) . the maximum value of @xmath22 in @xmath21/f.u is also indicated . ( b ) idem adding a cooling and re - heating to a temperature @xmath28 below @xmath3 while recording @xmath26(@xmath16 ) . @xmath27 , @xmath28 and @xmath30 correspond to @xmath16=5 , 80 and 300k respectively ; see main text . the insert shows the cooling rate dependence of the ac - susceptibility of ncmo , recorded vs. temperature using a small ac - field , after similar fast and slow cooling . @xmath19=20 oe , @xmath18=125hz . the simple line corresponds to fast cooling , and the dotted line to slow cooling . ] vs. @xmath15 up to high magnetic fields recorded after fast ( continuous line ) and slow ( dotted line ) cooling of ncmo ; @xmath16=35k . the insert shows the corresponding @xmath22 [ = @xmath23 @xmath24 @xmath25 difference plot . ] is measured along the c - axis and both zfc ( markers ) and fc ( simple line ) are measured with fast ( continuous line ) and slower ( dotted line ) cooling rates . the insert shows the temperature dependence of the irreversibility observed in the magnetization curves of ncmo and gcmo . ] for ncmo . intermediate temperature stops are made during the cooling and heating at @xmath44=35k ( see main text ) ( a ) shows the results plotted vs. temperature , as well as the reference @xmath38(@xmath16 ) curves from fig . 1(a ) in dotted lines . curve * a * is plotted separately for clarity ; @xmath38 was only recorded vs. time in this experiment . ( b ) shows the the relaxation of @xmath38 during @xmath40 at @xmath41=35k . ]

the low temperature phase of single crystals of nd@xmath0ca@xmath0mno@xmath1 and gd@xmath0ca@xmath0mno@xmath1 manganites is investigated by squid magnetometry . nd@xmath0ca@xmath0mno@xmath1 undergoes a charge - ordering transition at @xmath2=245k , and a long range ce - type antiferromagnetic state is established at @xmath3=145k . the dc - magnetization shows a cooling rate dependence below @xmath3 , associated with a weak spontaneous moment . the associated excess magnetization is related to uncompensated spins in the ce - type antiferromagnetic structure , and to the presence in this state of fully orbital ordered regions separated by orbital domain walls . the observed cooling rate dependence is interpreted to be a consequence of the rearrangement of the orbital domain state induced by the large structural changes occurring upon cooling .

You are an expert at summarizing long articles. Proceed to summarize the following text: the construction of background field formalism for n=2 super - yang - mills theory ( sym ) in projective hyperspace ( @xmath0 ) @xcite is an open problem . such a formalism is desirable for any ( non-)supersymmetric theory as it simplifies ( loop ) calculations and even intermediate steps respect gauge covariance . a major obstacle in solving this problem for the n=2 case seems to be the lack of knowledge relating the gauge connections to the tropical hyperfield @xmath1 , which describes the sym multiplet for all practical purposes @xcite . we note that the closely related @xcite n=2 harmonic superspace ( @xmath2 ) @xcite does nt encounter this issue as the hyperfield , @xmath3 describing the sym multiplet is itself a connection , @xmath4 . in fact , background field formalism in harmonic superspace has quite a straightforward construction @xcite . although the construction has some subtleties , it has been refined in a series of papers along with relevant calculations @xcite . in this paper , we solve the problem of constructing the background field formalism in projective superspace without the need for knowing the connections explicitly in terms of @xmath1 . this is possible by choosing the background fields to be in a ` real ' representation ( @xmath5 ) and the quantum fields to be in the ` analytic ' representation ( @xmath6 ) . this is reminiscent of the quantum - chiral but background - real representation used in n=1 superspace @xcite . what this does is make the effective action independent of @xmath7 and dependent on background fields ( like @xmath8 ) with ` dimension ' greater than @xmath9 ( since the lowest one is a spinor ) . non - existence of @xmath9-dimension background fields ( like @xmath7 ) is a crucial requirement for the non - renormalization theorems to hold as discussed in @xcite . this directly leads to a proof of finiteness beyond 1-loop . ( a different approach for proof of finiteness has been discussed in @xcite . ) the coupling of quantum fields to background fields comes through the former s projective constraint alone , which simplifies the vertex structure a lot . the calculations are also simplified at 1-hoop as most @xmath10-integrals turn out to be trivial since the background fields have trivial @xmath10-dependence . this means that the @xmath10-integration effectively vanishes from the effective action and as expected from the supergraph rules , only one @xmath11-integration survives at the end of the calculations . we also work in fermi - feynman gauge so there are no ir issues to worry about while evaluating the super - feynman graphs . another important aspect is the ghost structure of the theory in this background gauge . apart from the expected faddeev - popov ( fermionic @xmath12 ) and nielsen - kallosh ( bosonic @xmath13 ) ghosts , we require two more extra ghosts , namely real bosonic @xmath14 and complex fermionic @xmath15 . this is in contrast to n=1 sym but very similar to the harmonic treatment of n=2 theory . heuristically , we can even see that such a field content would give a vanishing @xmath16-function for n=4 . moreover , we will see that the loop contributions of @xmath1 and extra ghosts have spurious divergences arising due to multiple @xmath17 s . these are very similar to the ` coinciding harmonic ' singularities in the @xmath2 case , which manifest themselves at 1-loop level via the subtleties regarding regularization of similar looking determinants . however , in @xmath0 case , we do not encounter such striking similarities . only the divergences turn out to be similar , leading to a cancellation between the vector hyperfield s contribution and that of the extra ghosts . the finite pieces in the effective action are contributed by these extra ghosts only . this section is mostly built on the ordinary projective superspace construction of sym detailed in @xcite . we review it briefly below for the sake of continuity . we also use the 6d notation to simplify some useful identities involving background covariant derivatives and moreover , the results carry over to n=1 6d sym in a trivial manner with this notation . the projective hyperspace comprises of usual spacetime coordinates ( @xmath18 ) , four fermionic ones ( @xmath11 ) and a complex coordinate on cp@xmath19 ( @xmath10 ) . the full n=2 superspace requires four more fermionic coordinates ( @xmath20 ) in addition to these projective ones . the super - covariant derivatives corresponding to these extra @xmath20 s define a projective hyperfield ( @xmath21 ) via the constraint @xmath22 . the algebra of the covariant derivatives will be given below but we note here that in the ` real ' representation ( called ` reflective ' in @xcite and the one we use extensively in this paper ) the @xmath23 s are @xmath10-dependent . their anti - commutation relation at different @xmath10 s is all that we need here : @xmath24 the scalar hypermultiplet is described by an ` arctic ' hyperfield ( @xmath25 ) that contains only non - negative powers of @xmath10 and the vector hypermultiplet by a ` tropical ' @xmath1 , which contains all powers of @xmath10 . to construct the relevant actions , the integration over this internal coordinate is defined to be the usual contour integration , with the contour being a circle around the origin ( for our purposes in this paper ) . so , the projective measure simply reads : @xmath26 ( with the usual factor of @xmath27 being suppressed ) . now , we are ready to delve into the details of the background field formalism . the gauge covariant derivatives , @xmath28 , describing n=2 sym satisfy the following ( anti- ) commutation relations ( written in 6d notation ) : @xmath29=-_{}w_a^{}\,,\\ \{_{a},w_b^{}\}={{\cal d}}_{ab}_{}^{}-\tfrac{{\dot{\iota}}}{2}c_{ab}f_{}^{}\,,\\ [ _{},^{}]=f_^{[}_^{]}\,,\\ [ _{},_y]=_{}\,,\quad [ _{},_y]=0\ , , \label{dyd}\end{gathered}\ ] ] where the su(2 ) index @xmath30 , @xmath31 and @xmath32 are the field strengths , and @xmath33 are the triplet of auxiliary scalars . the 4d scalar chiral field strength , @xmath34 is related to the spinor field strength via appropriate spinor derivatives . we solve the commutation relation for @xmath35 by writing @xmath36 , where @xmath37 is an unconstrained complex hyperfield . we can do a background splitting of @xmath37 ( similar to n=1 superspace ) such that @xmath38 with @xmath39 being the background covariant derivative . we can now choose ` real ' representation for the background derivatives independently such that @xmath40 . this simplifies the @xmath10-dependence of the connections : @xmath41 since these connections have simple @xmath10-dependence , the @xmath10-integrals in the effective action can be trivially done . moreover , the quantum part of the full covariant derivatives then can be chosen to be in ` analytic ' representation , @xmath42 , @xmath43 and @xmath44 . the projective ( analytic ) constraint on hyperfields ` lifts ' to @xmath45 so we can now define a background projective hyperfield @xmath46 as @xmath47 such that @xmath48 . then , the scalar hypermultiplet s action reads : @xmath49 the vector hyperfield @xmath1 s action looks the same as in the ordinary case ; the difference being that the @xmath1 appearing below is only the quantum piece and is background projective : @xmath50 we know from @xcite that this action should give an expression for @xmath7 and hence the ` analytic ' representation for quantum hyperfields is a consistent choice . the background dependence of @xmath1 comes through the projective constraint and the background covariant derivatives only . the following identities will be useful in showing that and deriving other results in the following sections : @xmath51_^4\,,\\ _{1}^4_{2}^4=\left[y_{12}{{\cal d}}_{}+\tfrac{1}{2}y_{12}^2{\check{}}+\tfrac{1}{2}y_{12}^3{\left(}_{,}^{}_{,}+w_{}^ _{,}+2{{\cal d}}_{}{\right)}+y_{12}^4_{2}^4\right]_{2}^4\,,\end{gathered}\ ] ] where @xmath52 is the gauge - covariant dalembertian and @xmath53 . as the quantum connections do not appear explicitly in the calculations , we will drop the usage of curly fonts to denote the background fields ( as has been done above ) and also the subscript ` @xmath54 ' on @xmath55 from now on . the quantization procedure in the background gauge proceeds similar to the ordinary case . the ordinary derivatives are now background - covariant derivatives so @xmath56 gets replaced by @xmath57 ( or @xmath58 ) everywhere . moreover , we need extra ghosts for the theory to be consistent in this formalism as we elaborate further in the following subsections . the scalar hypermultiplet is background projective but the structure of its action is still the same as in the ordinary case . that means the kinetic operator appearing in the equations of motion is @xmath59 , @xmath42 , @xmath60 still holds . so the derivation of the propagator performed in @xcite goes through after employing these changes : @xmath61 and @xmath62 : @xmath63 the gauge - fixing for the vector hypermultiplet leading to faddeev - popov ( fp ) ghosts is still similar to the ordinary case and we just quote the results with suitable modifications : @xmath64v_2\,;\label{sgfinv}\\ { { \cal s}}_{fp}&=-{\text{tr}}\int dx\,d^4\theta\,dy\,\left[\bar{b}\,c+\bar{c}\,b+(y\,b+\bar{b})\frac{v}{2}\left(c+\frac{\bar{c}}{y}\right)+ ... \right].\label{fpaction}\end{aligned}\ ] ] the propagators for the fp ghosts are similar to the scalar hypermultiplet and will be written down later . we will always work in fermi - feynman gauge ( @xmath65 ) but let us derive the propagator for @xmath1 with arbitrary @xmath66 as this technique will be useful later . we first combine the terms quadratic in @xmath1 from the above equation and the vector hypermultiplet action to get @xmath67_{1}^4 v_2\nonumber\\ = & -\frac{{\text{tr}}}{2g^2}\int dx\,d^4\,dy_1\,dy_2\,v_1\frac{1}{y_{12}^2}\left[1+\frac{1}{}{\left(}-1+\frac{y_1+y_2}{2}(y_{12}){\right)}\right]y_{12}^2\left(\frac{1}{2}{\check{}}+\cdots\right ) v_2\nonumber\\ = & -\frac{{\text{tr}}}{2g^2}\int dx\,d^4\,dy_1\,dy_2\,v_1\left[1+\frac{-1+y_1(y_{12})}{}\right]\left(\frac{1}{2}{\check{}}+\cdots\right ) v_2\,.\end{aligned}\ ] ] then , we add a generic real source @xmath68 to the quadratic gauge - fixed vector action : @xmath69\frac{1}{2y_{12}^2}v_2-dx\,d^8\,dy_2\,j_2v_2\right\}\nonumber\\ = & -\frac{{\text{tr}}}{g^2}\left\{\int dx\,d^4\,dy_{1,2}\,v_1\left[1+\frac{-1+y_1(y_{12})}{}\right]\frac{_{1}^4}{2y_{12}^2}v_2-dx\,d^4\,dy_2\,{{\cal j}}_2v_2\right\}.\label{svj}\end{aligned}\ ] ] here , @xmath70 is now defined to be ( background ) projective . now , equation of motion for @xmath1 reads @xmath71\frac{_{2}^4}{y_{12}^2}={{\cal j}}_2\,,\label{eomf}\ ] ] which we can solve to write @xmath1 in terms of @xmath70 . this amounts to inverting the kinetic operator for @xmath1 as we will see . assuming the following ansatz for @xmath1 : @xmath72 and demanding it satisfy ( [ eomf ] ) , we are led to @xmath73 because @xmath74\left[1+\frac{-1+y_1(y_{12})}{}\right]=(y_{02})\,.\ ] ] plugging ( [ eomf ] ) and ( [ fjk ] ) in the action ( [ svj ] ) , we get @xmath75 which leads to the required propagator , first derived ( for the ordinary case ) in @xcite @xmath76 this expression simplifies @xcite for @xmath65 to @xmath77 as does the quadratic part of the vector action @xmath78 in background field gauge , the gauge fixing function leads to additional ghosts apart from the fp ghosts , which contribute to the 1-loop calculations . to see that , consider the effective action @xmath79 defined by the following functional : @xmath80 where @xmath81 is found by the normalization condition @xmath82 . it gives @xmath83 so ( [ expg ] ) simplifies to @xmath84 we can rewrite the last factor as @xmath85 where @xmath86 are unconstrained hyperfields . proceeding similar to the harmonic case @xcite , we redefine @xmath87 and introduce nielsen - kallosh ( nk ) ghost @xmath13 to account for the resulting jacobian . this means the 1-loop contribution for n=2 sym coupled to matter simplifies to : @xmath88 for n=4 , the scalar hypermultiplet is in adjoint representation and its contribution will cancel the joint fp and nk ghosts contributions . the remaining two terms have spurious divergences due to multiple @xmath17 s but their joint contribution has to be finite , which will turn out to be the case as we develop this section further . to incorporate the effect of @xmath86 fields directly in the path integral , we choose to introduce a real scalar @xmath14 and a complex fermion @xmath15 as follows : @xmath89 where @xmath90 so the background field requires 3 fermionic ghosts @xmath91 and 2 bosonic ghosts @xmath92 and the full quantum action for n=2 sym coupled to matter reads : @xmath93+s_{fp}(v , b , c)+s_{nk}(v , e)+s_{xr}(v , x , r)+s_{}(v,).\ ] ] the fp and nk ghosts are background projective hyperfields . the actions for these ghosts look the same as those in the case of non - background gauge . the action for fp ghosts is given in equation ( [ fpaction ] ) and that for nk ghost is similar to the scalar hypermultiplet s action . that means their propagators are straightforward generalizations and read @xmath94 now , we focus on the new ingredient of the background field formalism : the extra ghosts . in the same vein as the vector hypermultiplet , we can simplify the actions of these ghosts . let us just concentrate on the scalar ghost action as the fermionic ghost can be treated similarly : @xmath95x_2\\ = & \,-\frac{{\text{tr}}}{4}dx d^4dy_{1,2}\,x_1\left[{\left(}\frac{y_1}{y_{21}}+\frac{y_2}{y_{12}}{\right)}\frac{1}{y_{12}^2}{\widehat{}}\right]x_2\\ = & \,-\frac{{\text{tr}}}{4}dx d^4dy_{1,2}\,x_1\left[\frac{-1+y_1{\left(}y_{12}{\right)}}{y_{12}^2}{\widehat{}}\right]x_2\,.\end{aligned}\ ] ] the @xmath14 propagator can then be derived in a similar way as the vector propagator with arbitrary @xmath66 . lets add a source term to the action for x ghost : @xmath96x_2+{\text{tr}}dx\,d^8\,dy_2\,j_2x_2\nonumber\\ = & -\frac{{\text{tr}}}{4}dx\,d^4\,dy_{1,2}\,x_1{\left(}\frac{-1+y_1(y_{12})}{y_{12}^2}{\right)}{\widehat{}}x_2+{\text{tr}}dx\,d^4\,dy_2\,{{\cal j}}_2x_2\,.\end{aligned}\ ] ] the equation of motion for @xmath14 now reads @xmath97 adopting an ansatz for @xmath14 ( similar to what was done for @xmath1 before ) , @xmath98\frac{1}{\frac{1}{2}{\widehat{}}^2}2{{\cal j}}_0\,,\ ] ] we find that @xmath99 and @xmath100 satisfy ( [ xeom ] ) . collecting all the results , the action reduces to @xmath101 which leads to the required propagator @xmath102 the propagator for the fermionic @xmath15 ghost has a similar expression . given this new construction of the background field formalism for sym , we can now employ it to calculate contributions to the effective action coming from different hypermultiplets . the general rules for constructing diagrams in the background field formalism are similar to the ordinary case discussed in @xcite . however , as expected in this formalism , the quantum propagators form the internal lines of the loops and the external lines correspond to the background fields . the @xmath58 and @xmath57 operators in the propagators need to be expanded around @xmath103 ( the connection - independent part of @xmath56 ) , which will generate the vertices with the vector connection and background fields . for the extra ghosts , we can further simplify the nave rules by noticing that the vertices have @xmath104-factor and the propagator will generate such a factor in the numerator due to the presence of @xmath105 . thus , we can remove them from the very start and work with the revised propagator and vertex for the purpose of calculating diagrams . let us now collect all the relevant feynman rules below . @xmath106{\left(}{\widehat{}}-_0{\right)}\end{aligned}\ ] ] [ [ scalar ] ] scalar + + + + + + the one - loop contribution from the scalar hypermultiplet to the effective action can not be written in a fully gauge covariant form with a projective measure . thus , the diagrammatic calculation required to get this contribution ( which includes the uv - divergent piece too ) is not accessible via the formalism constructed here . we note that such an issue appears in the n=1 background formalism too when the scalar multiplets in complex representation are considered . the calculations can not be performed covariantly and explicit gauge fields appear in addition to the connections . [ [ vector ] ] vector + + + + + + the contribution to one - loop n - point diagrams from vector hypermultiplet running in the loop would be given by the following : @xmath107 where the numerical subscript on @xmath108 denotes the external momenta dependence . as usual , to kill the extra @xmath109-function , at least four @xmath110 should be available from the vertices and so @xmath111 . the first non - vanishing contribution is from the 4-point diagram : @xmath112 too many @xmath17 s lead to spurious @xmath113 singularity , similar to ` coinciding harmonic ' singularities in @xmath2 . these will cancel when we take into account the @xmath114 ghosts . [ [ extra - ghosts ] ] extra ghosts + + + + + + + + + + + + their combined contribution to one - loop n - point diagrams reads : @xmath115\nonumber\\ & { \left(}w^(1)_{,}+ ... {\right)} ... \,_{n}^4^8(_{n1 } ) \frac{(y_{nb,1b})}{y_{nb}}\frac{1}{k_n^2}\left[{\left(}-1+y_{na } (y_{na , nb}){\right)}\right]{\left(}w^(n ) _{,}+ ... {\right)}\nonumber\\ \sim&-d^4kd^4_ndy_{1a, ... ,1b}_{1b}^4^8(_{n1})\frac{{\left(}-1+y_{1a}(y_{1a,2b}){\right)}}{y_{1a}}\frac{1}{k_1 ^ 2}\nonumber\\ & { \left(}w^(1 ) _{,}+ ... {\right)} ... \,\frac{{\left(}-1+y_{na}(y_{nb,1b}){\right)}}{y_{1b}}\frac{1}{k_n^2}{\left(}w^(n ) _{,}+ ... {\right)}.\label{fxn}\end{aligned}\ ] ] again , the first non - vanishing contribution is from @xmath116 that has the same @xmath117 singularity structure as the vector in ( [ 4pdiv ] ) leading to a cancellation , in addition to the following finite part : @xmath118 the last line follows because only @xmath10-independent pieces of @xmath119 s can survive the @xmath10-integrals . till here , we have treated @xmath119 s as fields depending on individual external momenta and eq . ( [ 4pfinite ] ) is the complete 4-point effective action . assuming them to be momentum independent , we can further simplify this expression in case of the u(1 ) gauge group and perform the integral over loop - momentum to get @xmath120 where we used the reduction to 4d for @xmath121 . using this and the fact that @xmath122 is related to @xmath123 , we get the same non - holomorphic 4-point contribution ( with the full superspace measure @xmath124 ) to n=4 sym action rather directly when compared to the calculation done in @xcite ( for similar calculations in @xmath2 see , for example , @xcite ) . [ [ loops ] ] 2-loops + + + + + + + we can also see that there are no uv divergences at two - loops . the proof is similar to that given in the ordinary case , _ i.e. _ , absence of sufficient @xmath125 s . only 3 diagrams shown in fig . [ v2hbg ] are supposed to contribute at 2-loops . all of them will vanish due to the @xmath126-algebra unless we get at least 4 @xmath110 s from the expansion of the propagators . this , as we have seen before , brings in 4 more @xmath56 s making these 2-loop diagrams convergent . furthermore , we note that the arguments of @xcite apply in our case since there is no background connection @xmath7 , there can not be any divergences at 2 or more loops from just power counting . this situation is different than @xmath2 where such ` 0-dimensional ' connections are present and arguments similar to the one given above involving number of @xmath35 s have to be used and at higher loops they can be quite involved @xcite . we have formulated the background field formalism for n=2 , 4d projective superspace . the crucial ingredient was to recognize that different representations for background and quantum pieces of the hypermultiplets are required . choosing real representation for the background fields allowed non - renormalization theorems to be applicable here as the lowest - dimensional fields available were spinors . the usual choice of analytic representation for the quantum fields allowed us to make a simple extension of the existing ` ordinary ' super - feynman rules to the background covariant rules . moreover , there are extra ghosts required ( apart from fp and nk ghosts ) to evaluate the full sym effective action . these extra ghosts also appear in the harmonic case but in projective case , they cancel the spurious ` harmonic ' divergences coming from vector hypermultiplet in a straightforward manner and the resultant finite pieces are as expected for n=4 . the uv divergent parts come only from the usual ( fp and nk ) ghosts and scalar hypermultiplet . however , their contribution can not be directly calculated in the formalism developed here for reasons mentioned in section [ examples ] . we also gave a diagrammatic 2-loops argument for finiteness of n=2 sym coupled with matter . this is easily supplanted by the power counting argument of @xcite in general , which directly leads to a proof for finiteness beyond 1-loop . for n=1 background formalism , there exist improved rules as showcased in @xcite and our hope is that such techniques could be applied to what we have developed in this paper . that would lead to a further simplification of the higher - loop calculations while also allowing explicit inclusion of the scalar hypermultiplet s 1-loop contribution . this research work is supported in part by nsf grant no . phy-0969739 . 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we introduce a new background field method for n=2 superspace . ( we treat projective hyperspace , but similar remarks apply for the harmonic case . ) in analogy to n=1 , background gauge fields are in the real representation , so the lowest - dimension potentials are spinor and the usual non - renormalization theorems are manifest . another consequence is that the r - coordinates disappear from the effective action .

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Proceed to summarize the following text: the solar wind ( sw ) plasma , an ideal laboratory for the study of collisionless plasma dynamics , is mostly found in a turbulent state @xcite . subproton - scale ( `` dissipation range '' ) turbulence in the sw has become a major research topic over the past decade , both for in - situ satellite measurements @xcite and for numerical @xcite and theoretical @xcite studies . spacecraft observations provide important constraints on turbulent spectra , revealing the presence of breaks in the electromagnetic fluctuations around the proton kinetic scales @xcite . at subproton scales , typical slopes for the magnetic energy spectrum are found to be in the range @xmath2 $ ] , while preliminary results about its electric counterparts are in the range @xmath3 $ ] . from a theoretical point of view , possible explanations for the observed spectra are the development of a kinetic alfvn wave ( kaw ) cascade and/or a whistler cascade @xcite . however , the predicted energy spectra are the same for the two cases and thus auxiliary methods have been suggested in order to identify the exact nature of turbulent fluctuations @xcite . observational evidence points towards a kaw - dominated scenario for a @xmath4 plasma @xcite ( @xmath1 is the ratio between the thermal and the magnetic pressures ) , although contradictory results have also been reported @xcite . theoretical studies , on the other hand , have suggested that oblique kaws and whistlers could coexist as the plasma parameters vary in space and time @xcite . so far , numerical simulations have focused only on one scenario at a time , not on a possible coexistence or a transition between those cascades , leaving such a question as an open problem in sw turbulence research . in this letter , we wish to tackle the fundamental question of a possible dependence of the physics of subproton - scale kinetic turbulence on the plasma @xmath1 parameter by carrying out high - resolution 2d3v simulations of forced plasma turbulence as described by a hybrid vlasov maxwell ( hvm ) model with fluid electrons . while not retaining electron kinetic effects , this approach allows for both kaws and whistlers to be present , and it fully captures the ion kinetic physics . besides , this 2d3v setting allows us to include large `` fluid '' scales while still fully resolving subproton scales , which is not currently possible in 3d3v due to computational limitations . due to the intrinsic anisotropy of the turbulent mhd cascade , and to the strong damping of the parallel modes via resonances @xcite , we also expect such `` 2.5d '' simulations to retain some important dynamical features of the fully 3d case . in the hvm model , fully kinetic ions are coupled with massless fluid electrons @xcite . the hvm equations normalized with respect to the ion mass @xmath5 , the ion gyrofrequency @xmath6 , the alfvn speed @xmath7 and the ion skin depth @xmath8 are given by @xmath9 where @xmath10 is the ion distribution function , @xmath11 and @xmath12 are the electric and magnetic fields , respectively , and @xmath13 is the current density . we assume quasi - neutrality @xmath14 . the number density @xmath15 and the ion mean velocity @xmath16 are computed as the velocity moments of @xmath17 . an isothermal equation of state is assumed for the scalar electron pressure @xmath18 , with a given initial electron - to - ion temperature ratio @xmath19 . @xmath20 is a @xmath21-correlated in time , external forcing that injects momentum in the system with a prescribed average power density @xmath22 . its correlation tensor in fourier space reads @xmath23 $ ] , where brackets denote ensemble averaging , @xmath24 is a wave vector , @xmath25 is a scalar function depending on the amplitude of the wavenumber only , and @xmath26 and @xmath27 , respectively , quantify the relative degrees of incompressibility and compressibility of the forcing . in all simulations presented in this letter , we use @xmath28 . while it may overestimate the actual compressible component of the driving in the sw context , this choice can be justified by the lack of scale separation in the simulations between the driving and ion scales , at which a mixture of compressible and incompressible fluctuations is found in the solar wind ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and by the desire to not artificially direct energy into a particular mode at large scale . our numerical implementation of this forcing is a direct transposition of a widely used hydrodynamic technique @xcite . equations ( [ eq : hvm_vlasov])-([eq : hvm_maxwell ] ) are solved in a 2d3v phase space using an eulerian algorithm @xcite , with fully three - dimensional vector fields . the initial condition is a maxwellian plasma in a constant perpendicular magnetic field @xmath29 . the system is initially perturbed by random , 3d , large - scale , small - amplitude magnetic fluctuations , @xmath30 ( with wavenumbers @xmath31 in the range @xmath32 ) . the driving procedure and amplitude ( in code units ) is identical for all the cases documented below . the average power input of @xmath33 is @xmath34 and the forcing acts on the smallest wave numbers of the system , @xmath35 , thus injecting energy only at the largest scales admitted by our numerical box . in the following , we consider three different initial plasma beta values ( @xmath36 , @xmath37 and @xmath38 ) and a temperature ratio @xmath39 , i.e. , typical of sw parameters . we use @xmath40 uniformly distributed grid points to discretize a squared simulation box with @xmath41 , corresponding to a resolution @xmath42 . doubly periodic boundary conditions are imposed , and the spectral domain spans a perpendicular wavenumber range @xmath43 . spectral filters @xcite on the electromagnetic fields are applied during the simulation , in order to avoid spurious numerical effects at the smallest scales : this determines the cutoff in the energy spectra at @xmath44 . the velocity domain is limited by @xmath45 in each @xmath46-direction , with @xmath47 uniformly distributed grid points , so @xmath48 . the time step is constrained by the cfl conditions @xcite . we first investigate the spectral properties of the statistically quasi - steady turbulent state and whether they reproduce the phenomenology expected for kaws or whistlers . the analysis is performed at about @xmath49 , where @xmath50 is the outer - scale nonlinear time ( estimated from a kolmogorov argument ) . in this regime , the average modulus of the in - plane magnetic field , @xmath51 , remains relatively low , @xmath52 . nevertheless , larger values ( up to @xmath53 ) are observed locally in space and time , and coherent magnetic structures are formed . on the one hand , non - negligible in - plane magnetic fluctuations allow for finite @xmath54 ( @xmath55 is the local unit vector along @xmath12 ) , i.e. , for parallel kinetic effects and oblique waves with non - zero @xmath56 . on the other hand , the local in - plane magnetic field turns out to be randomly oriented in the fully turbulent regime , and the spectra are globally isotropic in the @xmath57-plane . therefore a shell - averaging technique can be adopted in that spectral plane ( @xmath58-reduction ) , without being polluted by any preferential direction . moreover , spectra are time - averaged over about @xmath59 . in fig . [ fig : spectra_comparison ] we plot the total magnetic and electric energy spectra , @xmath60 and @xmath61 . the @xmath62 spectrum at @xmath63 exhibits a slope close to @xmath64 , although this result should be treated with caution because of the vicinity of the injection scale and of the small extent of the range ( especially at @xmath65 ) . and @xmath61 ( left and right panel , respectively ) , for @xmath66 , blue , green and red color ( grey scale ) , respectively.,scaledwidth=105.0% ] and @xmath61 ( left and right panel , respectively ) , for @xmath66 , blue , green and red color ( grey scale ) , respectively.,scaledwidth=105.0% ] at @xmath67 , the spectral index changes for all three cases and lies between @xmath68 and @xmath69 , in general agreement with spacecraft observations @xcite . on the one hand , at @xmath36 and @xmath37 , the @xmath70 ( not shown ) and @xmath62 spectra for @xmath67 appear to be fitted better with a @xmath68 slope ( fig . [ fig : spectra_kaw - or - whistler_a ] ) , while the @xmath71 ( and @xmath72 , at @xmath73 ) spectrum is well fitted by a @xmath74 slope ( fig . [ fig : spectra_kaw - or - whistler_b ] ) . a @xmath68 slope would be in agreement with theory for fluctuations forming two - dimensional structures ( * ? ? ? * ; coherent structures are indeed visible in our simulations ) , whereas @xmath74 is the prediction of the standard theories of kaw and whistler turbulence @xcite . on the other hand , at @xmath65 , all the spectra are steeper , fitted by a @xmath69 slope . steepening of the spectra are possibly due to features not included in the standard theories , such as compressibility and/or wave damping effects ( * ? ? ? * ; * ? ? ? * ; see also fig . [ fig : damping_kaw - whistler ] ) . in the electric energy , @xmath75 , at @xmath65 and @xmath76 , a power law steeper than @xmath64 is seen at @xmath77 , whereas a spectral index between @xmath78 and @xmath79 is observed at @xmath67 . at @xmath73 , instead , a bump is present at @xmath80 that makes the spectrum appear steeper , with a slope of @xmath81 and it only partially agrees with the other two cases at @xmath82 ( fig . [ fig : spectra_comparison ] ) . in all the @xmath1 cases , the @xmath75 spectrum at @xmath77 is dominated by the mhd term , @xmath83 , whereas at @xmath67 it is dominated by the hall term , @xmath84 ( cf . ( [ eq : hvm_ohm ] ) ) . the electron pressure term , @xmath85 , is always found to be sub - dominant with respect to @xmath86 . finally , the electric energy overcomes its magnetic counterpart at @xmath87 , regardless of the @xmath88-scale position . we now investigate the nature of turbulent fluctuations in the different @xmath1 regimes . first , we compare the levels of magnetic and density spectra , @xmath62 and @xmath89 ( with @xmath90[1+{\beta_{\rm i}}(1+\tau)/2]$ ] ) , a method to distinguish between kaw ( @xmath91 ) and whistler ( @xmath92 ) turbulence @xcite . second , we check if the relation @xmath93 ( with @xmath94 ^ 2 $ ] ) between the density and parallel magnetic spectra , expected for kaw fluctuations , is satisfied @xcite . we stress that the two methods are not conclusive if taken separately , but are complementary to each other and must therefore be inspected accordingly . in fig . [ fig : spectra_kaw - or - whistler_a ] we compare the magnetic and the normalized density spectra , @xmath62 and @xmath89 , as obtained in our simulations . the main result is that the turbulence is mediated by magnetosonic / whistler ( ms / w ) fluctuations at @xmath36 , whereas the dynamics at @xmath73 appear to be dominated by alfvn wave / kinetic alfvn wave ( aw / kaw ) turbulence . at @xmath65 , instead , there is a signature of a transition at @xmath95 , from a ms to a kaw regime . in order to confirm this scenario , in fig . [ fig : spectra_kaw - or - whistler_b ] we show the comparison between @xmath71 and @xmath96 . in particular , at @xmath36 , a significant disagreement between the two quantities remains even at @xmath44 , thus providing a confirmation of the whistler - dominated regime inferred from fig . [ fig : spectra_kaw - or - whistler_a ] . at @xmath73 , we find @xmath93 through the entire @xmath58 range , thus confirming the kaw - dominated scenario . at @xmath65 , @xmath71 and @xmath96 differ by more than an order of magnitude for @xmath97 , whereas the relation @xmath93 holds well for @xmath67 . this supports the interpretation of a transition from an ms dynamics at large scales to a kaw regime at smaller scales , for @xmath65 ( see also sec . [ subsec : interpret ] and fig . [ fig : damping_kaw - whistler ] ) . further evidence leading to the above conclusions is provided by inspecting the magnetic compressibility , @xmath98 , and by the predominantly perpendicular heating of the ions at low @xmath1 , @xmath99 ( not shown here ) . we finally caution that these results may be dependent on the details of how the turbulence is driven . while we concluded from several test simulations ( not shown ) that the qualitative results presented here do not depend significantly on the resolution and/or on the forcing amplitude , we also found that ms / w waves are not excited in complementary test simulations conducted with a purely incompressible ( perhaps somewhat idealized ) driving ( @xmath100 , @xmath101 ) . studying the detailed dependence of this kind of turbulence on the driving lies outside the scope of the present paper , but may also be relevant to the solar wind context and will therefore be worth exploring in the future . + there are several examples of wave - supporting turbulent systems where linear physics leaves an imprint on the nonlinear dynamics even in strong turbulence regimes @xcite . while it may not apply quantitatively in such regimes , linear theory may still provide some interesting physical insights into the dynamics at work in that case . , for the aw / kaw ( black ) and ms / w ( cyan ) branches at @xmath102 ( left panel ) and @xmath103 ( central panel ) , for a propagation angle of @xmath104 , and at @xmath105 ( right panel ) , for @xmath106.,scaledwidth=115.0% ] , for the aw / kaw ( black ) and ms / w ( cyan ) branches at @xmath102 ( left panel ) and @xmath103 ( central panel ) , for a propagation angle of @xmath104 , and at @xmath105 ( right panel ) , for @xmath106.,scaledwidth=115.0% ] , for the aw / kaw ( black ) and ms / w ( cyan ) branches at @xmath102 ( left panel ) and @xmath103 ( central panel ) , for a propagation angle of @xmath104 , and at @xmath105 ( right panel ) , for @xmath106.,scaledwidth=115.0% ] a possible interpretation for the transition reported above is in terms of the linear properties of the magnetosonic / whistler ( ms / w ) and of the alfvn / kinetic alfvn ( aw / kaw ) modes . in fig . [ fig : damping_kaw - whistler ] we display the ratio of the damping rate to the real frequency , @xmath107 , for the aw / kaw and the ms / w branches of the hvm system , eqs . ( 1)-(3 ) , within our simulation parameters . a representative propagation angle of @xmath108 has been estimated by @xmath109 . at @xmath36 , the aw / kaw is weakly damped for @xmath110 and undergoes a complete resonant absorption as @xmath111 for @xmath112 . the ms / w mode is instead practically undamped for @xmath110 , except for a well - separated series of peaks representing the crossing of the resonant surfaces @xmath113 ( @xmath114 , @xmath115 , @xmath116 , @xmath117 ) . then , for @xmath112 , the peaks form a quasi - continuum of wave damping , but still more than one order of magnitude lower than that of the aw / kaw counterpart . this would suggest a complete absorption of kaws for @xmath118 at @xmath36 , leaving this regime whistler - dominated . at @xmath73 , the frequency - normalized damping rates of the two modes are comparable and an extrapolation to the turbulent state is not obvious ( a comparison of just @xmath119 would show a slightly higher damping of the ms / w branch , but still of the same order of magnitude ) . however , in this regime the aw / kaw mode is not completely absorbed anymore by the ion cyclotron resonance , consistent with a kaw - dominated cascade inferred from the simulations . at @xmath65 , instead , the frequency - normalized damping rates exhibit a transition at @xmath120 : for @xmath121 , the aw / kaw branch is more damped than the ms / w counterpart , whereas at @xmath122 the contrary holds ( this transition is much more pronounced in the pure damping rates , @xmath119 ) . this reflects the behavior shown in fig . 23 at @xmath65 , from which a transition from an ms - regime at @xmath123 to a kaw - dominated scenario for @xmath122 was inferred . we point out that the electron damping on both the aw / kaw and the ms / w modes is missing in the hvm system . this represents a limitation of this model , which should be properly investigated as appropriate numerical resources become available . nevertheless , we note that the interpretation proposed above is in qualitative agreement with previous linear studies in a full - kinetic framework @xcite and with observations about the relevance of cyclotron - resonant dissipation mechanisms in some regimes of sw turbulence @xcite . we presented the first high - resolution simulations of 2d3v forced hybrid - kinetic turbulence ranging from magnetohydrodynamic scales to scales well below the ion gyroradius . the spectral properties of the simulated turbulence , such as power - law exponents and spectral breaks at ion scales , are in agreement with the existing theory of subproton - scale turbulence and close to the observed sw spectra . moreover , we find that small - scale turbulence in this driven 2d3v setup mainly involves magnetosonic / whistler fluctuations at low @xmath1 , and kaws at somewhat higher @xmath1 . we found that this transition correlates with a change in the relative strength of the damping of the underlying wave modes , suggesting that cyclotron - resonant damping may be relevant in this context . we point out that this scenario is not mutually exclusive of other important effects involving nonlinearities , such as the presence of coherent structures also spotted in the simulations , and they can in fact be coupled with each other . while the model used in this paper presents some limitations and does not accommodate all the dynamical complexity of the sw , the results suggest a possible dependence of subproton - scale kinetic turbulence on the plasma @xmath1 parameter that may be relevant to the time and space variability of the sw . high - resolution simulations in three spatial dimensions , also including electron kinetic effects and different forms of driving , appear necessary to further our understanding of this problem , but will have to wait until computational capabilities become available . the authors acknowledge useful discussions with j. m. tenbarge , a. a. schekochihin , w. dorland , m. kunz , r. bruno and f. pegoraro . we gratefully acknowledge the anonymous referee , whose in - depth comments helped to significantly improve the presentation and discussion of the results . the research leading to these results has received funding from the european research council under the european union s seventh framework programme ( fp7/2007 - 2013)/erc grant agreement no . this project has received funding from the euratom research and training programme 2014 - 2018 . this work was facilitated by the max - planck / princeton center for plasma physics . the simulations were performed on fermi ( cineca , italy ) and on hydra ( rechenzentrum garching , germany ) .

a long - lasting debate in space plasma physics concerns the nature of subproton - scale fluctuations in solar wind ( sw ) turbulence . over the past decade , a series of theoretical and observational studies were presented in favor of either kinetic alfvn wave ( kaw ) or whistler turbulence . here , we investigate numerically the nature of the subproton - scale turbulent cascade for typical sw parameters by means of unprecedented high - resolution simulations of forced hybrid - kinetic turbulence in two real - space and three velocity - space dimensions . our analysis suggests that small - scale turbulence in this model is dominated by kaws at @xmath0 and by magnetosonic / whistler fluctuations at lower @xmath1 . the spectral properties of the turbulence appear to be in good agreement with theoretical predictions . a tentative interpretation of this result in terms of relative changes in the damping rates of the different waves is also presented . overall , the results raise interesting new questions about the properties and variability of subproton - scale turbulence in the sw , including its possible dependence on the plasma @xmath1 , and call for detailed and extensive parametric explorations of driven kinetic turbulence in three dimensions .

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Proceed to summarize the following text: with the discovery of the higgs boson @xcite , the particle content of the standard model ( sm ) is well established . the future aim of the large hadron collider ( lhc ) lies in uncovering clues for beyond the sm ( bsm ) physics - we expect these would manifest in the form of tev scale resonances . to make connection with theoretical models , we would need to measure both the mass of such a resonance and its coupling to the sm particles . however , it is not straightforward to measure the couplings of a massive particle in a hadron collider environment , where the momenta of the interacting partons are not known for any single event . generally , it is easier to measure ratios of couplings or production rate times branching ratios ( brs ) . from these , in principle , it is possible to measure any coupling if the total decay width of the particle is also known . however , since unlike the brs , measuring the width of a particle accurately can be quite difficult , this way of measuring couplings may lead to large uncertainties . in this note we present a method of extracting couplings at a hadron collider that depends only on the brs but not on the measured width or the production mechanisms of the particle in question . motivation for this method comes from the simple observation that the decay of an off - shell particle is sensitive to the coupling involved . this idea , in itself , is not new and has been used , for example , in studies about constraining the higgs width @xcite and also in ref . @xcite where it was hinted that it can be used to measure new physics couplings . what we propose in this note is a systematic way of extracting unknown couplings using this observation by identifying a set of new physical variables that are sensitive to the coupling of interest . when a massive unstable particle decays to lighter particles , the invariant mass of the daughter particles shows a distribution peaked about the mass of the particle . the shape of the distribution is well approximated by the famous breit - wigner distribution around the peak ( resonance ) and its width gives us the decay rate . discovery of a new unstable particle usually entails looking at this distribution , designing cuts to isolate regions close to the peak as its position gives us the mass of the particle . however , the same distribution can also be used to extract information about the coupling involved in the decay since both the width and the height of the distribution depend on it . if instead of looking near the resonance we look at regions away from it , the cross - sections increase with the coupling as the width grows . at the same time , the height of the peak decreases with the increasing coupling keeping the total cross - section roughly the same . in other words , when the coupling involved in the decay increases , the distribution spreads out keeping the area covered roughly constant . this tells us it may be possible to learn about the coupling involved in the decay if instead of the total cross - section we focus on parts of the phase - space . to begin with , let us consider the single production of a heavy particle @xmath0 within a toy model at the lhc . the @xmath0 decays to two sm particles , @xmath1 and @xmath2 , with the decay controlled by an unknown coupling @xmath3 . the actual method of production of @xmath0 does not concern us here as we are interested in its decay and the final state kinematics . our analysis would hold even if @xmath0 is produced in association with a different particle or produced in pairs . the width - dependent part of the amplitude for the process @xmath4 can be written as : m ( qq^ ) _ qq /,where @xmath5 is the momentum of @xmath0 . from this , it is clear that for @xmath6 , i.e. , near the resonance , the cross - section scales as @xmath7 . if @xmath8 is the dominant decay mode of @xmath0 , then @xmath9 and so @xmath10 . on the other hand , in the kinematic region where @xmath11 , the cross - section scales like @xmath12 . our minimal assumptions at this point are that both @xmath13 and br(@xmath14 ) are known fairly accurately from experiments . is equivalent to measuring its decay width . hence the recent studies about constraining the higgs width @xcite are actually similar in spirit to this study . ] below , we outline the main steps that one could follow to obtain @xmath3 . 1 . the first step is to numerically simulate the process for many different values of @xmath15 using @xmath16 obtained from the theoretically computed partial width and the experimentally measured br as @xmath17 . this way , one can avoid using the measured width but rely on the br that can be measured more accurately . the regions _ outside _ the resonance can be accessed with an invariant mass cut on @xmath18 defined by a new parameter @xmath19 that parametrizes the degree of off - shellness of the intermediate @xmath20 : |m(qq)-m_| & _ o m _ .[eq : phioff - cut ] we note that the above cut is similar to the one used in @xcite . for each @xmath21 and for various choices of @xmath19 , one could compute the following ratio : r(_qq^i)|__r,_o = , where @xmath22 is the _ on - shell _ cross - section , computed using a cut that isolates the resonance : |m(qq)-m_| & _ r m_.[eq : phion - cut ] the advantage of working with the ratio @xmath23 instead of either cross - section is that most uncertainties associated with the production mechanism involving pdfs etc . get canceled in the ratio . and more importantly , any other unknown coupling present in the production of @xmath20 , i.e. , @xmath24 , also gets canceled . now , for each combination of @xmath25 , one can prepare a `` calibration curve '' ( cc ) by interpolating between @xmath26 s and thus prepare a family of such curves . 3 . from the experiment one can measure the ratio @xmath27 for all the @xmath25 combinations considered to prepare the ccs . here , we have accounted for errors in the cross - section measurements by a factor @xmath28 . now , in principle , one could simply match @xmath29 for any single combination of @xmath25 with the corresponding theoretical cc and read off the coupling . however , there are some difficulties associated with this procedure . the efficiency of coupling extraction is not uniform for a particular cc since it depends on the steepness ( or slope ) of the curve - for instance , if a particular cc is almost flat , it is not possible to extract a unique value of the coupling by matching on to the experiment . _ a priori _ , it is not clear how to select the optimal combination of @xmath25 . one could also take the average of the couplings extracted from different ccs but the issue of assigning proper weight factors for averaging may become an ambiguous issue . instead , one can extract @xmath3 from a simultaneous fit using all the ccs . this can be done , e.g. , by maximizing a likelihood function defined as : l = _ k , [ eq : likelihood ] where the index @xmath30 runs over all the combinations of \{@xmath31}. the @xmath3 thus extracted will correspond to the value for which @xmath32 best describes @xmath33 for all the \{@xmath31 } combinations . for these outlined steps to work optimally , the process in question should have a large enough cross - section so that there are an appreciable number of events left over after the off - shell cuts . if the cross - section is small it may not be possible to measure @xmath33 reasonably accurately for many \{@xmath31 } combinations which in turn will increase the error in fitting . also , the method works better with a broad resonance . thus far , our discussion has been purely about the signal process . however , in practice , handling the sm backgrounds is an important issue . the preparation of the ccs should thus also include the effect of the sm background and its interference with the signal . inclusion of these effects could change the dependence of the ccs on the coupling ( due to the interference term ) . however , since the method does not rely on the nature of the dependence and only on the fact that there is one , we expect the outlined steps to work . we demonstrate this within the context of an illustrative example in the next section . to demonstrate the above method with an example , we consider a simple model with a new , heavy color triplet @xmath34 quark with electric charge @xmath35 . we make the simplifying assumption that it decays to @xmath36 via : @xmath37 in the lagrangian above , @xmath38 is the unknown coupling that we want to extract . at the lhc , if the @xmath39 is not too heavy , qcd mediated pair production will be the dominant production channel . there will also be the @xmath40 mediated single production channel of @xmath41 , like @xmath42 . following the steps described before one can use the single production process to probe @xmath38 . however , as this process is weak interaction mediated , its cross - section will be quite small and it will be more so if @xmath38 is also small . hence , to take advantage of the enhanced cross - sections , here we make use of pair production and simply apply off - shell cuts to one of the two final state particles - this reiterates our earlier point that this analysis depends only on the kinematics of the final state particles and we are free to choose the particular production process that would serve our purpose well . we note , however , that in our analysis below we will include all possible contributions to the @xmath43 process , although the pair production will dominate . in fig . [ fig : dist ] we show the invariant mass distributions of the @xmath36 pair in @xmath44 for different values of @xmath38 . as @xmath45 increases with increasing @xmath38 , the distribution spreads out and , with the increasing width , the height of the distribution decreases so that the area contained by the distribution remains approximately constant , i.e , the total cross - section is not very sensitive to @xmath38 , reiterating our point at the beginning of the last section . the invariant mass distribution of the @xmath36 pair in the process @xmath46 for different values of @xmath38 ( events generated with with madgraph5 @xcite ) . here , we fix @xmath47 tev and br@xmath48 . ] but as can be seen , there _ are _ parts of the phase space that are sensitive to @xmath38 . for example , if we look away from the mass peak , i.e. , when the @xmath36 pair is not coming from a close to on - shell @xmath39 , the cross - section becomes highly sensitive to @xmath38 , in keeping with our discussion earlier . of course , the price to pay for looking away from the resonance is the reduced cross - section compared to the resonant production . however , we can mitigate this to a large extent by considering `` mixed states '' where one @xmath39 is produced resonantly . to extract @xmath38 , we follow the steps outlined before . assuming @xmath49 and @xmath50 are known from experiments , we generate events for @xmath51 in madgraph5 @xcite ( including both @xmath39 and sm contributions as well as their interference ) for different values of @xmath38 where we compute @xmath52 as @xmath53 . for this example , we set @xmath54 tev and @xmath55 and generate events for the 14 tev lhc . since in this case we are considering pair production , rather than using the cut defined in eq . [ eq : phioff - cut ] , we modify it to include two cuts on the @xmath36 pairs , one on - shell and one off - shell : @xmath56 where @xmath57 or @xmath58 and the numbers imply that the particles are @xmath59-ordered . condition ( i ) reconstructs the resonant @xmath39 while condition ( ii ) accesses the @xmath38 sensitive off - shell region . our expression for @xmath60 is given by : r(_tw^i)|__r,_o = , [ eq : bprr ] where @xmath61 is now computed by applying cuts as in eq . [ eq : on - cut ] to reconstruct two resonant @xmath62s . we produce a family of ccs for @xmath63 s with various combinations of \{@xmath31}. the dependence of @xmath64 on @xmath38 with @xmath65 cuts defined in eqs . ( [ eq : on - cut ] ) & ( [ eq : off - cut ] ) . the curves marked with only @xmath66 show @xmath67 . ] in fig . [ fig : cali ] we plot @xmath68 ( including sm and bsm contributions ) with respect to @xmath38 before and after cuts for three different choices of @xmath25 . we also show @xmath69 in the same figure . the total cross - section @xmath68 without any cut is mostly insensitive to @xmath38 it remains almost constant for smaller @xmath38 , but slightly decreases in the higher @xmath38 region ( mostly ) because of the larger widths . in this example the conditions [ eq : on - cut ] and [ eq : off - cut ] cuts become competitive in nature as they are applied on different @xmath36 pairs simultaneously . applying the cut in eq . [ eq : on - cut ] we reconstruct the resonant @xmath39 , i.e. , we accept the @xmath36 pairs that fall _ inside _ the @xmath66 mass window and so an increasing @xmath70 means accepting more events ( larger cross - section ) . whereas , with eq . [ eq : off - cut ] we look away from the peak ( we accept the @xmath36 pairs that fall _ outside _ the @xmath19 mass window ) and so an increasing @xmath19 results in reduced cross - section . with this in mind , it becomes easier to understand the curves with the @xmath25 cuts . for example , for a fixed @xmath66 , we expect the cross - section to go down with increasing @xmath19 . this we can see by comparing the curves with \{0.10 , 0.01 } and \{0.10 , 0.10 } the second one is smaller than the first for all values of @xmath38 . similarly for a fixed @xmath19 , if we decrease @xmath66 we should get a smaller cross - section which can be seen from the curves with \{0.10 , 0.01 } and \{0.04 , 0.01}. we also observe that with increasing @xmath38 , the @xmath68 curves first increase then decrease . this is because for a fixed @xmath66 , as we increase @xmath38 ( and hence @xmath71 ) , we miss more and more @xmath36 pairs that come from the resonant @xmath39 , i.e. , reconstruction efficiency of the resonant @xmath39 decreases and as a result the cross - section decreases . this also explains why the @xmath72 curves , for which we are reconstructing two resonant @xmath39 s , fall ( and fall faster compared to @xmath68 curves ) with increasing @xmath38 . when the coupling @xmath73 0 , the difference between the @xmath74 curves and the total cross - section is accounted for by the sm background . to see how the method performs we have considered four different test couplings @xmath75 and use @xmath76 in place of @xmath33 ( eq . [ eq : bprr ] ) , assuming all @xmath76 s have uniform 10% errors for all combinations of @xmath25 . we have considered eight different values of @xmath70 between @xmath77 and @xmath78 and ten different values of @xmath19 between @xmath79 and @xmath80 . we present the results of our analysis in table [ tab : fit ] . we see that we are able to extract the couplings with percent - level accuracies . we note , however , that ours is a parton level analysis . doing full detector simulations and considering sm backgrounds properly are likely to modify the extraction efficiency . these issues are currently under investigation @xcite . .[tab : fit ] the different choices for @xmath81 and the corresponding values obtained by maximizing the likelihood function defined in eq . [ eq : likelihood ] . the errors in the fitted values correspond to @xmath82 . [ cols="^,^,^",options="header " , ] in this note , we presented a novel method of extracting the couplings of new , heavy states to sm particles . the essential point underlying the method is a rather simple observation that when a particle decays inside a collider , the invariant mass distribution of its decay products retains information of the coupling involved in the decay . with the off - shell cuts , this coupling extraction procedure actually becomes sensitive to the shape of the invariant mass distribution . the method is largely free of modeling assumptions apart from the fact it uses theoretically computed partial decay width to avoid the use of the measured total width of the new particle , and thus avoids the errors associated with width measurement . of course , computing the total width using the theoretically computed partial width and the experimentally measured br means that some errors could come from these sources and ideally one should account for both . however , since generally these errors are expected to be much less severe than the ones in measuring widths experimentally , we neglect these for the time being . similarly we neglect any error in measuring the mass of the new particle . the method does not depend on the exact production mechanism of the new particle . the coupling extraction method outlined here is completely general and applicable to any new fermionic or bosonic state . it also has the advantage of being independent of collider details , or the actual values of the mass and the brs of the particle in question . in this note , we have ignored the issue of considering the complete sm backgrounds which in reality is a very crucial one . since realistic background can only be considered in a case - by - case basis , inclusion of simple parton level background in the example shown should only be considered as an illustration . however , we hope that after the discovery of a new particle , its background will also be fairly well known and thus can be dealt with . since our aim here is simply to outline the methodology , we postpone a more complete demonstration with detector level simulations for both signal and background to a future publication @xcite . to extract any coupling the cross - section of the process in question needs to be large enough to start with so we have enough events left after all the cuts . it is interesting to contrast this method with the usual discovery procedure . if the new state has a significant width ( comparable to its mass ) , discovery is rendered more difficult as we would not have the typical sharp peak . on the other hand , a large width guarantees that regions away from resonance do not have negligibly small cross - sections and hence coupling extraction becomes more feasible . of course , one need not restrict attention to new physics alone . in principle , this method can be applied to extract the sm ckm matrix element @xmath83 from top pair production . in fact , it can be applied to extract any coupling , be it sm or bsm , as long as enough events remain after the cuts . we thank heather logan and biswarup mukhopadhyaya for helpful comments on the draft , shrihari gopalakrishna and erich varnes for useful discussions . bc is supported by the department of energy under grant de- fg02 - 13er41976 . tm is partially supported by funding from the dae , for the recapp , hri . sm acknowledges financial support from the cnrs . bc acknowledges the hospitality of imsc where this project was started . 99 g. aad _ et al . _ [ atlas collaboration ] , phys . b * 716 * , 1 ( 2012 ) [ arxiv:1207.7214 [ hep - ex ] ] ; 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in this note , we present a novel method of extracting the couplings of a heavy particle to the standard model states . contrary to the usual discovery process which involves studying the on - shell production , we look at regions away from resonance to take advantage of the simple scaling of the cross - section with the couplings . we illustrate the procedure with the example of a heavy quark .

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Proceed to summarize the following text: in the non perturbative regime of strongly interacting theories effective lagrangians play a dominant role since they efficiently describe the non perturbative dynamics in terms of the relevant degrees of freedom . symmetries , anomalous and exact , are used to constrain the effective lagrangians . an important point is that the effective lagrangian approach is applicable to any region of the qcd or qcd - like phase diagram whenever the relevant degrees of freedom and the associated symmetries are defined . at zero temperature and quark chemical potential the simplest effective lagrangian describing a relevant part of the nonperturbative physics of the yang - mills ( ym ) theory is the glueball lagrangian whose potential is : @xmath0 \ . \end{aligned}\ ] ] the latter is constrained using trace anomaly and @xmath1 $ ] with @xmath2 the gluon field stress tensor . it describes the vacuum of a generic yang - mills theory . a similar effective lagrangian ( using superconformal anomalies ) can be written for the non perturbative super yang - mills ( sym ) theory . this is the celebrated veneziano - yankielowicz @xcite lagrangian . in yang mills and super yang - mills theories no exact continuous global symmetries are present which can break spontaneously and hence no goldstones are present . the situation is different when flavors are included in the theory . here the spontaneous breaking of chiral symmetry leads to a large number of goldstone s excitations . we note that in we were able , using string techniques , to derive a number of fundamental perturbative and non perturbative properties for supersymmetric qcd such as the beta function , fermion condensate as well as chiral anomalies . recently in @xcite we constructed effective lagrangians of the veneziano - yankielowicz ( vy ) type for two non - supersymmetric but strongly interacting theories with a dirac fermion either in the two index symmetric or two index antisymmetrix representation of the gauge group . these theories are planar equivalent , at @xmath3 to sym @xcite . in this limit the non - supersymmetric effective lagrangians coincide with the bosonic part of the vy lagrangian . we departed from the supersymmetric limit in two ways . first , we considered finite values of @xmath4 . then @xmath5 effects break supersymmetry . we suggested the simplest modification of the vy lagrangian which incorporates these @xmath5 effects , leading to a non - vanishing vacuum energy density . we analyzed the spectrum of the finite-@xmath4 non - supersymmetric daughters . for @xmath6 the two - index antisymmetric representation ( one flavor ) _ is one - flavor qcd_. we showed that in this case the scalar quark - antiquark state is heavier than the corresponding pseudoscalar state , the @xmath7 . second , we added a small fermion mass term which breaks supersymmetry explicitly . the vacuum degeneracy is lifted , the parity doublets split and we evaluated this splitting . the @xmath8-angle dependence and its implications were also investigated . this new effective lagrangian provides a number of fundamental results about qcd which can be already tested either experimentally or via lattice simulations . this new type of expansion in the inverse of number colors in which the quark representation is the two index antisymmetric representation of the gauge group at any given @xmath4 may very well be more convergent then the ordinary @xmath5 expansion . in the ordinary case one keeps the fermion in the fundamental representation of the gauge group while increasing the number of colors . indeed recently in @xcite we have studied the dependence on the number of colors ( while keeping the fermions in the fundamental representation of the gauge group ) of the leading pi pi scattering amplitude in chiral dynamics . we have demonstrated the existence of a critical number of colors for and above which the low energy pi pi scattering amplitude computed from the simple sum of the current algebra and vector meson terms is crossing symmetric and unitary at leading order in a @xmath5 expansion . the critical number of colors turns out to be @xmath9 and is insensitive to the explicit breaking of chiral symmetry . this means that the ordinary @xmath5 corrections for the real world are large . at nonzero temperature the center of the @xmath10 gauge group becomes a relevant symmetry @xcite . however except for mathematically defined objects such as polyakov loops the physical states of the theory are neutral under the center group symmetry . a new class of effective lagrangians have been constructed to show how the information about the center group symmetry is efficiently transferred to the actual physical states of the theory and will be reviewed in detail elsewhere . via these lagrangians we were also able to have a deeper understanding of the relation between chiral restoration and deconfinement @xcite for quarks in the fundamental and in the adjoint representation of the gauge group . i will focus here on the two basic effective lagrangians developed for color superconductivity . more specifically the lagrangian for the color flavor locked phase ( cfl ) of qcd at high chemical potential and the 2 flavor color superconductive effective lagrangian . [ uno ] a color superconducting phase is a reasonable candidate for the state of strongly interacting matter for very large quark chemical potential . many properties of such a state have been investigated for two and three flavor qcd . in some cases these results rely heavily on perturbation theory , which is applicable for very large chemical potentials . some initial applications to supernovae explosions and gamma ray bursts can be found in @xcite and @xcite respectively , see also @xcite . the interested reader can find a discussion of the effects of color superconductivity on the mass - radius relationship of compact stars in @xcite for @xmath11 light flavors at very high chemical potential dynamical computations suggest that the preferred phase is a superconductive one and the following ansatz for a quark - quark type of condensate is energetically favored : @xmath12 a similar expression holds for the right transforming fields . the greek indices represent spin , @xmath13 and @xmath14 denote color while @xmath15 and @xmath16 indicate flavor . the condensate breaks the gauge group completely while locking the left / right transformations with color . the final global symmetry group is @xmath17 , and the low energy spectrum consists of @xmath18 goldstone bosons . here we seek insight regarding the relevant energy scales of various physical states of the color flavor locked phase ( cfl ) , such as the vector mesons and the solitons @xcite . our results do not support the naive expectation that all massive states are of the order of the color superconductive gap , @xmath19 . our strategy is based on exploiting the significant information already contained in the low energy effective theory for the massless states . we transfer this information to the massive states of the theory by making use of the fact that higher derivative operators in the low energy effective theory for the lightest state can also be induced when integrating out heavy fields . for the vector mesons , this can be seen by considering a generic theory containing vector mesons and goldstone bosons . after integrating out the vector mesons , the induced local effective lagrangian terms for the goldstone bosons must match the local contact terms from operator counting . we find that each derivative in the ( cfl ) chiral expansion is replaced by a vector field @xmath20 as follows @xmath21 this relation allows us to deduce , among other things , that the energy scale for the vector mesons is @xmath22 where @xmath23 is the vector meson mass . our result is in agreement with the findings in . we shall see that this also suggests that the ksrf relation holds in the cfl phase . in the solitonic sector , the cfl chiral lagrangian gives us the scaling behavior of the coefficient of the skyrme term and thus shows that the mass of the soliton is of the order of @xmath24 which is contrary to naive expectations . this is suggestive of a kind of duality between vector mesons and solitons in the same spirit as the duality advocated some years ago by montonen and olive for the @xmath25 georgi - glashow theory @xcite . this duality becomes more apparent when considering the product @xmath26 which is independent of the scale , @xmath19 . in the present case , if the vector meson self - coupling is @xmath27 , we find that the skyrme coefficient , @xmath28 , can be identified with @xmath27 . thus , the following relations hold : @xmath29 in this notation the electric - magnetic ( i.e. vector meson - soliton ) duality is transparent . since the topological wess - zumino term in the cfl phase is identical to that in vacuum , we identify the soliton with a physical state having the quantum numbers of the nucleon . we expect that the product of the nucleon and vector meson masses will scale like @xmath30 for any non - zero chemical potential for three flavors . interestingly , quark - hadron continuity can be related to duality @xcite . testing this relation can also be understood as a quantitative check of quark - hadron continuity . it is important to note that our results are tree level results and that the resulting duality relation can be affected by quantum corrections . our results have direct phenomenological consequences for the physics of compact stars with a cfl phase . while vector mesons are expected to play a relevant role , solitons can safely be neglected for large values of the quark chemical potential . when diquarks condense for the three flavor case , we have the following symmetry breaking : @xmath31 \times su_l(3 ) \times su_r(3 ) \times u_b(1 ) \rightarrow su_{c+l+r}(3 ) \ . \nonumber\end{aligned}\ ] ] the gauge group undergoes a dynamical higgs mechanism , and nine goldstone bosons emerge . neglecting the goldstone mode associated with the baryon number and quark masses ( which will not be important for our discussion at lowest order ) , the derivative expansion of the effective lagrangian describing the octect of goldstone bosons is : @xmath32\equiv \frac{f^2_{\pi}}{2 } { \rm tr}\left[p_{\mu}p^{\mu}\right ] \ , \end{aligned}\ ] ] with @xmath33 , @xmath34 , @xmath35 and @xmath36 is the octet of goldstone bosons . @xmath37 transforms linearly according to @xmath38 and @xmath39 while @xmath40 transforms non - linearly : @xmath41 this constraint implicitly defines the matrix , @xmath42 . here , we wish to examine the cfl spectrum of massive states using the technique of integrating in / out at the level of the effective lagrangian . @xmath43 is the goldstone boson decay constant . it is a non - perturbative quantity whose value is determined experimentally or by non - perturbative techniques ( e.g. lattice computation ) . for very large quark chemical potential , @xmath43 can be estimated perturbatively . it is found to be proportional to the fermi momentum , @xmath44 , with @xmath45 the quark chemical potential @xcite . since a frame must be fixed in order to introduce a chemical potential , spatial and temporal components of the effective lagrangians split . this point , however , is not relevant for the validity of our results . when going beyond the lowest - order term in derivatives , we need a counting scheme . for theories with only one relevant scale ( such as qcd at zero chemical potential ) , each derivative is suppressed by a factor of @xmath43 . this is not the case for theories with multiple scales . in the cfl phase , we have both @xmath43 and the gap , @xmath19 , and the general form of the chiral expansion is @xcite : @xmath46 following @xcite , we distinguish between temporal and spatial derivatives . chiral loops are suppressed by powers of @xmath47 , and higher - order contact terms are suppressed by @xmath48 where @xmath49 is the momentum . thus , chiral loops are parametrically small compared to contact terms when the chemical potential is large . there is also a topological term which is essential in order to satisfy the thooft anomaly conditions at the effective lagrangian level . it is important to note that respecting the thooft anomaly conditions is more than an academic exercise . in fact , it requires that the form of the wess - zumino term is the same in vacuum and at non - zero chemical potential . its real importance lies in the fact that it forbids a number of otherwise allowed phases which can not be ruled out given our rudimentary treatment of the non - perturbative physics . as an example , consider a phase with massless protons and neutrons in three - color qcd with three flavors . in this case chiral symmetry does not break . this is a reasonable realization of qcd for any chemical potential . however , it does not satisfy the thooft anomaly conditions and hence can not be considered . were it not for the thooft anomaly conditions , such a phase could compete with the cfl phase . gauging the wess - zumino term with to respect the electromagnetic interactions yields the familiar @xmath50 anomalous decay . this term @xcite can be written compactly using the language of differential forms . it is useful to introduce the algebra - valued maurer - cartan one form @xmath51 which transforms only under the left @xmath52 flavor group . the wess - zumino effective action is @xmath53 = c\,\int_{m^{5}}{\rm tr}\left [ \alpha ^{5}\right ] \ . \label{wz}\end{aligned}\ ] ] the price which must be paid in order to make the action local is that the spatial dimension must be augmented by one . hence , the integral must be performed over a five - dimensional manifold whose boundary ( @xmath54 ) is ordinary minkowski space . in the constant @xmath55 has been shown to be the same as that at zero density , i.e. @xmath56 where @xmath57 is the number of colors ( three in this case ) . due to the topological nature of the wess - zumino term its coefficient is a pure number . it is well known that massive states are relevant for low energy dynamics . consider , for example , the role played by vector mesons in pion - pion scattering in saturating the unitarity bounds . more specifically , vector mesons play a relevant role when describing the low energy phenomenology of qcd and may also play a role also in the dynamics of compact stars with a cfl core @xcite . in order to investigate the effects of such states , we need to know their in medium properties including their gaps and the strength of their couplings to the cfl goldstone bosons . except for the extra spontaneously broken @xmath58 symmetry , the symmetry properties of the cfl phase have much in common with those of zero density phase of qcd . this fact allows us to make some non perturbative but reasonable estimates of vector mesons properties in medium . we have already presented the general form of the chiral expansion in the cfl phase . as will soon become clear , we are now interested in the four derivative ( non topological ) terms whose coefficients are proportional to @xmath59 this must be contrasted with the situation at zero chemical potential , where the coefficient of the four derivative term is always a pure number before quantum corrections are taken into account . in vacuum , the tree - level lagrangian which simultaneously describes vector mesons , goldstone bosons , and their interactions is : @xmath60+\frac{m^2_{v}}{2 } { \rm tr } \left[\left(\rho_{\mu } + \frac{v_{\mu}}{\widetilde{g}}\right)^2\right ] \nonumber \\ & -&\frac{1}{4}{\rm tr } \left[f_{\mu\nu}(\rho)f^{\mu\nu}(\rho)\right ] \ , \end{aligned}\ ] ] where @xmath61 mev and @xmath62 is the one form @xmath63 with @xmath64 and @xmath65 $ ] . at tree level this lagrangian agrees with the hidden local symmetry results @xcite . when the vector mesons are very heavy with respect to relevant momenta , they can be integrated out . this results in the field constraint : @xmath66 substitution of this relation in the vector meson kinetic term ( i.e. , the replacement of @xmath67 by @xmath68 ) gives the following four derivative operator with two time derivatives and two space derivatives @xcite : @xmath69 ^ 2\right ] \ .\label{4d}\end{aligned}\ ] ] the coefficient is proportional to @xmath70 . it is also relevant to note that since we are describing physical fields we have considered canonically normalized fields and kinetic terms . this lagrangian can also be applied to the cfl case . in the vacuum , @xmath27 is a number of order one independent of the scale at tree level . this is no longer the case in the cfl phase . here , by comparing the coefficient of the four derivative operator in eq . ( [ 4d ] ) obtained after having integrated out the vector meson with the coefficient of the same operator in the cfl chiral perturbation theory we determine the following scaling behavior of @xmath27 : @xmath71 by expanding the effective lagrangian with the respect to the goldstone boson fields , one sees that @xmath27 is also connected to the vector meson coupling to two pions , @xmath72 , through the relation @xmath73 in vacuum @xmath74 and @xmath75 are quantities of order one . since @xmath62 is essentially a single derivative , the scaling behavior of @xmath27 allows us to conclude that each derivative term is equivalent to @xmath76 with respect to the chiral expansion . for example , dropping the dimensionless field @xmath37 , the operator with two derivatives becomes a mass operator for the vector meson @xmath77 this demonstrates that the vector meson mass gap is proportional to the color superconducting gap . this non - perturbative result is relevant for phenomenological applications . it is interesting to note that our simple counting argument agrees with the underlying qcd perturbative computations of ref . @xcite and also with recent results of ref . @xcite . in @xcite , at high chemical potential , vector meson dominance is discussed . however , our approach is more general since it does not rely on any underlying perturbation theory . it can be applied to theories with multiple scales for which the counting of the goldstone modes is known . since @xmath78 , we find that @xmath79 scales with @xmath27 suggesting that the ksrf relation is a good approximation also in the cfl phase of qcd . the low energy effective theory supports solitonic excitations which can be identified with the baryonic sector of the theory at non - zero chemical potential . in order to obtain classically stable configurations , it is necessary to include at least a four derivative term ( containing two temporal derivatives ) in addition to the usual two derivative term . such a term is the skyrme term : @xmath80 ^ 2\right ] \ .\end{aligned}\ ] ] since this is a fourth order term in derivatives not associated with the topological term we have : @xmath81 this term is the same as that which emerges after integrating out the vector mesons ( see eq . ( [ 4d ] ) ) , and one concludes that @xmath82 @xcite . the simplest complete action supporting solitonic excitations is : @xmath83 + l^{\rm skyrme}\right ] + \gamma_{wz } \ .\end{aligned}\ ] ] the wess - zumino term in eq . ( [ wz ] ) guarantees the correct quantization of the soliton as a spin @xmath84 object . here we neglect the breaking of lorentz symmetries , irrelevant to our discussion . the euler - lagrangian equations of motion for the classical , time independent , chiral field @xmath85 are highly non - linear partial differential equations . to simplify these equations skyrme adopted the hedgehog _ ansatz _ which , suitably generalized for the three flavor case , reads @xcite : @xmath86 where @xmath87 represents the pauli matrices and the radial function @xmath88 is called the chiral angle . ansatz _ is supplemented with the boundary conditions @xmath89 and @xmath90 which guarantee that the configuration posseses unit baryon number . after substituting the _ ansatz _ in the action one finds that the classical solitonic mass is , up to a numerical factor : @xmath91 and the isoscalar radius , @xmath92 . interestingly , due to the non perturbative nature of the soliton , its mass turns to be dual to the vector meson mass . it is also clear that although the vector mesons and the solitons have dual masses , they describe two very distinct types of states . the present duality is very similar to the one argued in @xcite . indeed , after introducing the collective coordinate quantization , the soliton ( due to the wess - zumino term ) describes baryonic states of half - integer spin while the vectors are spin one mesons . here , the dual nature of the soliton with respect to the vector meson is enhanced by the fact that , in the cfl state , @xmath93 is expected to be substantially reduced with respect to its value in vacuum . once the soliton is identified with the nucleon ( whose density dependent mass is denoted with @xmath94 ) we predict the following relation to be independent of the matter density : @xmath95 in this way , we can relate duality to quark - hadron continuity . we considered duality , which is already present at zero chemical potential , between the soliton and the vector mesons a fundamental property of the spectrum of qcd which should persists as we increase the quark chemical potential . should be noted that differently than in @xcite we have not subtracted the energy cost to excite a soliton from the fermi sea . since we are already considering the lagrangian written for the excitations near the fermi surface we would expect not to consider such a corrections . in any event this is of the order @xmath45 @xcite and hence negligible with respect to @xmath96 . we have shown that the vector mesons in the cfl phase have masses of the order of the color superconductive gap , @xmath19 . on the other hand the solitons have masses proportional to @xmath97 and hence should play no role for the physics of the cfl phase at large chemical potential . we have noted that the product of the soliton mass and the vector meson mass is independent of the gap . this behavior reflects a form of electromagnetic duality in the sense of montonen and olive @xcite . we have predicted that the nucleon mass times the vector meson mass scales as the square of the pion decay constant at any nonzero chemical potential . in the presence of two or more scales provided by the underlying theory the spectrum of massive states shows very different behaviors which can not be obtained by assuming a naive dimensional analysis . qcd with 2 massless flavors has gauge symmetry @xmath98 and global symmetry @xmath99 at very high quark density the ordinary goldstone phase is no longer favored compared with a superconductive one associated to the following type of diquark condensates : @xmath100 if parity is not broken spontaneously , we have @xmath101 , where we choose the condensate to be in the 3rd direction of color . the order parameters are singlets under the @xmath102 flavor transformations while possessing baryon charge @xmath103 . the vev leaves invariant the following symmetry group : @xmath104 \times su_{l}(2)\times su_{r}(2)\times \widetilde{u}_{v}(1)\ , \ ] ] where @xmath105 $ ] is the unbroken part of the gauge group . the @xmath106 generator @xmath107 is the following linear combination of the previous @xmath108 generator @xmath109 and the broken diagonal generator of the @xmath98 gauge group @xmath110 : @xmath111 . the quarks with color @xmath112 and @xmath113 are neutral under @xmath107 and consequently so is the condensate . the spectrum in the 2sc state is made of 5 massive gluons with a mass of the order of the gap , 3 massless gluons confined ( at zero temperature ) into light glueballs and gapless up and down quarks in the direction ( say ) 3 of color . the relevant coset space @xmath114 with @xmath115 is parameterized by : @xmath116 where @xmath117 @xmath118 belong to the coset space @xmath114 and are taken to be @xmath119 for @xmath120 while @xmath121 @xmath122 are the standard generators of @xmath123 . the coordinates @xmath124 via @xmath125 describe the goldstone bosons which will be absorbed in the longitudinal components of the gluons . the vevs @xmath126 and @xmath127 are , at asymptotically high densities , proportional to @xmath128 . @xmath129 transforms non linearly : @xmath130 with @xmath131 it is convenient to define the following differential form : @xmath132 with @xmath133 the gluon fields while @xmath134 is the strong coupling constant . @xmath135 transforms according to : @xmath136 we decompose @xmath137 into @xmath138 \quad { \rm and}\quad \omega _ { \mu } ^{\perp } = 2x^{i}{\rm tr}\left [ x^{i}\omega _ { \mu } \right ] \ , \end{aligned}\ ] ] @xmath139 are the unbroken generators of @xmath140 , while @xmath141 and @xmath142 . the most generic two derivative kinetic lagrangian for the goldstone bosons is : @xmath143 + f^{2}a_{2}{\rm tr}\left [ \,\omega _ { \mu } ^{\perp } \,\right ] { \rm tr } \left [ \,\omega ^{\mu \perp } \,\right ] \ . \label{dt}\end{aligned}\ ] ] the double trace term is due to the absence of the condition for the vanishing of the trace for the broken generator @xmath144 . it emerges naturally in the non linear realization framework at the same order in derivative expansion with respect to the single trace term . in the unitary gauge these two terms correspond to the five gluon masses @xcite . for the fermions it is convenient to define the dressed fermion fields @xmath145 transforming as @xmath146 . @xmath147 has the ordinary quark transformations ( i.e. is a dirac spinor ) . -1.5 cm -.6 cm pictorially @xmath148 can be viewed as a constituent type field or alternatively as the bare quark field @xmath147 immersed in the diquark cloud represented by @xmath129 . the non linearly realized effective lagrangian describing in medium fermions , gluons and their self interactions , up to two derivatives is : @xmath149 + f^{2}a_{2}{\rm tr}\left [ \,\omega _ { \mu } ^{\perp } \,\right ] { \rm tr}\left [ \,\omega ^{\mu \perp } \,\right ] \nonumber \\ & + & b_{1}\overline{\widetilde{\psi } } i\gamma ^{\mu } ( \partial _ { \mu } -i\omega _ { \mu } ^{\parallel } ) \widetilde{\psi } + b_{2}\overline{\widetilde{\psi } } \gamma ^{\mu } \omega _ { \mu } ^{\perp } \widetilde{\psi } \nonumber \\ & + & m_{m}\overline{\widetilde{\psi } ^{c}}_{i}\gamma ^{5}(it^{2})\widetilde{\psi } _ { j}\varepsilon ^{ij}+{\rm h.c.}\ , \end{aligned}\ ] ] where @xmath150 , @xmath151 are flavor indices and @xmath152 here @xmath153 and @xmath154 are real coefficients while @xmath155 is complex . from the last two terms , representing a majorana mass term for the quarks , we see that the massless degrees of freedom are the @xmath156 . the latter possesses the correct quantum numbers to match the t hooft anomaly conditions @xcite . the @xmath157 gauge symmetry does not break spontaneously and confines . calling @xmath140 a mass dimension four composite field describing the scalar glueball we can construct the following lagrangian @xcite : @xmath158 \right . \nonumber \\ & - & \left . \frac{b}{2 } h\log\left[\frac{h}{\hat{\lambda}^4}\right ] \right\ } \ . \label{g - ball}\end{aligned}\ ] ] this lagrangian correctly encodes the underlying @xmath157 trace anomaly . the glueballs move with the same velocity @xmath159 as the underlying gluons in the 2sc color superconductor . @xmath160 is related to the intrinsic scale associated with the @xmath157 theory and can be less than or of the order of few mevs @xcite is @xmath161 with @xmath162 the intrinsic scale of @xmath157 after the coordinates have been appropriately rescaled to eliminate the @xmath159 dependence from the action . ] once created , the light @xmath157 glueballs are stable against strong interactions but not with respect to electromagnetic processes @xcite . indeed , the glueballs couple to two photons via virtual quark loops . @xmath163 \approx 1.2\times 10^{-2 } \left[\frac{m_h}{1~{\rm mev}}\right]^5~{\rm ev } \ , \end{aligned}\ ] ] where @xmath164 . for illustration purposes we consider a glueball mass of the order of @xmath112 mev which leads to a decay time @xmath165 . this completes the effective lagrangian for the 2sc state which corresponds to the wigner - weyl phase . using this lagrangian one can estimate the @xmath157 glueball melting temperature to be @xcite : @xmath166{\frac{90 { v}^3}{2\,e\ , \pi^2}}\ , { \hat{\lambda } } < t_{csc } \ .\end{aligned}\ ] ] where @xmath167 is the color superconductive transition temperature . the deconfining / confining @xmath157 phase transition within the color superconductive phase is second order . the superconductive phase for @xmath168 possesses the same global symmetry group as the confined wigner - weyl phase . the ungapped fermions have the correct global charges to match the t hooft anomaly conditions as shown in @xcite . specifically the @xmath169 global anomaly is correctly reproduced in this phase due to the presence of the ungapped fermions . this is so since a quark in the 2sc case is surrounded by a diquark medium ( i.e. @xmath170 ) and behaves as a baryon . @xmath171 the validity of the thooft anomaly conditions at high matter density have been investigated in . a delicate part of the proof presented in @xcite is linked necessarily to the infrared behavior of the anomalous three point function . in particular one has to show the emergence of a singularity ( i.e. a pole structure ) . this pole is then interpreted as due to a goldstone boson when chiral symmetry is spontaneously broken . one might be worried that , since the chemical potential explicitly breaks lorentz invariance , the gapless ( goldstone ) pole may disappear modifying the infrared structure of the three point function . this is not possible . thanks to the nielsen and chadha theorem @xcite , not used in @xcite , we know that gapless excitations are always present when some symmetries break spontaneously even in the absence of lorentz invariance . since the quark chemical potential is associated with the barionic generator which commutes with all of the non abelian global generators the number of goldstone bosons must be larger or equal to the number of broken generators . besides all of the goldstones must have linear dispersion relations ( i.e. are type i @xcite ) . this fact not only guarantees the presence of gapless excitations ( justifying the analysis made in @xcite on the infrared behavior of the form factors ) but demonstrates that the pole structure due to the gapless excitations needed to saturate the triangle anomaly is identical to the zero quark chemical potential one in the infrared . it is also interesting to note that the explicit dependence on the quark chemical potential is communicated to the goldstone excitations via the coefficients of the effective lagrangian ( see @xcite for a review ) . for example @xmath43 is proportional to @xmath45 in the high chemical potential limit and the low energy effective theory is a good expansion in the number of derivatives which allows to consistently incorporate in the theory the wess - zumino - witten term @xcite and its corrections . the validity of the anomaly matching conditions have far reaching consequences . indeed , in the three flavor case , the conditions require the goldstone phase to be present in the hadronic as well as in the color superconductive phase supporting the quark - hadron continuity scenario @xcite . at very high quark chemical potential the effective field theory of low energy modes ( not to be confused with the goldstone excitations ) has positive euclidean path integral measure @xcite . in this limit the cfl is also shown to be the preferred phase with the aid of the anomaly conditions . since the fermionic theory has positive measure only at asymptotically high densities one can not use this fact to show that the cfl is the preferred phase for moderate chemical potentials . this is possible using the anomaly constraints . while the anomaly matching conditions are still in force at nonzero quark chemical potential @xcite the _ persistent mass _ condition @xcite ceases to be valid . indeed a phase transition , as function of the strange quark mass , between the cfl and the 2sc phases occurs . we recall that we can saturate the thooft anomaly conditions either with massless fermionic degrees of freedom or with gapless bosonic excitations . however in absence of lorentz covariance the bosonic excitations are not restricted to be fluctuations related to scalar condensates but may be associated , for example , to vector condensates @xcite . i thank a.d . jackson for stimulating collaboration on some of the recent topics presented here and for careful reading of the manuscript . i also thank r. casalbuoni , p.h . damgaard , z. duan , m. harada , d.k . hong , s.d . hsu , r. marotta , a. mcsy , f. pezzella , j. schechter , m. shifman , k. splittorff and k. tuominen for their valuable collaboration and/or discussions on some of the topics i presented here . k. tuominen is thanked also for careful reading of the manuscript . 999 j. schechter , phys . d * 21 * ( 1980 ) 3393 . c. rosenzweig , j. schechter and g. trahern , phys . rev . * d21 * , 3388 ( 1980 ) ; p. di vecchia and g. veneziano , nucl . b171 * , 253 ( 1980 ) ; e. witten , ann . of phys . * 128 * , 363 ( 1980 ) ; p. nath and a. arnowitt , phys . rev . * d23 * , 473 ( 1981 ) ; a. aurilia , y. takahashi and d. townsend , phys . lett . * 95b * , 65 ( 1980 ) ; k. kawarabayashi and n. ohta , nucl . phys . * b175 * , 477 ( 1980 ) . a. a. migdal and m. a. shifman , phys . b * 114 * , 445 ( 1982 ) ; j. m. cornwall and a. soni , phys . d * 29 * , 1424 ( 1984 ) ; phys . d * 32 * , 764 ( 1985 ) . a. salomone , j. schechter and t. tudron , phys . rev . * d23 * , 1143 ( 1981 ) ; j. ellis and j. lanik , phys . lett . * 150b * , 289 ( 1985 ) ; h. gomm and j. schechter , phys . lett . * 158b * , 449 ( 1985 ) ; f. sannino and j. schechter , phys . d * 60 * , 056004 ( 1999 ) [ hep - ph/9903359 ] . g. veneziano and s. yankielowicz , phys . b * 113 * , 231 ( 1982 ) . r. marotta , f. nicodemi , r. pettorino , f. pezzella and f. sannino , jhep * 0209 * , 010 ( 2002 ) [ arxiv : hep - th/0208153 ] . r. marotta and f. sannino , phys . b * 545 * , 162 ( 2002 ) [ arxiv : hep - th/0207163 ] . m. harada , f. sannino and j. schechter , `` large n(c ) and chiral dynamics , '' arxiv : hep - ph/0309206 . to appear in phys . d. a. mocsy , f. sannino and k. tuominen , phys . lett . * 91 * , 092004 ( 2003 ) [ arxiv : hep - ph/0301229 ] . a. mocsy , f. sannino and k. tuominen , `` induced universal properties and deconfinement , '' arxiv : hep - ph/0306069 . a. mocsy , f. sannino and k. tuominen , `` confinement versus chiral symmetry , '' arxiv : hep - ph/0308135 . see t. schafer , `` quark matter , '' arxiv : hep - ph/0304281 and references therein for a concise review on the @xmath45 dependence of @xmath172 . f. sannino , phys . b * 480 * , 280 ( 2000 ) [ arxiv : hep - ph/0002277 ] . s. d. hsu , f. sannino and m. schwetz , mod . a * 16 * , 1871 ( 2001 ) [ arxiv : hep - ph/0006059 ] . f. sannino , anomaly matching and low energy theories at high matter density arxiv : hep - ph/0301035 . proceedings for the review talk at the electroweak and strong matter conference , heidelberg 2002 . f. sannino , phys . d * 67 * , 054006 ( 2003 ) [ arxiv : hep - ph/0211367 ] . f. sannino and w. schfer , phys . b * 527 * , 142 ( 2002 ) hep - ph/0111098 . f. sannino and w. schfer , hep - ph/0204353.j . t. lenaghan , f. sannino and k. splittorff , phys . d * 65 * , 054002 ( 2002 ) [ arxiv : hep - ph/0107099 ] .

i briefly discuss effective lagrangians for strong interactions while concentrating on two specific lagrangians for qcd at large matter density . i then introduce spectral duality in qcd a la montonen and olive . the latter is already present in qcd in the hadronic phase . however it becomes transparent at large chemical potential . finally i show the relevance of having exact non perturbative constraints such as thooft anomaly conditions at zero and nonzero quark chemical potential on the possible phases of strongly interacting matter . an important outcome is that for three massless quarks at any chemical potential the only non trivial solution of the constraints is chiral symmetry breaking . this also shows that for three massless flavors at large quark chemical potential cfl is the ground state .

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Proceed to summarize the following text: in cosmological models based on cold dark matter ( cdm ) , the first stars are believed to form within small protogalaxies , with virial temperatures @xmath2 @xcite . cooling within these protogalaxies is dominated by molecular hydrogen , @xmath0 , which forms via the gas - phase reactions @xmath3 and @xmath4 even in the absence of dust . although the fractional abundance of @xmath0 that forms in this way is small , it is sufficient to allow for effective cooling and the formation of stars @xcite . as soon as massive stars form , however , they immediately begin to photoionize and photodissociate this @xmath0 . photoionization requires photons with energies greater than @xmath5 , which are strongly absorbed by neutral hydrogen , and is only of importance within regions . photodissociation , by contrast , occurs through the absorption of photons in the lyman - werner band system @xcite , with energies in the range @xmath6 @xmath7 . these photons are not strongly absorbed by neutral hydrogen and can readily escape into the intergalactic medium @xcite . initially , many of these photons will be absorbed by intergalactic @xmath0 , but its abundance is small and it is rapidly photodissociated ( see section [ ther_chem ] below ) . consequently , the onset of star formation is soon followed by the appearance of an ultraviolet background radiation field . this ultraviolet background acts to suppress further star formation by photodissociating @xmath0 within newly - forming protogalaxies . the effects of this background have been studied by a number of authors @xcite . in particular , @xcite study the coupled problem of the evolution of the ultraviolet background and its feedback on the global star formation rate using a simple galaxy formation model based on the press - schechter formalism @xcite . they find that cooling ( and hence star formation ) within small protogalaxies is completely suppressed prior to cosmological reionization . taken at face value , their results suggest that star formation within small , @xmath0-cooled protogalaxies is a transient phenomenon , with little impact on later stages of galaxy formation . these conclusions , however , rest on the assumption that the only free electrons present in the protogalactic gas come from the small residual fraction remaining after cosmological recombination . this is important , as free electrons ( and protons ) catalyze @xmath0 formation , as we can see from equations [ h2f1 ] to [ h2f4 ] . if the free electron abundance were significantly higher , then the @xmath0 formation rate would also be higher , offsetting the effects of photodissociation . at the very least , this would delay the suppression of star formation , and in principle could entirely negate it . it is therefore important to determine whether there is any way in which an enhanced level of ionization could be produced . one possible source is the residual ionization that would remain after the recombination of regions produced by an earlier generation of stars . this has recently been studied by @xcite , who find that it can be an effective source of @xmath0 and can dramatically reduce the effectiveness of photodissociation feedback . however , their conclusions are still somewhat uncertain , both because their simulations are under - resolved ( see figures 57 in * ? ? ? * ) and because they neglect the effects of supernovae , which would act to disperse the gas and significantly lengthen the recombination timescale , thereby delaying the formation of @xmath0 . an alternative possibility is ionization by a high redshift x - ray background . at x - ray energies , the optical depth of the intergalactic medium ( igm ) is small and any x - ray sources present will naturally generate an x - ray background . moreover , x - rays can penetrate to large depths within newly - formed protogalaxies , allowing them to raise the fractional ionization throughout the gas . the potential importance of such a background was first highlighted by @xcite , with high redshift quasars suggested as a possible source . using a very simple model for a quasar - produced x - ray background , they showed that if quasars contribute more than 10% of the uv background , then the ionization produced by the associated x - ray background is sufficient to negate the effects of uv photodissociation . indeed , they found evidence that such a background could actually promote cooling within the dense gas in the centres of protogalaxies . this scenario has not yet been firmly ruled out , but observational evidence suggests that quasars are unlikely to be present in sufficient number at high redshift @xcite . quasars , however , are not the only potential source of x - rays . star formation also leads to the production of x - rays , primarily through the formation of massive x - ray binaries @xcite , but also through bremsstrahlung and inverse compton emission from supernova remnants . these sources generate only a small fraction of the present - day x - ray background @xcite , but may become dominant at high redshifts . in this paper , we examine the effects of the x - rays produced by these sources , both on the cooling of gas within virialized protogalaxies and also on the chemical and thermal evolution of the igm . the outline of the paper is as follows . in section [ back ] , we discuss the sources responsible for producing the uv and x - ray backgrounds , and show how the build - up of these backgrounds can be computed . in section [ meth ] , we outline the method used to study the effect of this radiation on the primordial gas , and in section [ res ] apply it for a number of different x - ray source models . we present our conclusions in section [ conc ] . for an observer at redshift @xmath8 , we can write the mean specific intensity of the radiation background at an observed frequency @xmath9 as @xcite @xmath10 where @xmath11 , @xmath12 is the proper space - averaged volume emissivity , @xmath13 is the optical depth at frequency @xmath9 due to material along the line of sight from redshift @xmath8 to @xmath14 and @xmath15 is the cosmological line element . to solve this equation , we need to know how the emissivity and opacity evolve with redshift . for simplicity , we write the space - averaged emissivity in terms of the global star formation rate ( sfr ) as @xmath16 where @xmath17 is the luminosity density ( i.e. the luminosity per unit frequency ) per unit star formation rate ( in solar masses per year ) , and @xmath18 is the global star formation rate , with units @xmath19 . in principle , @xmath17 may be a complicated function of frequency and redshift . in practice , however , lyman - werner band emission is dominated by massive , short - lived ob stars and declines rapidly once star formation comes to an end ( see , for example , the instantaneous starburst models of * ? ? ? * in which the lyman - werner flux declines by an order of magnitude within 4@xmath20 ) . similarly , x - ray emission is dominated by short - lived sources such as massive x - ray binaries and supernova remnants which are end products of the same massive stars . both kinds of emission are therefore strongly correlated with the star formation rate ; to a first approximation , we can assume that they are directly proportional to it , and that any redshift dependence of @xmath17 can be neglected . with this simplification , determining the emissivity breaks down into two independent problems : determining the global star formation rate as a function of redshift , and determining the luminosity density as a function of the star formation rate . although we have observational constraints on the star formation rate up to @xmath21 , we have no direct constraints ( and few indirect ones ) at higher redshift . consequently , any model of high redshift star formation must inevitably be highly theoretical . moreover , this lack of constraints motivates us to choose as simple a model as possible ; more complicated ( and realistic ) models can always be considered once our observational knowledge improves . a good example of this kind of simple model is the one used by @xcite ; we adopt the same model here . we assume that star formation proceeds primarily through starbursts , of duration @xmath22 years , that are triggered when galaxies form . during the starburst , the star formation rate is assumed to be constant . the global star formation rate in this model is given by @xmath23 where @xmath24 is the star formation efficiency , @xmath25 is the baryon fraction ( ie the ratio of baryons to dark matter ) , and where @xmath26 is the cosmological density of matter in newly - formed galaxies ( with units of @xmath27 ) . we assume that the value of @xmath25 in the protogalaxies is the same as in the igm , or in other words that @xmath28 . for the cosmological model adopted in section [ res ] , this corresponds to @xmath29 . we further assume that the rate of change of @xmath30 is approximately the same as the rate of change of @xmath31 , the total fraction of matter in halos with virial temperatures greater than a critical temperature @xmath32 , or in other words that @xmath33 where @xmath34 is the cosmological matter density . here , @xmath32 represents the minimum virial temperature required for efficient cooling ; to a first approximation , halos with @xmath35 are unable to cool , while those with @xmath36 cool rapidly and can form stars . various different definitions of @xmath32 are in use in the literature ( see , for example * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) ; we discuss our particular choice in a later section . making this approximation is equivalent to assuming that the growth in @xmath31 is dominated by the formation of new halos with @xmath37 ( either by monolithic collapse or by the merger of smaller objects ) rather than by the accretion of matter by existing halos with @xmath36 . this is justified at high redshift when such halos are rare and @xmath31 is dominated by objects near @xmath32 , but becomes less accurate at lower redshifts . for this reason ( and others , to be discussed later ) we do not attempt to simulate the evolution of the background below @xmath38 . to solve equation [ diff_rho ] we need to know the value of @xmath32 . in general , this will depend both on redshift ( see , eg * ? ? ? * ) and on the intensities of the uv and x - ray backgrounds . indeed , understanding the evolution of @xmath32 with redshift is one of the main goals of this paper . our procedure for determining @xmath32 is discussed at length in section [ meth ] ; for now , we assume that it is known . in this case , we can calculate @xmath31 using the press - schechter formalism @xcite : @xmath39,\ ] ] where @xmath40 is the critical density threshold for collapse , @xmath41 is the square root of the variance of the cosmological density field , as smoothed on a mass scale @xmath42 , and where @xmath43 is the mass of a protogalaxy with virial temperature @xmath32 . although both @xmath44 and @xmath41 depend upon the choice of cosmological model , their behaviour is well - known , and the problem of determining @xmath45 reduces to the relatively simple one of relating @xmath43 to @xmath32 . to do this , we need to know the protogalactic density profile . hydrodynamical simulations suggest that it is approximately isothermal @xcite , but representing it as a singular isothermal sphere is physically unrealistic due to the latter s infinite central density . accordingly , we follow @xcite and represent it as a truncated isothermal sphere @xcite . with this choice , we find that @xmath46 to determine @xmath47 , we must integrate equation [ diff_rho ] over a redshift interval @xmath48 , corresponding to the duration of the starburst ; hence , @xmath49 this simple model has a number of shortcomings . for instance , it assumes that the star formation efficiency @xmath24 and starburst duration @xmath22 are both constant , independent of redshift or galaxy mass . it also assumes that each galaxy forms its stars in a single starburst and thus ignores the effects of continuous star formation and of subsequent , merger - triggered starbursts . . ] nevertheless , it has the virtue of simplicity , and is a good point from which to start our examination of the effects of the x - ray background . the ultraviolet flux of a star - forming galaxy is dominated by emission from young , massive o and b - type stars . these are short - lived , with the most massive having lifetimes of only a few myr , and thus the ultraviolet luminosity density is closely correlated with the star formation rate . its value depends upon the spectral properties of the newly - formed stellar population , and thus on their initial mass function ( imf ) , metallicity and age . in a recent paper , @xcite presents values for the photon flux in the lyman - werner bands calculated for a number of different metal - free stellar populations . if we assume that the spectrum within the bands is flat ( a reasonable approximation ) , then we can convert this photon flux into a luminosity density . for a salpeter imf with minimum mass @xmath50 and maximum mass @xmath51 ( model a in * ) , we find that @xmath52 reducing @xmath53 to the more conventional value of @xmath54 reduces this luminosity to @xmath55 as we form a greater number of low mass stars that do not contribute significantly to the lyman - werner flux . both of these results assume that the dissociative flux has stabilized at its equilibrium value and is therefore proportional to the star formation rate . this equilibrium is typically established after only 2 3 myr , so this is generally a good approximation , even for starbursts of relatively short duration . the above figures are appropriate so long as we are dealing with stars formed out of _ entirely _ metal - free gas . such stars are somewhat unusual , however , as the absence of carbon means that they are unable to generate energy via the cno cycle , which otherwise would dominate energy production in stars of mass @xmath56 . as a result , metal - free stars are hotter than their metal - enriched counterparts @xcite and have harder spectra . a surprising consequence of this fact is that a low - metallicity stellar population will produce a larger dissociative flux than a metal - free population the lower effective temperatures move the peak in the thermal emission from the most massive stars closer to the lyman - werner band , causing the ionizing flux to fall but the dissociative flux to rise . we can use the data presented in @xcite for a stellar population with @xmath57 to examine the difference that this effect makes to the lyman - werner flux . using the same imf as in equation [ flux_dis ] , we find that @xmath58 thus , raising the metallicity increases the lyman - werner flux , but only by about 6070% ; as we will see in section [ res_nox ] , this has little effect on the evolution of @xmath32 . clearly , there are many possible models other than those considered here . indeed , there is growing evidence that the imf of population iii stars is strongly biased towards high masses @xcite . however , this remains uncertain , and in this paper we have chosen to err on the side of caution and assume that the high - redshift imf is similar to that at the present day . in a recent study , @xcite collate data on a number of local starburst galaxies and compare the 2 10 @xmath59 x - ray fluxes measured by asca with the 8 1000 @xmath60 infrared fluxes measured by iras . they find that a clear correlation exists , with the total x - ray and infrared fluxes related by @xmath61 similar correlations have previously been reported by @xcite and @xcite for x - rays in the 0.5 4.5 @xmath59 and 2 30 @xmath59 energy bands respectively . theoretically , we would expect such a correlation , with both the x - ray and infrared emission tracing the underlying star formation rate . for the x - rays , this occurs because the emission is dominated by massive x - ray binaries ( mxrbs ) : binary systems consisting of a massive ob star accreting onto a compact companion ( a neutron star or black hole ) . x - ray emission from such systems generally switches on a few million years after the formation of the compact object , and the lifetime of the emitting phase is short , typically of the order of ( 2 5@xmath62 @xcite . these short timescales tie the emission closely to the underlying star formation rate @xcite . the far - infrared flux , on the other hand , tracks star formation far more directly , being dominated by emission from dust heated by short - lived , massive stars . to use this observed correlation to determine the x - ray luminosity of a star - forming galaxy as a function of its star formation rate , we use the result from the starburst models of @xcite that @xmath63 together with equation [ x_to_ir ] to write the x - ray luminosity as @xcite @xmath64 we then assume that this relationship between x - ray luminosity and star formation rate remains valid as we move to higher redshifts . note that the same need not be true for the relationship between x - ray luminosity and infrared luminosity , or infrared luminosity and star formation rate : although they serve to establish the correlation between x - ray luminosity and star formation rate at @xmath65 , the assumption that this correlation remains valid at higher redshift does not imply that these other correlations also remain valid . indeed , we would expect the infrared luminosity of dust - free protogalaxies to be very much lower than would be predicted by equation [ ir ] . evidence that the correlation between x - ray luminosity and star formation rate does indeed remain valid at high redshift is provided by the recent stacking analysis of individually undetected lyman break galaxies in the _ chandra _ deep field - north @xcite . this analysis finds that the average rest frame luminosity of the lyman break galaxies in the 2 8 @xmath66 energy band is @xmath67 . assuming a typical star formation rate of @xmath68 for these galaxies @xcite , this corresponds to an x - ray luminosity of @xmath69 consistent with the value derived above . is measured in a slightly different energy band from that of equation [ lx ] , so the agreement between the two values is not quite as good as may at first appear . nevertheless , the necessary correction is small , and the values agree to within 50% . ] although far from conclusive , this result suggests that we can extrapolate the locally observed correlation to at least as far as @xmath70 . comparing our determination of @xmath71 with a recent calculation by @xcite , we find a difference of a factor of ten in our results . some of this disagreement is due to the difference in x - ray energy bands considered ( @xmath72 @xmath73 in @xcite , compared to @xmath74 @xmath75 here ) , but some must surely be due to intrinsic scatter in the observational data , suggesting that equation [ lx ] should properly be regarded as an order of magnitude estimate of the true x - ray luminosity . given equation [ lx ] for the x - ray luminosity as a function of the star formation rate , we calculate the x - ray luminosity density by assuming a template spectrum of power - law form @xmath76 where @xmath77 , and requiring that @xmath78 where @xmath79 and @xmath80 . @xcite find that a weighted average of the galaxies in their sample gives a value for the spectral index of @xmath81 . adopting this value and solving for @xmath82 , we find that @xmath83 altering @xmath84 changes @xmath82 , but never by more than 50% for values consistent with the @xcite measurement . clearly , individual galaxies may have spectra that differ markedly from this simple template , but it should be a reasonable approximation when averaging over a large number of galaxies . the above model is simple , and empirically motivated , but does assume that the x - ray emission of high - redshift star - forming galaxies is very similar to that of starbursts observed locally . this is a reasonable assumption in the absence of evidence to the contrary , and is probably valid as long as massive x - ray binaries continue to dominate the galactic x - ray emission . however , it is quite possible that at high redshift some other type of source will come to dominate the emission , particularly if the number of binary systems is small , as is suggested by recent simulations @xcite . it is therefore prudent to consider the effects of other potential sources of x - rays . the obvious candidates are supernova remnants ( snr ) ; next to x - ray binaries , they are the most significant galactic sources @xcite . they can emit x - rays through a variety of different emission mechanisms , but at high redshift the most significant will be thermal bremsstrahlung emission and non - thermal inverse compton emission . thermal bremsstrahlung is produced by the hot gas within the snr . detailed modeling properly requires a hydrodynamical treatment ( see , for example , * ? ? ? * ) , but for our purposes a simple parameterization suffices . if we assume that the hot gas has a single characteristic temperature @xmath85 , then we can write the luminosity density per unit star formation rate as @xmath86 where @xmath87 and where @xmath88 is constant . moreover , we can write the total x - ray luminosity as @xmath89 where @xmath90 is the typical supernova explosion energy ( in units of @xmath91 ) , @xmath92 is the fraction of this energy radiated as bremsstrahlung , and where @xmath93 is the number of supernovae that explode per solar mass of stars formed . the value of @xmath93 depends on the imf ; for the standard salpeter imf adopted previously , @xmath94 . finally , since @xmath95 we can fix the value of @xmath88 ; we find that @xmath96 for typical supernova parameters ( @xmath97 and an ambient density @xmath98 ) , @xcite find that a fraction @xmath99 of the explosion energy is radiated , at a characteristic temperature @xmath100 . on the other hand , the higher mean density at high redshift , together with the comparative weakness of outflows from low metallicity stars @xcite suggest that the typical ambient density may be very much higher . in particular , if it is as high as @xmath101 , then a supernova remnant will radiate its energy extremely rapidly , before the ejecta have time to thermalize @xcite . in this case , the fraction of energy radiated as bremsstrahlung is very much higher ( @xmath102 ) , as is the characteristic temperature ( @xmath103 ) . we examine both of these models in section [ res ] , with the understanding that the true picture lies somewhere in between . in addition to this thermal emission , supernova remnants also produce non - thermal x - rays . these are generated as the relativistic electrons produced by the snr gradually lose energy through synchrotron radiation , non - thermal bremsstrahlung and the inverse compton scattering of photons from the cosmic microwave background . at high redshift , the latter is likely to dominate @xcite . the spectrum of the resulting emission depends upon the energy spectrum of the relativistic electrons , but at the energies of interest is well represented by a power law : @xmath104 . the intensity of the emission depends upon the fraction of the supernova energy transferred to the electrons ; this is not well constrained , with estimates ranging from 0.1% to 10% . accordingly , we model the emission as @xmath105 where @xmath87 , @xmath106 is the fraction of energy deposited in the electrons , and where we have assumed that @xmath97 and @xmath107 as in the thermal bremsstrahlung case . this expression assumes a high - energy cutoff for the x - ray spectrum at @xmath73 but is only logarithmically dependent on the value of this cutoff . in section [ res ] , we examine results for models with @xmath108 and @xmath109 , which bracket the range of plausible values . much more detail on high redshift inverse compton emission , including a discussion of potential observational tests , is given in @xcite . the opacity @xmath13 can be separated into two distinct components absorption by dust and gas within the emitting protogalaxy , which we term intrinsic absorption , and absorption by gas along the line of sight through the igm . intrinsic absorption is difficult to model with any degree of accuracy as it depends upon a number of variables the size and shape of the galaxy , its ionization state , the position of the sources within it , the dust content etc . rather than attempt to model these in detail a significant undertaking in itself we instead adopt a highly approximate representation . we assume that the emitted x - rays are attenuated by absorption by a neutral hydrogen column density of @xmath110 plus an associated neutral helium column density @xmath111 . this absorption is assumed to be the same for all sources . these values are chosen because they are representative of the column densities of the protogalaxies studied in this paper ( which presumably contain the bulk of the x - ray sources ) . reducing @xmath112 ( as would be appropriate if much of the surrounding were photoionized and/or dispersed by the progenitors of the x - ray sources ) has little or no effect on @xmath32 , as gas in the igm and within the protogalaxy itself quickly come to dominate the total absorption . increasing @xmath112 , on the other hand , has more significant effects : an order of magnitude increase in @xmath112 produces similar results to an order of magnitude decrease in the strength of the x - ray background , which , as we will see in section [ res ] , is generally sufficient to render x - ray feedback ineffective . consequently , we will overestimate the effect of the background if the bulk of the x - ray sources reside within massive galaxies . turning to the lyman - werner bands , we note that intrinsic absorption will generally be negligible within small protogalaxies , as their @xmath0 content is rapidly photodissociated @xcite . moreover , we also assume that the effects of dust absorption are negligible . in galaxies of primordial composition , this is obviously true ; in metal - poor galaxies , it should also be a good approximation , as very large column densities are required for significant dust obscuration ( for instance , @xmath113 for @xmath114 gas , if the dust - to - gas ratio is similar to that in the milky way ) . again , these assumptions break down if the majority of sources are to be found in massive , metal - rich galaxies , but we expect such galaxies to be extremely rare at the redshifts of interest in this paper . compared to intrinsic absorption , the effects of absorption due to the igm are much simpler to treat , particularly if we can assume that the bulk of the gas remains at an approximately uniform density . this assumption proves reasonable at high redshift for photons with mean free paths much greater than the typical clumping scale , as is the case for both lyman - werner band photons and x - rays . our treatment of igm absorption is discussed in the following sections . the continuum opacity of metal - free gas is very small @xcite and for our purposes can be neglected . consequently , the only significant sources of opacity encountered by lyman - werner photons are absorption by the lyman series lines of neutral hydrogen , and by the lyman - werner lines of molecular hydrogen . the lyman series lines have the effect of absorbing any lyman - werner photons of the same frequency , and reprocessing them to lyman-@xmath84 photons plus associated softer photons . as lyman-@xmath84 lies outside of the lyman - werner band , the net effect is to block from view any sources at redshifts higher than some maximum , @xmath115 , given by @xmath116 where @xmath117 is the frequency of the appropriate lyman series line and @xmath9 and @xmath8 are the observed frequency and redshift . clearly , the size of @xmath115 depends upon the distance between @xmath9 and @xmath117 , and thus more sources are seen at frequencies that are a long way from a line . as a result , the spectrum develops a characteristic ` sawtooth ' shape ( see figure 1 in @xcite ) , with the effect becoming more pronounced as one nears the lyman limit . absorption by molecular hydrogen is rather more complicated , due to the large number of lyman - werner lines that contribute to the opacity . if we approximate the lines as delta functions then an individual line produces an opacity @xmath118 where @xmath119 and @xmath120 are the oscillator strength and wavelength of the transition , @xmath121 is the number density of @xmath0 molecules in the level giving rise to the line at @xmath122 , the redshift of absorption , and @xmath123 is the hubble constant at @xmath122 . the value of @xmath122 is given by @xmath124 where @xmath125 . if we assume that all of the photons that are absorbed in the lines are permanently removed from the lyman - werner band , then the total opacity @xmath13 is simply given by the sum over all lines with @xmath126 : @xmath127 this sum potentially involves a very large number of lines , but can be greatly simplified by assuming that all of the @xmath0 is to be found in its ortho or para ground state ; at the redshifts of interest , the population of excited states will be negligible . in deriving this expression , we have assumed that every absorption permanently removes a lyman - werner band photon . this is not entirely correct . on average , only 15% of absorptions are followed by photodissociation of the @xmath0 molecule @xcite ; the rest of the time , the molecule decays back to a bound state , emitting a photon . in their treatment of this problem , @xcite assumed that the excited @xmath0 molecule would always decay directly back into the original state , and would thus emit a photon of the same energy as the one initially absorbed . in fact , this is not correct ( t. abel , private communication ) ; most decays occur initially to highly excited vibrational states , producing photons redwards of the lyman - werner bands . only a small fraction of decays ( about 5% ) take place directly into the original state , while a slightly larger fraction ( about 15% ) produce photons that lie elsewhere in the lyman - werner band system @xcite . we do not include the effect of these photons , however ; an accurate treatment would be quite complicated and is almost certainly unnecessary as we shall see in section [ res ] , @xmath0 in the igm is rapidly destroyed by the growing lyman - werner background and is completely negligible by the time that negative feedback begins . at x - ray energies , the opacity of the intergalactic gas is dominated by the ionization of neutral hydrogen and helium ; prior to recombination , the @xmath128 abundance is small and can be neglected . the x - ray opacity can thus be written as @xmath129 \frac{{\,{\rm d}l}}{{\,{\rm d}z } } { \,{\rm d}z},\ ] ] where @xmath130 and where @xmath131 and @xmath132 are the absorption cross - sections of neutral hydrogen and helium respectively , with @xmath133 and @xmath134 being the corresponding number densities . as long as the fractional ionization of the igm remains small ( i.e. a few percent or less ) , the ratio between @xmath135 to @xmath136 can be accurately approximated by its primordial value @xmath137 where @xmath138 is the helium mass fraction , and we can write equation [ tauigm ] purely in terms of @xmath136 as @xmath139 n_{{\rm{h } } } \frac{{\,{\rm d}l}}{{\,{\rm d}z } } { \,{\rm d}z}.\ ] ] this integral is readily computable by means of numerical integration . in the previous section , we showed that , given a simple star formation model , it is relatively easy to calculate the evolution of the lyman - werner and x - ray backgrounds . two of the parameters of our star formation model the star formation efficiency @xmath24 and starburst lifetime @xmath22 we treat as free parameters ( although they can be constrained to some extent see , e.g. @xcite ) . the remaining parameter , @xmath32 , the temperature at which efficient cooling becomes possible , is determined by the strength of the backgrounds themselves . this clearly presents us with a problem : the evolution of @xmath32 is coupled to that of the backgrounds , and to know one we must first know the other . fortunately , this difficulty is easily avoided . we know that at high redshift the number of protogalaxies , and hence the star formation rate , must be very small . consequently , there must be some redshift above which the external radiation field will become too weak to affect galaxy formation . the precise redshift at which this occurs is model dependent , but for the models examined in this paper we typically find that radiative feedback is negligible above @xmath140 . by choosing an initial redshift @xmath141 , therefore , we can be sure that in our initial simulation the background radiation will have no effect . given this starting point , we next proceed incrementally to lower redshifts via the following procedure : 1 . given @xmath142 , we calculate the background radiation field at @xmath143 , assuming that @xmath144 . 2 . using the computed background , we simulate the chemical and thermal evolution of a protogalaxy with @xmath145 ; the details of this simulation are outlined in sections [ comp ] to [ halt ] below . the main aim of this simulation is to determine whether the protogalactic gas can cool efficiently . if the protogalactic gas cools , then our assumed value of @xmath32 is correct ; we store this result , and return to step one to proceed to the next redshift . if the gas fails to cool , we continue to step four . we increment our assumed value of @xmath146 by a small amount @xmath147 , and recalculate the background radiation field . we assume that @xmath32 varies linearly over @xmath148 . given the new background , we return to step two . provided that @xmath148 and @xmath147 are both small , the error in @xmath146 will also be small ; this is particularly the case once emission from larger protogalaxies ( which cool via lyman-@xmath84 radiation ) begins to dominate the background . this approach reduces the coupled problem to the simpler one of determining whether a protogalaxy with virial temperature @xmath149 and formation redshift @xmath150 will cool when exposed to a particular background radiation field . to answer this question , we need to be able to model the thermal and chemical evolution of the protogalaxy . our approach to this problem is outlined in the following sections . ideally , we would like to use a high - resolution hydrodynamical simulation to follow the thermal and chemical evolution of the protogalactic gas ( see , e.g. * ? ? ? unfortunately , including the effects of radiative transfer , particularly of photons in the lyman - werner bands , into such a simulation is not currently feasible . we are thus forced to approximate . in choosing an appropriate approximation , we are also motivated by the desire to minimize the computational requirements of our simulations , so that we can explore the effects of a variety of different source models . we make three main approximations : 1 . we assume spherical symmetry . this is a reasonable approximation for the first generation of protogalaxies , but clearly is incorrect in detail ( see , e.g. figure 2 of * ? ? ? we assume that the protogalactic gas is _ static _ , at least on the timescale of the simulation . this allows us to ignore the hydrodynamical evolution of the gas , and also substantially simplifies the treatment of radiative transfer . this assumption clearly breaks down once the gas begins to cool strongly and loses its pressure support , but as we are only interested in the evolution up to this point , this is not a significant problem . we assume that all of the @xmath0 molecules remain in the rotational and vibrational ground state , in either ortho or para form . this simplification allows us to evolve the chemistry and radiative transfer on the timescale on which the total @xmath0 abundance changes ( typically @xmath151 @xmath152 ) rather than that on which the individual level populations change ( @xmath153 @xmath154 ) . it also simplifies our treatment of the radiative transfer . we discuss this approximation in more detail in section [ pd ] . together , these approximations allow us to solve for the chemical and thermal evolution of a model protogalaxy in a matter of minutes on a fast desktop computer . this allows us to study the redshift evolution of @xmath32 at high resolution in both temperature and redshift and for a number of different x - ray source models . however , this approach has an obvious drawback we can not be sure that our approximations give a fair representation of the real protogalaxy . of particular concern is the neglect of the hydrodynamic evolution of the gas , and the consequent error in the density profile . this is potentially significant because the @xmath0 cooling rate , along with many of the chemical reaction rates , scales as the square of the density . small errors in the density can thus lead to larger ones in the computed temperature . these concerns are mitigated to some extent , however , by the close agreement between the results of detailed numerical simulations and previous semi - analytic treatments . for instance , the values of @xmath32 obtained from the smoothed - particle hydrodynamics simulations of @xcite agree well with the results of @xcite , despite the highly approximate uniform density profile adopted by the latter group . similarly , the results of @xcite , obtained with a three - dimensional adaptive mesh hydrodynamical code broadly agree with those of @xcite , who use a static model similar to that presented here . together , these results suggest that @xmath32 is insensitive to the precise details of the density profile , but clearly this remains an area of concern . our computational method can be broken down into three main stages initialization of the density profile and the chemical abundances , computation of the thermal and chemical evolution of the gas , and termination of the simulation at a suitable point . these are described below . the protogalactic density profile is modeled as a truncated isothermal sphere , with central overdensity @xmath155 and truncation radius @xmath156 the virial temperature and redshift of formation of the protogalaxy completely specify its density profile . we subdivide this profile into @xmath157 spherical shells of uniform thickness and compute the mean density within each shell . we have run a number of test simulations with different values of @xmath157 , and find that setting @xmath158 provides sufficient spatial resolution to accurately determine @xmath32 . we assume that the initial chemical composition of the protogalactic gas is the same as that of the intergalactic medium . at our initial redshift @xmath159 , we take this from @xcite . at lower redshifts , the chemical evolution of the igm is influenced by the lyman - werner and x - ray backgrounds . we therefore calculate the intergalactic abundances explicitly , using the chemical model outlined in section [ chem ] , by solving the chemical rate equations : @xmath160 where @xmath161 and @xmath162 are source and sink terms for @xmath163 . at the same time we also solve for the temperature of the intergalactic gas @xcite @xmath164 where @xmath165 is the hubble constant , @xmath166 is the total particle number density and where @xmath167 and @xmath168 are the net heating and cooling rates ( see section [ cool ] ) . we solve this set of ordinary differential equations with the stifbs integrator of @xcite . as the size of the required timestep is generally much smaller than the redshift interval @xmath148 that separates our individual determinations of @xmath32 , we compute intermediate values by interpolation and from these determine the strength of the radiation background and hence the photochemical rates . although our main aim in following this chemistry is to determine the correct initial abundances for our simulations of protogalactic evolution , the results are of interest in their own right and are presented and discussed in section [ ther_chem ] . our treatment of the chemistry of the igm does not include the effects of the ionizing photons from stars ( and/or quasars ) that are ultimately responsible for the reionization of the intergalactic gas . this is justified at early epochs , as these photons are confined within small regions surrounding the luminous sources , but this simplification restricts the validity of our results to the period prior to cosmological reionization . the post - reionization epoch , and the effect of reionization on galaxy formation , have received extensive study elsewhere ( see * ? ? ? * and references therein ) . to simulate the chemical evolution of the protogalactic gas , we adopt a chemical model consisting of thirty reactions between nine species : @xmath169 , @xmath170 , @xmath171 , @xmath172 , @xmath128 , @xmath173 , @xmath0 , @xmath174 and free electrons . the reactions included in the model are summarized in table [ chemtab ] , together with the source(s ) of the data used . this model is based in large part on that of @xcite , but has been modified to improve its accuracy when applied to optically thick gas . aside from a number of updates to the reaction coefficients in the light of new data , the main differences are as follows : 1 . we include the contribution to the hydrogen ionization rate arising from the ionizing photons produced by @xmath128 recombination , in line with the discussion in chapter 2 of @xcite . although commonly a small correction to the total rate , this can become significant when x - ray photoionization dominates . 2 . to enable us to accurately determine the @xmath128 abundance , we find that we need to include the effects of charge transfer between @xmath128 and @xmath169 ( reaction 20 ) , as this can be comparable to the recombination rate when the fractional ionization is small . for completeness we also include the inverse reaction ( no . 21 ) , although this is unimportant at @xmath175 . we include the contribution to the ionization rates of hydrogen and helium arising from secondary ionization by energetic photoelectrons , based on the recent calculations of @xcite . the contribution of secondary ionization to the other photoionization rates is small and can be neglected . 4 . we do not include the photodissociation of @xmath0 by photons above the lyman limit ( reaction 28 in @xcite ) , as in optically thick gas this will be negligible compared to the effects of @xmath0 photoionization ( reaction 26 ) . on the other hand , we do include the effects of dissociative photoionization ( reaction 27 ) , which becomes significant for photon energies greater than 30@xmath176 . more information about all of these points , and the chemical model generally , can be found in @xcite . .a list of the reactions included in our chemical model of protogalactic gas . values for the rate coefficients ( or radiative cross - sections where appropriate ) are given in @xcite . references are to the primary source(s ) of the data whenever possible ; in many cases , we have also used analytical fits to this data from @xcite , @xcite or @xcite . [ cols= " < , < , < " , ] given the initial temperature and chemical abundances , plus the set of chemical reaction rates , actually solving for the thermal and chemical evolution is relatively easy . as in the igm case , we simply solve the coupled set of chemical rate equations @xmath160 @xmath177 using the stifbs integrator of @xcite . at the start of each timestep ( hereafter time @xmath178 ) , we compute the photochemical rates as outlined in sections [ rtmethod ] . we then use stifbs to solve for the new chemical abundances and new temperature at the end of the timestep ( time @xmath179 ) , repeating this for each shell in turn . we next store these values , return to time @xmath178 , and recalculate them using the same procedure , but with _ two _ timesteps of length @xmath180 . we recalculate the photochemical rates at the intermediate point . we next test for convergence by comparing our two sets of results . if any of the chemical abundances or temperatures of any of the shells differ by more than 0.1% , then we reject the results and begin again from time @xmath178 with a smaller timestep . otherwise , we check to see whether we need to halt the simulation , using the criteria discussed below , and , if we do not , we begin the computations for a new timestep starting from @xmath179 . one final approximation that we find useful in practice is to fix the @xmath170 and @xmath174 abundances at their equilibrium values . this allows the integrator to take much larger timesteps than would otherwise be possible , but introduces very little error into the computed @xmath0 abundances . using the method outlined in the preceding sections , we compute the chemical and thermal evolution of the protogalaxy until one of two conditions is met : either the protogalactic gas begins to cool strongly , or we exceed a preset time limit , @xmath181 . to assess whether gas cooling is ` strong ' enough requires an objective cooling criterion . a number of different possibilities have been suggested in the literature @xcite . in our simulations , we adopt the criterion used by @xcite : we require that the elapsed time exceeds the cooling time , as computed at the edge of the protogalactic core , at a distance @xmath182 from the centre of the protogalaxy . the advantage of this choice is that it avoids giving us a false positive result in cases where @xmath183 drops briefly below @xmath184 at early times , but remains so for a time @xmath185 . as an additional sanity check , we also require that the final temperature be smaller than the initial temperature . if the protogalactic gas does not cool strongly , then the simulation will terminate when it reaches @xmath181 . this pre - set time limit is required on purely practical grounds , to prevent simulations in which the gas does not cool from running for excessive amounts of time , but also has a physical justification . in our simulations , we treat protogalaxies as isolated objects , uninfluenced by external events . in reality , they are part of a dynamically evolving mass distribution , and the majority will only survive for a limited time before merging with other protogalaxies of a similar or larger size . it is possible to use the press - schechter formalism to calculate the distribution of survival times as a function of mass @xcite , but for rare objects the mean survival time is typically of the order of the hubble time and thus for simplicity we set @xmath186 . in the following sections , we present results from a number of simulations that examine the effects of the x - ray backgrounds produced by the various source models discussed in sections [ def_xray ] and [ alt_xray ] . unless otherwise noted , all of these simulations assume the same parameters for the star formation model : a standard salpeter imf , with @xmath187 and @xmath188 , a star formation efficiency @xmath189 and a starburst lifetime @xmath190 . additionally , all of the simulations use the same cosmological model , the @xmath168cdm concordance model of @xcite , which has parameters @xmath191 . before examining the effect of an x - ray background on protogalactic evolution , we first briefly study the evolution of @xmath32 in its absence . as well as allowing us to determine the sensitivity of our results to variations in the uv source model , this also provides us with a necessary baseline against which to compare our other results . in figure [ baseline ] , we plot the evolution of @xmath32 with redshift for the three different star formation models discussed in section [ uvdensity ] . all three models assume a salpeter imf , with maximum stellar mass @xmath51 , as well as our standard star formation efficiency , starburst lifetime and cosmological model , described previously . our basic model assumes a metal - free stellar population , with a minimum stellar mass of @xmath187 ; the corresponding results are given by the dotted line in figure [ baseline ] . the dashed line illustrates the effect of increasing the minimum mass to @xmath50 ; the solid line assumes the same @xmath53 , together with a metallicity @xmath57 . figure [ baseline ] demonstrates that although the strength of the lyman - werner background increases by almost a factor of five as we move from our basic model to the metal - enriched model , this has less effect on @xmath32 than we might expect : the difference in @xmath32 between the three models is never more than 50% , and the qualitative details of its evolution are very similar in all three models . this suggests that the uncertainty introduced by our lack of knowledge of the properties of the primordial stellar population need not be unduly limiting . however , it is also clear that some uncertainty remains , and this will place a lower limit on the magnitude of any effect that we can reliably claim to detect , as small variations in @xmath32 due to the x - ray background will be swamped the by the error resulting from the uncertainty in the lyman - werner background . in the work that follows , we take as our baseline the results of our basic , metal - free , low @xmath53 model ; as figure [ baseline ] demonstrates , this minimizes the strength of the lyman - werner background , and thus will tend to maximize the effectiveness of the x - ray background . in figure [ base_xray ] , we plot the evolution of @xmath32 in the presence of the x - ray background generated by the massive x - ray binary model described in section [ def_xray ] . for the purposes of comparison , we also plot the results of our basic x - ray free model . initially , the evolution of @xmath32 is the same in both models , implying that the x - ray background has little or no effect on the gas . at a redshift @xmath192 and critical temperature @xmath193 , however , the models begin to diverge significantly . in the x - ray free model , the critical temperature continues to increase rapidly until it reaches its maximum value of @xmath194 . in the x - ray binary model , by contrast , the rate of increase of @xmath32 is significantly slower , and it fails to reach its maximum value by the end of the simulation at @xmath195 . because at lower redshift we expect the effects of ionizing radiation from both stellar sources and quasars to become increasingly important and thus our results to become unreliable . clearly if reionization occurs at @xmath196 , the same is true for some portion of the results plotted here . ] it is clear from figure [ base_xray ] that the presence of the x - ray background significantly affects the evolution of gas in the larger of the @xmath0-cooled protogalaxies . in small protogalaxies , on the other hand , the negative feedback caused by the lyman - werner background remains as strong as ever . a simple way to judge the importance of this effect is to examine the difference it makes to the fraction of gas in the universe that can collapse and cool . as we saw in section [ sfr ] , we can use the press - schechter formalism to express the cool gas fraction as @xmath39.\ ] ] using the relationship between @xmath43 and @xmath32 derived in that section , it is straightforward to calculate the evolution of @xmath45 in both the x - ray binary and x - ray free models . to better highlight the difference between the two models , we plot in figure [ fcoll ] the ratio of the cooled gas fraction in the x - ray binary model to that in the x - ray free model . at high redshift , the evolution of @xmath32 is the same in both models , and thus the ratio is one . below @xmath197 , the behaviour of the models begins to diverge , but this has little effect on the ratio until the models begin to diverge sharply at @xmath198 . thereafter , the ratio rises sharply to a peak at @xmath199 , and subsequently declines as the growth in the cooled mass fraction becomes dominated by the formation of larger protogalaxies that cool via lyman-@xmath84 emission . figure [ fcoll ] shows us that by ignoring the effect of the x - ray background we underestimate @xmath45 by at most a factor of two . in fact , the difference is likely to be even smaller : we have assumed that all of the gas in a protogalaxy with @xmath200 will cool , but this need not be the case if an x - ray background is present , as x - ray heating will prevent gas from cooling in the low - density outer layers of the protogalaxy . whether this will affect the star formation rate within individual protogalaxies is not clear , but in any case the difference in the globally - averaged star formation rate is unlikely to be greater than a factor of two . note , however , that this extra star formation occurs entirely in low - mass systems , from which ionizing photons @xcite and supernova ejecta @xcite can more readily escape . doubling the star formation rate may thus significantly increase the feedback of primordial star formation on the igm . finally , given the uncertainties in the data underlying our simple massive x - ray binary model ( see section [ def_xray ] ) , it is of interest to investigate the sensitivity of our results to changes in the strength of the x - ray background . accordingly , we have run additional simulations in which the strength of the x - ray sources was increased or decreased by a factor of ten . the results are plotted in figure [ alt_norm ] , together with the results of our basic x - ray binary model and of the x - ray free model . unsurprisingly , increasing or decreasing the strength of the x - ray background alters the evolution of @xmath32 . decreasing it by an order of magnitude increases @xmath32 to the point where its evolution is little different from that in the x - ray free model . increasing it by an order of magnitude , on the other hand , systematically lowers @xmath32 , although never by more than a factor of two . as we discussed in section [ alt_xray ] , it is possible that massive x - ray binaries are much less abundant at high redshift than at the present day . if so , then the x - ray emission of star - forming galaxies will be dominated by supernova remnants ( snr ) . these will generate x - rays through two main emission mechanisms : thermal bremsstrahlung from hot gas , and inverse compton scattering of the cmb by relativistic electrons . we consider the effects of these mechanisms separately . we examined two possible variants of the thermal bremsstrahlung model . in one , we assumed that the characteristics of the emission are broadly the same as those observed locally , with a fraction @xmath99 of the supernova energy being radiated as x - rays with a characteristic temperature @xmath201 @xcite . in the other model , we assumed that all high - redshift supernovae explode in extremely dense surroundings , producing x - ray bright , ultra - compact remnants with @xmath202 and @xmath203 @xcite . realistic models should lie somewhere between these two extremes . however , we found that in neither of these cases does the x - ray background affect the evolution of @xmath32 : at our level of temperature resolution , the results are identical to those obtained for the x - ray free model . the obvious conclusion is that the background produced by bremsstrahlung is simply too weak to be effective . our inverse compton model fares somewhat better . in this model , the strength of the x - ray background is proportional to the mean fraction @xmath106 of the supernova explosion energy that is transferred to relativistic electrons within the remnant . this value is not known accurately and so we considered two possible cases , one with @xmath108 and another with @xmath109 , these being conservative lower and upper bounds on the true value . in the former case , we again saw no significant effect on the evolution of @xmath32 . in the latter case , on the other hand , we saw results very similar to those obtained for the x - ray binary model , as illustrated in figure [ sn_ic ] . in principle , therefore , inverse compton emission from supernova remnants could be as important an x - ray source as emission from x - ray binaries . ultimately , however , its importance depends upon the value of @xmath106 , and no firm conclusions are possible until this value is better constrained . in this context , the possible observational tests suggested by @xcite could prove extremely valuable . as well as determining the evolution of @xmath32 , our simulations also allow us to study the thermal and chemical evolution of the diffuse intergalactic medium , as outlined in section [ model_pg ] . as a simple example , we plot in figure [ h2_in_igm ] the evolution of the fractional @xmath0 abundance in the igm for the x - ray free model ( solid line ) and the x - ray binary model ( dashed line ) . in both cases , the fractional abundance rapidly decreases from its primordial value due to photodissociation by the ultraviolet background , reaching @xmath204 by @xmath205 . subsequently , its decline slows , in part because the rate of increase in the strength of the ultraviolet background also slows . below @xmath206 , the behaviour of the two models diverges . in the x - ray free model , @xmath207 continues to decline until the end of the simulation . in the x - ray binary model , on the other hand , the increasing ionization of the igm boosts the @xmath0 formation rate to the point where it overtakes the photodissociation rate and the @xmath0 abundance , after reaching a minimum at @xmath208 , begins to climb . nevertheless , it remains extremely small at the end of the simulation , readily justifying our assertion in section [ lwabs ] that absorption by intergalactic @xmath0 does not play a significant role in determining the strength of the lyman - werner background . in figure [ igm_ion ] , we plot the evolution of the fractional ionization of the igm . in the x - ray free model , this remains approximately constant over the lifetime of the simulation , as the recombination timescale is significantly longer than the hubble time . in the x - ray binary model , on the other hand , photoionization by the growing x - ray background eventually overcomes the very small recombination rate and drives the fractional ionization upwards , increasing it by just over an order of magnitude by the end of the simulation . even so , it remains small , demonstrating that the x - ray background does not contribute significantly to cosmological reionization . the effect on the fractional ionization of helium ( the ratio of @xmath128 to @xmath172 ) is rather more striking . the post - recombination @xmath128 abundance is extremely small @xcite , and in the x - ray free model remains at this low level throughout the simulation . in the x - ray binary model , on the other hand , it increases dramatically over the course of the simulation , reaching @xmath209 by @xmath38 . this is illustrated in figure [ igm_heion ] . finally , in figure [ igm_heat ] we plot the evolution with redshift of the temperature of the igm . in the x - ray free model , adiabatic cooling dominates the thermal evolution , and the temperature falls off approximately as @xmath210 . in the x - ray binary model , on the other hand , photo - electric heating begins to heat the igm strongly at @xmath211 , driving the temperature up to @xmath212 by the end of the simulation . thus , although the x - ray background does not contribute significantly to the reionization of the igm , it does produce substantial reheating prior to reionization . moreover , given the large mean free path of the x - ray photons , this reheating occurs almost uniformly throughout the igm , rather than being localized to the vicinity of star - forming galaxies . one consequence of this reheating is that the formation of very small - scale structure will be suppressed , as the increased temperature of the igm leads to an increased jeans mass . this is unlikely to affect the global star formation rate , however , as star formation within these small structures would in any case be strongly suppressed by the ultraviolet background . nevertheless , it will reduce the mean clumping factor of the igm below the level that we would otherwise predict , which may in turn speed up reionization @xcite . reheating also affects the visibility of the igm in the redshifted @xmath213 line of neutral hydrogen . @xcite show that scattered lyman-@xmath84 emission from high redshift galaxies efficiently couples the spin temperature of the hyperfine levels to the kinetic temperature of the gas . if the gas temperature is smaller than the cmb temperature , this results in @xmath213 line absorption ; if it is greater , then it results in emission . absorption is easier to detect than emission @xcite , but the heating produced by the x - ray background implies that absorption occurs only at @xmath214 , and that concentrating on detecting @xmath213 emission may be the more viable strategy . the results of the previous section allow us to assess the impact of the high redshift x - ray background that is produced by star - forming galaxies . if we assume that the x - ray emission of these galaxies is similar to that observed locally , and that the same correlation between x - ray luminosity and star formation rate applies , then we find that the background produced is strong enough to partially offset the effects of uv photodissociation in large ( @xmath215 ) , @xmath0-cooled protogalaxies . however , local emission is dominated by massive x - ray binaries , which may not form in large numbers at high redshift . therefore , we have also explored the effect of an x - ray background produced by emission from supernova remnants . if this emission is dominated by inverse compton scattering and if the fraction of the supernova energy transferred to the relativistic electrons powering this emission is large , then the resulting background has very similar effects to one produced by x - ray binaries . on the other hand , if the fraction of energy transferred is small , then the background has little or no effect . in addition to inverse compton emission , we have also examined the effect of thermal bremsstrahlung emission from hot gas in the remnants , and find that even if all supernovae were to form x - ray bright , ultra - compact remnants , the resulting x - ray background would still be too weak to significantly affect protogalactic evolution . finally , none of these models produces an x - ray background that is strong enough to balance uv photodissociation in small protogalaxies , with virial temperatures @xmath216 . in these protogalaxies , negative feedback always dominates . how significant are these results ? one simple way to assess this is to study the evolution of the mass fraction of cooled gas , which represents the total amount of matter available to form stars . comparing its evolution in the x - ray binary model to that in the absence of an x - ray background , we find that it is increased by approximately a factor of two . given our star formation model , this corresponds to an increase in the global star formation rate by the same amount . however , this is small compared to the order of magnitude increase that would result if we were simply to ignore the effect of the uv background ( see figure 7 of * ? ? ? * ) . in reality , the difference between the two models may be greater than this because the additional star formation takes place entirely in low - mass systems , from which ionizing photons and supernova - produced metals can readily escape @xcite . doubling the star formation rate may therefore have more than double the impact on the intergalactic medium . however , to properly assess the ultimate importance of this effect requires more detailed modelling , beyond the scope of this paper . ultimately , understanding the history of star formation in the small protogalaxies studied in this paper remains important even if they do not contribute to the reionization or enrichment of the igm to any great degree . this is simply because , in a hierarchical universe , these protogalaxies are the building blocks from which larger galaxies form and therefore set the initial conditions for later stages of galaxy formation . in particular , very little metal enrichment of the primordial gas is required in order to allow the cno cycle to operate and population ii ( rather than population iii ) stars to form , and yet this can have a profound effect on the predicted spectral energy distribution of an early stellar population . although the main purpose of our study was to examine the effects of the x - ray background on the thermal and chemical evolution of gas within protogalaxies , our approach also allows us to examine the effects of the background on the diffuse igm . our main results are three - fold : 1 . we confirm the rapid destruction of @xmath0 in the intergalactic medium noted by @xcite , but also show that when an x - ray background is present the @xmath0 abundance does not continue to decline indefinitely , but eventually stabilizes and may even begin to increase . however , it never becomes large enough to significantly affect the lyman - werner background . we show that although photoionization by the x - ray background significantly increases the fractional ionization of the igm ( and in particular the fractional ionization of helium ) , the bulk of the gas remains mostly neutral , demonstrating that the contribution of the x - ray background to cosmological reionization is small . we find that the x - ray background will also heat the intergalactic gas , raising its temperature to @xmath217 by @xmath38 ( compared to @xmath218 in the x - ray free model ) . this will suppress the formation of structure on the smallest scales by increasing the jeans mass . it is unlikely to affect the global star formation rate , since @xmath219 , but may speed up the process of reionization by reducing the mean clumping factor of the igm . a number of other authors have studied the effects of radiative feedback on the formation of @xmath0-cooled protogalaxies @xcite . most of these studies have concentrated solely on the effects of the lyman - werner background , generally finding that it suppresses cooling ( and hence star formation ) by @xmath19830 , prior to cosmological reionization . our x - ray free simulations support this conclusion . in particular , comparison with the results of @xcite , who use a very similar method but with a different implementation of radiative transfer , gas chemistry and cooling , shows good agreement . rather less work has been done on the effects of the high redshift x - ray background . @xcite examined the formation of molecular hydrogen in a constant density primordial cloud illuminated by a power - law uv spectrum extending into the hard x - rays , and found that in some cases , the background could enhance @xmath0 formation . this lead @xcite to examine the effects of a quasar - produced x - ray background , using a very simple model in which the x - ray flux is a fixed fraction of the lyman - werner band flux , modulated by absorption by a fixed column density of @xmath169 and @xmath172 . they found that an x - ray to uv flux ratio of 10% was enough to overcome negative feedback , and that a higher flux ratio could potentially produce positive feedback . recently , @xcite have studied the effects of similar model backgrounds using an adaptive - mesh hydrodynamics code . they also find that x - rays reduce the effectiveness of negative feedback , but that the latter still dominates . they do not find evidence for the positive feedback predicted by @xcite . however , their simulations do not include the effects of @xmath0 self - shielding , and thus potentially underestimate the amount of @xmath0 that forms ( although see * ? ? ? * for a different view ) . there are several significant differences between our work and these previous investigations . firstly , we do not assume a fixed spectrum or intensity for the x - ray background ; rather , we specify the properties of the x - ray _ sources _ and subsequently compute the build - up of the background in a self - consistent fashion . moreover , we consider source models where the x - ray emission is proportional to the star formation rate , as is observed to be the case for star - forming galaxies at low redshift ; the relationship between star formation rate and x - ray emission in quasar - based models is far less clear . as a result , we generally consider x - ray backgrounds signficantly fainter than those studied in the papers cited above . this makes direct comparison of our results difficult . however , we note that @xcite find x - ray feedback to be ineffective below @xmath220 , regardless of the strength of the x - ray background ; at the redshifts they consider , this corresponds to a virial temperature @xmath221 , and thus agrees well with the similar result obtained in this paper . in closing , we note that a number of uncertainties still remain in our treatment of this problem . some of these the high redshift star formation rate or the appropriate population iii initial mass function , for instance we simply do not know at the present time . fortunately , changes to our assumed values can be readily incorporated in the framework laid out in this paper . other uncertainties arise from our method of simulation ; in particular , from our assumption of a static density profile for the protogalactic gas . we hope to address these issues in future work . the authors would like to acknowledge detailed comments by the anonymous referee which helped to significantly improve this paper . scog would also like to acknowledge useful discussions on various aspects of this work with tom abel , omar almaini , rennan barkana , marie machacek and si peng oh . financial support for much of this work was provided by a pparc studentship . additional support was provided by nsf grant ast99 - 85392 .

the first generation of stars ( commonly known as population iii ) are expected to form in low - mass protogalaxies in which molecular hydrogen is the dominant coolant . radiation from these stars will rapidly build up an extragalactic ultraviolet background capable of photodissociating @xmath0 , and it is widely believed that this background will suppress further star formation in low - mass systems . however , star formation will also produce an extragalactic x - ray background . this x - ray background , by increasing the fractional ionization of protogalactic gas , promotes @xmath0 formation and reduces the effectiveness of ultraviolet feedback . in this paper , we examine which of these backgrounds has the dominant effect . using a simple model for the growth of the uv and x - ray backgrounds , together with a detailed one - dimensional model of protogalactic chemical evolution , we examine the effects of the x - ray backgrounds produced by a number of likely source models . we show that in several cases , the resulting x - ray background is strong enough to offset uv photodissociation in large @xmath0-cooled protogalaxies . on the other hand , small protogalaxies ( those with virial temperatures @xmath1 ) remain dominated by the uv background in all of the models we examine . we also briefly investigate the effects of the x - ray background upon the thermal and chemical evolution of the diffuse igm . cosmology : theory galaxies : formation molecular processes radiative transfer

You are an expert at summarizing long articles. Proceed to summarize the following text: ferromagnetic nanowires possess interesting properties that might be exploited in spintronic devices and more specifically in non - volatile memory ( mram ) and magnetic logic devices @xcite . the mermin - wagner @xcite theorem forbids ( heisenberg - type ) magnetism in systems of dimension ( @xmath1 ) with short - range interactions . hence , ferromagnetic nanowires being a quasi - one dimensional system displaying magnetic properties represent an interesting system from the fundamental point of view . in addition to their interest in fundamental magnetism , they have many applications in microwave devices such as circulators , superconducting single - photon ghz detectors and counters @xcite , mass information storage ( perpendicular recording ) as read - write recording heads , magneto - electronics ( wire bending angle dependent gmr ) as well as in quantum computing and telecommunication . they are simpler than nanotubes since their physical properties do not depend on chirality and they can be grown with a variety of methods : molecular beam epitaxy , electrochemical methods ( template synthesis , anodic alumina filters ) , chemical solution techniques ( self - assembly , sol - gel , emulsions ... ) etc ... ordered nanowire arrays may be of paramount importance in areas such as extremely high - density information recording of patterned media such as the quantum magnetic disk @xcite and in novel high - frequency communication or signal - processing devices based on the exploitation of spin - waves ( magnonic devices ) @xcite to transfer and process information of spin - currents with no disspative joule effect . in this work , fmr measurements are performed to detect the uniform resonance mode and extract the effective anisotropy field @xmath2 versus angle from arrays of ni nanowires with diameters ranging from 15 nm to 100 nm using conventional angular - dependent ferromagnetic resonance fmr in the x - band ( 9.4 ghz ) as a function of temperature ranging from liquid helium ( 4.2 k ) to room temperature . we find that magneto - elastic effects play an important role in those systems that might be exploited in novel storage concepts . this work is organised as follows : in section 2 , fmr measurements are presented and later analysed in section 3 . we conclude the work in section 4 and appendix a details the fmr angular fitting procedure while appendix b covers dipolar effects . individual ni wires inside the array are aligned parallel to each other within a deviation of a few degrees . they have a length of 6 @xmath3 m and are characterized by a cylindrical shape with a typical variation in diameter of less than 5% and with a low - surface roughness . , applied field @xmath4 and corresponding angles @xmath5 they make with the nanowire axis that can be considered as an ellipsoid - shaped single domain with characteristic lengths @xmath6 and @xmath7 . when the aspect ratio @xmath8 is large enough the ellipsoid becomes an infinite cylinder.,width=240 ] previous experiments on magnetization reversal mechanisms in ni arrays using alternating gradient - force magnetometry and superconducting quantum interference device ( squid ) magnetometry reveal that at room temperature the magnetic anisotropy is dominated by the shape anisotropy . fmr experiments are performed with the microwave pumping field @xmath9 has a frequency of 9.4 ghz with a dc bias field @xmath10 oriented arbitrarily making an angle @xmath11 with the nanowire axis . despite the fact several studies have considered reversal modes by domain nucleation and propagation ( see for instance henry _ et al . _ @xcite for an extensive discussion of the statistical determination of reversal processes and distribution functions of domain nucleation and propagation fields ; see also ferr _ et al . _ @xcite and hertel @xcite who showed the existence of domains with micromagnetic simulations ) , we do not consider domain nucleation and propagation in this work and rather concentrate on the single domain case that might be expected due to the smallness of ni magnetocristalline with respect to shape anisotropy . the angular dependence of @xmath0 in the uniform mode is obtained by considering an ellipsoid with an effective uniaxial anisotropy , despite the fact ni has cubic anisotropy @xcite . this description is sufficient to describe the fmr spectra we find ( see de la torre medina _ et al . _ @xcite for a description fully based on a cubic anisotropy approach ) . the energy is therefore comprised of a small second - order uniaxial anisotropy contribution @xmath12 , shape demagnetization energy and a zeeman term : @xmath13 \label{energy}\ ] ] with @xmath14 the angle the mgnetization makes with the nanowire axis . the resonance frequency @xmath15 is obtained from the smit - beljers @xcite formula that can be derived from the landau - lifshitz equation of motion with a damping term @xmath16 and that requires calculating the angular second derivatives of the total energy : @xmath17 ^ 2=\frac{(1+\alpha ^2)}{\sin ^2\theta } \left [ { \frac{\partial ^2e}{\partial \theta ^2}\frac{\partial ^2e}{\partial \phi ^2}- \left [ { \frac{\partial ^2e}{\partial \theta \,\partial \phi } } \right]^2 } \right ] \label{smit1}\ ] ] this provides a relationship between the effective anisotropy field and the external field at the resonance frequency . the frequency - field dispersion relation obtained from the smit - beljers equation is : @xmath18 [ h_{eff } \cos^2\theta + h \cos(\theta -\theta_h ) ] } \label{smit2}\ ] ] where @xmath19 . at the resonance frequency @xmath20 , @xmath21 and the applied field @xmath22 in the saturated case . in the unsaturated case the magnetization angle @xmath23 and one determines it from energy minimization . calling at equilibrium , magnetization orientation @xmath24 ( taking @xmath25 ) we determine it by evaluating the derivative @xmath26 . we get : @xmath27 where@xmath28 . equations [ min ] and [ smit2 ] are used simultaneously to determine the resonance field @xmath0 versus angle @xmath11 at any temperature . from the measured resonance field @xmath0 versus field angle @xmath11 the @xmath29-factor , saturation magnetization @xmath30 and cubic anisotropy constant @xmath12 are determined with a least - squares fitting method ( see appendix a ) . measured fmr absorption derivative spectra @xcite are similar for ( 50 , 80 and 100 nm ) diameters but differ from the 15 nm case . angular @xmath0 curves versus @xmath11 show a minimum at @xmath31@xmath32for the large diameters ( 50 , 80 and 100 nm ) and a minimum at @xmath33@xmath32for the 15 nm case ( see fig . [ fit ] ) . from this angular variation , we infer that 15 nm samples behave differently from larger diameter samples with a transition observed about 50 nm in agreement with the hysteresis loop vsm measurements ( as displayed in fig . [ vsm ] ) . previously neilsch _ et al . _ @xcite mentioned a change of behaviour in nanowire arrays because of the existence of a coherence diameter in ni to be @xmath34 40 nm . the coherent diameter @xmath35 separates coherent ( stoner - wohlfarth style or homogeneous ) from inhomogeneous reversal ( reversal by curling ) obtained by equating nucleation fields in both cases @xcite : @xmath36 where @xmath37 is the exchange stiffness constant ( for ni , it is about 1.5@xmath38 erg / cm @xcite ) . approximating the nanowire by an infinitely long cylinder , the demagnetizing factor along the minor axis @xmath39 , thus : @xmath40 with @xmath41 the first positive zero of the first kind bessel function @xmath42 derivative @xmath43 . consequently neilsch _ et al . _ @xcite estimate is recovered . nevertheless in our case , the change in behaviour is probably due to change in the values of anisotropy constant @xmath12 and saturation magnetization @xmath30 with the diameter as seen in table [ fit_table ] . [ cols="^,^ " , ] theoretical and experimental values of the resonance field @xmath0 versus temperature are compared in fig . [ fig5 ] after accounting for the magneto - elastic anisotropy @xmath44 contribution in the form @xmath45 . the agreement as a function of temperature for both nanowire diameters and both orientations ( @xmath31@xmath32and @xmath33@xmath32 ) of the field is a strong indication of the presence of magneto - elastic effects . we have performed angular and temperature dependent fmr on ni nanowire arrays with variable diameter and shown with fmr that the easy axis orientation for the 15 nm diameter sample is perpendicular to the wire axis in sharp contrast with the 50 nm , 80 nm and 100 nm samples . note that we expect ( from bulk ni ) the easy axis along the wire axis by comparing the value of shape energy with respect to anisotropy energy . results obtained from the angular behavior of @xmath0 versus @xmath46 show that @xmath0 is minimum at 90@xmath32for the 15 nm sample whereas it is minimum at 0@xmath32for the larger diameter samples agree with hysteresis loops obtained from vsm measurements and confirm presence of the transition of easy axis direction from perpendicular at 15 nm to parallel to nanowire axis at 50 nm diameter . the transition observed at 50 nm is interesting because of several potential applications in race - track mram devices . _ et al . _ @xcite predicted that in permalloy nanowires of 50 nm and less , moving zero - mass domain walls may attain a velocity of several 100 m / s beating walker limit obeyed in permalloy strips with same lateral size . hence , nanowire cylindrical geometry in contrast to prismatic geometry of stripes bears important consequences on current injection in nanowires and its effects on domain wall motion with reduced ohmic losses @xcite . at low temperature , we find that @xmath2 increases when temperature is decreased . this may be attributed to the increase of @xmath47 , as temperature is decreased , affecting all anisotropy fields ( @xmath48 , @xmath49 , and @xmath50 ) that depend on @xmath47 . separating the various contributions to anisotropy by using thermal , frequency , diameter and angular variations we have been able to pinpoint the main contribution to anisotropy at low temperature as stemming from magneto - elastic effects between the nanowire array and the dm supporting it . magneto - elastic effects affect directly the thermal variation of @xmath2 that increase at low temperature due to the increasing difference between the thermal expansion of the metallic nanowire and the dm . thermal effects must be analyzed properly for building extremely high density storage devices . ordered arrays of nanowires are good candidates for patterned media and may also be used in plasmonic applications such as nanoantenna arrays or nanophotonic waveguides in integrated optics @xcite . recently , heat assisted magnetic perpendicular recording using plasmonic aperture or nano - antenna has been tested on patterned media in order to process large storage densities starting at 1 tbits / in@xmath51 and scalable up to 100 tbits / in@xmath51 . in thermally - assisted perpendicular magnetic recording , a waveguide delivers light to a plasmonic nanoantenna placed just above the disk platter surface creating an intense optical pattern in the near - field region . however , this heats the disk on the nanometer scale ( around 25 nm ) and consequently temperature dependent magneto - elastic effects analysed in this work might affect the physics of high - density information writing in these systems . + * acknowledgments * + low temperature measurements were kindly performed by dr . r. zuberek at the institute of physics of the polish academy of science , warsaw ( poland ) . we have developed a procedure based on a least squares minimization procedure of the curve @xmath0 versus @xmath11 to the set of @xmath52 experimental measurements @xmath53}_{i=1,n}$ ] where @xmath54 and @xmath55 . @xmath56 represents the set of parameters to fit @xmath57 . hence the set of three minimum equations for the data points are : @xmath58^{2 } \mbox { minimum } \nonumber \\ \label{eqfit}\end{aligned}\ ] ] where the values @xmath59 are the experimental values of @xmath0 corresponding to the angles @xmath60 and @xmath61 . the fitting method is based on the broyden algorithm , a generalization to higher dimension of the one - dimensional secant method @xcite that allows us to determine in a least - squares fashion , the set @xmath57 of unknowns . broyden method is selected because it can handle over or underdetermined numerical problems and that it works from a singular value decomposition point of view @xcite . this means it is able to circumvent singularities and deliver a practical solution to the problem at hand as an optimal set@xcite within a minimal distance from the real one . demagnetization energy is given by @xmath62 @xcite where the demagnetization coefficients are such that@xmath63 and @xmath64 . since the demagnetization field is given by @xmath65 , component @xmath66 of @xmath67 is @xmath68 . we have two limits : * single isolated nanowire ( for which @xmath69 and @xmath70 ) , @xmath71 and @xmath72 with @xmath73 the unit vector along @xmath74 . * thin film limit ( for which @xmath75 and @xmath76 ) , @xmath77 and @xmath78 . when the sample is saturated along @xmath74 ( nanowire axis ) @xmath79 , the demagnetization field in both cases has a single @xmath74 component : @xmath80 in the single wire case and @xmath81 in the thin film case . making a linear interpolation between these two limits , we get : @xmath82 . if we rather consider a 2d square lattice of nanowires with parameter @xmath83 the average nanowire separation , the porosity is given by : @xmath84 and we can calculate directly the dipolar energy as follows . 99 z. z. sun and j. schliemann , phys . lett . * 104 * , 037206 ( 2010 ) . m. yan , a. kakay , s. gliga and r. hertel , phys . 104 * , 057201 ( 2010 ) . c. t. boone , j. a. katine , m. carey , j. r. childress , x. cheng , and i. n. krivorotov , phys . lett . * 104 * , 097203 ( 2010 ) . n. d. mermin and h. wagner , phys . * 17 * , 1133 ( 1966 ) . j. k. w. yang , e. dauler , a. ferri , a. pearlman , a ; verevkin , g. goltsman , b. voronov , r ; sobolewski , w. e. keicher and k. k. berggren , * ieee * trans superconductivity , * 15 * , 626 ( 2005 ) . s. y. chou , proceedings of the ieee , * 85 * , 652 ( 1997 ) . v. v. kruglyak , s. o. demokritov and d. grundler , j. phys . phys . * 43 * , 264001 ( 2010 ) . y. henry , a. iovan , j .- george and l. piraux , phys . b. * 66 * , 184430 ( 2002 ) . r. ferr , k. ounadjela , j. m. george , l. piraux and s. dubois , phys . b. * 56 * , 14066 ( 1997 ) . r. hertel , j. appl . phys . * 90 * , 5752 ( 2001 ) . sixth - order cubic anisotropy energy is : @xmath111 where : @xmath112 are the cosines of the angles , the magnetization makes with the @xmath113 axes respectively ( see fig . [ fig1 ] ) . the following parameters at room temperature apply to bulk nickel : @xmath114 erg/@xmath115 , @xmath116 erg/@xmath115 whereas @xmath30=485 emu/@xmath115 . j. de la torre medina , m. darques and l. piraux , j. phys . * 41 * 032008 ( 2008 ) . j. smit and h. c. beljers , philips res 10 * , 113 ( 1955 ) u. ebels , j. duvail , p. wigen , l. piraux , l. d. buda , k. ounadjela , phys . b * 64 * , 144421 ( 2001 ) k. neilsch , r. b. wehrspohn , j. barthel , j. kirschner , u. gsele , s. f. fischer and h. kronmller , appl . phys . lett . * 79 * , 1360 ( 2001 ) . r. skomski , a. kashyap , k. d. sorge and d. j. sellmyer j. appl . phys . * 95 * , 7022 ( 2005 ) . g. bertotti , _ `` hysteresis in magnetism '' , _ academic press , new - york ( 1998 ) . a. encinas , m. demand , l. vila , l. piraux , and i. huynen , appl . . lett . * 81 * , 2032 ( 2002 ) w. j. carr , jr , phys . rev . * 109 * , 1971 ( 1958 ) e. r. callen , h. b. callen , phys . rev . * 129 * , 578 ( 1963 ) m. d. kuzmin , phys . lett . * 94 * , 107204 ( 2005 ) . r. m. bozorth , _ `` ferromagnetism '' _ , * ieee * press ( 1993 ) . o. a. tretiakov , y. liu , and a. abanov , phys . lett . * 105 * , 217203 ( 2010 ) . h. ditlbacher , a. hohenau , d. wagner , u. kreibig , m. rogers , f. hofer , f. r. aussenegg and j. r. krenn , phys . lett . * 95 * , 257403 ( 2005 ) . w. h. press , w. t. vetterling , s. a. teukolsky and b. p. flannery , _ `` numerical recipes in c : the art of scientific computing '' , _ second edition , cambridge university press ( new - york , 1992 ) . l. d. landau and e. m. lifshitz , _ electrodynamics of continuous media _ , pergamon , oxford ( 1975 ) .

ferromagnetic resonance ( fmr ) measurements are performed as a function of temperature on nickel nanowires with different diameters ( 15 nm , 50 nm , 80 nm and 100 nm ) from room temperature down to liquid helium ( 4.2 k ) . the resonance field @xmath0 measured as a function of field angle and temperature yields interesting information about the size and temperature variation of various anisotropy effects . with diameter increase from 15 nm to 100 nm we observe a transition at 50 nm in easy axis orientation from perpendicular to parallel to the nanowire array axis . the temperature variation of @xmath0 is analyzed and explained theoretically with presence of strong magneto - elastic effects in small ( 15 nm ) and large diameter ( 100 nm ) nanowire arrays .

You are an expert at summarizing long articles. Proceed to summarize the following text: the foundations of our present understanding of _ advection - dominated accretion _ were laid out in a series of papers by narayan & yi ( 1994 , 1995a , b , hereafter ny94 , ny95a , ny95b ) , abramowicz et al . ( 1995 ) and chen et al . ( 1995 ) , although some ideas were anticipated much earlier by ichimaru ( 1977 ) . the specific abbreviation adaf , which stands for _ advection - dominated accretion flow _ , was introduced by lasota(1996 ) and has become standard in the field . we review here the application of adafs to accreting black holes ( bhs ) , and in 4 we also consider their application to accreting neutron stars . the energy equation per unit volume of an accretion flow can be written compactly in the form @xmath4 where @xmath5 is the density , @xmath6 is the temperature , @xmath7 is the entropy per unit mass , @xmath8 is time , @xmath9 is the flow velocity , and @xmath10 and @xmath11 are the heating and cooling rates per unit volume . this equation states that the rate at which the entropy per unit volume of the gas increases is equal to the heating rate minus the cooling rate . since any entropy stored in the gas is advected with the flow , the left - hand side of equation ( 1 ) may be viewed as the effective _ advective cooling _ rate @xmath12 . equation ( 1 ) can then be rearranged to give @xmath13 which states that the heat energy released by viscous dissipation is partially lost by radiative cooling and partially by advective cooling . the standard thin accretion disk model ( shakura & sunyaev 1973 ; novikov & thorne 1973 ; frank , king & raine 2002 ) corresponds to the case when the accreting gas is _ radiatively efficient _ , so that we have @xmath14 since the gas cools efficiently , the sound speed is much less than the local keplerian speed @xmath15 and the disk is geometrically thin . also , the disk radiates about a tenth of the rest mass energy of the accreting gas , the precise fraction depending on the bh spin ( see shapiro & teukolsky 1983 ) . an adaf corresponds to the opposite regime . here the gas is _ radiatively inefficient _ and the accretion flow is underluminous . thus , an adaf is defined by the condition @xmath16 some papers in the literature define an adaf as a flow that corresponds exactly to a self - similar solution described in ny94 ( see 3.1 ) . this is a needless restriction . in our opinion , it is more fruitful to employ the general definition of an adaf as given in equation ( [ adafdef ] ) , and the flow is only marginally radiatively inefficient ( see fig . 4 ) . nevertheless , even here , the two solutions are very distinct from each other . ] . there are two distinct regimes of advection - dominated accretion . the first , the one that we focus on in this article is when the accreting gas is very tenuous and has a long cooling time ( ny94 ; ny95b ; abramowicz et al . this regime is sometimes referred to as a riaf a `` radiatively inefficient accretion flow '' and is defined by the condition @xmath17 here @xmath18 is the cooling time of the gas and @xmath19 is the accretion time . the second regime of advection - dominated accretion is when the particles in the gas have no trouble cooling , but the scattering optical depth of the accretion flow is so large that the radiation is unable to diffuse out of the system . this radiation - trapped regime was briefly discussed by begelman ( 1979 ) and was developed in detail by abramowicz et al . ( 1988 ) in their `` slim disk '' model . the defining condition for this regime of adafs is @xmath20 where @xmath21 is the diffusion time for photons . the present review is devoted exclusively to the adaf / riaf form of accretion ( see narayan , mahadevan & quataert 1998b ; kato , fukue & mineshige 1998 ; lasota 1999a , b ; quataert 2001 ; narayan 2002 , 2005 ; igumenshchev 2004 ; done , gierlinsky & kubota 2007 for reviews emphasizing various aspects of adafs ) . we will henceforth drop the modifier riaf and refer to these flows simply as adafs . as explained above , an adaf is by definition very different from the standard thin accretion disk . correspondingly , it is characterized by very distinct observational signatures . both adafs and thin disks have well - known counterparts in nature . in an adaf , since most of the energy released by viscous dissipation is retained in the gas , the pressure is large , and so is the sound speed : @xmath22 , where the keplerian velocity @xmath23 is equal to @xmath24 with @xmath25 being the radius in schwarzschild units , @xmath26 . the large pressure has several immediate consequences . first , the accretion flow becomes geometrically thick , with a vertical height @xmath27 of order the radius @xmath28 ( an adaf may be viewed as the viscous rotating analog of spherical bondi accretion ) . second , the flow has considerable pressure - support in the radial direction , so the angular velocity becomes sub - keplerian . third , the radial velocity of the gas is relatively large : @xmath29 , where @xmath30 ( see below ) is the standard dimensionless viscosity parameter ( shakura & sunyaev 1973 ) . fourth , the large radial velocity leads to a short accretion time : @xmath31 , where @xmath32 is the free - fall time . finally , the large velocity and large scale height cause the gas density to be very low , and so the cooling time is very long and the medium is optically thin . the above properties are nicely illustrated in the self - similar adaf solution derived by ny94 , in which all quantities in the accretion flow behave as power - laws in radius . ( a similar solution was obtained by spruit et al . 1987 in a different context . ) ny94 obtained a general solution for arbitrary viscosity parameter @xmath33 , adiabatic index @xmath34 , and advection parameter corresponds to a fully cooling - dominated ( no advection ) flow and @xmath35 corresponds to a fully advection - dominated ( no radiative cooling ) flow . ] @xmath36 . in the limit @xmath37 ( a good approximation ) and @xmath38 ( radiatively very inefficient flow ) , the solution simplifies to @xmath39v_k = -0.53 \alpha v_k , \\ \omega & = & \left[{2(5/3-\gamma)\over3(\gamma-5/9)}\right]^{1/2}\omega_k = 0.34 \omega_k , \\ c_s & = & \left[{2(\gamma-1)\over3(\gamma-5/9)}\right]^{1/2}v_k = 0.59 v_k.\end{aligned}\ ] ] the numerical coefficients on the right correspond to @xmath40 , a reasonable choice ( see quataert & narayan 1999 ) . the great virtue of the above self - similar solution is that it is analytic and provides an easy and transparent way of understanding all the key properties of an adaf . its biggest deficiency is that it is scale - free , which means that it is inappropriate near the inner or outer boundary of the flow . therefore , for detailed work , one must use global solutions of the adaf equations that satisfy appropriate boundary conditions ( e.g. , abramowicz et al . 1996 ; narayan , kato & honma 1997c ; chen , abramowicz & lasota 1997 ; manmoto , mineshige & kusunose 1997 ; popham & gammie 1998 ; manmoto 2000 ) . calculating global solutions is somewhat involved and one may wish to employ some short - cuts ( e.g. , yuan , ma & narayan 2008 ) . on the other hand , even the exact global solutions are somewhat limited since they solve a set of height - integrated equations . for greater realism , one might wish to work directly with numerical simulations ( e.g. , goldston , quataert & igumenshchev 2005 ; noble et al . 2007 ) . the adaf solution is gas pressure dominated . since @xmath41 , this means the gas temperature is nearly virial . under normal conditions , gas at such a high temperature will radiate copiously , especially at small radii where the temperature can approach @xmath0 k. thus , in order to have an adaf , the accreting gas generally has to be a _ two - temperature plasma _ ( at least at small radii ) , with electron temperature @xmath42 much less than the ion temperature @xmath43 ( ny95b ) . ( the only way to avoid this condition is by having an extremely low accretion rate below about @xmath44 of the eddington rate . ) typical adaf models have the two temperatures scaling as @xmath45 in order for gas in an adaf to be two - temperature , there must be weak coupling between electrons and ions . models generally assume that the coupling occurs via coulomb collisions , which is inefficient at the densities under consideration . begelman & chiueh ( 1988 ) investigated whether plasma instabilities might enhance the coupling and drive the plasma rapidly to a single temperature ; this would be problematic for the adaf solution . however , they were unable to identify a clear mechanism . for normal mass accretion rates , the electrons in a two - temperature adaf will have a thermal energy distribution ( but not necessarily the ions , see mahadevan & quataert 1997 ) . early work on two - temperature adafs assumed that viscous heating acts primarily on the ions ; for instance , ny95b took the ratio of electron heating to total heating , @xmath46 , to be zero , while esin , mcclintock & narayan ( 1997 ) assumed @xmath47 . however , an adaf does not _ require _ @xmath46 to be this small . because of a degeneracy in model parameters ( quataert & narayan 1999 ) , it is possible to have a viable adaf model with larger values of @xmath46 , provided the mass loss parameter @xmath48 , defined in 3.6 , is adjusted . more recent adaf models ( e.g. , yuan , quataert & narayan 2003 ) typically assume @xmath49 . various attempts have been made to estimate the value of @xmath46 from first principles by considering the effects of reconnection ( bisnovatyi - kogan & lovelace 1997 ; quataert & gruzinov 1999 ) or mhd turbulence ( quataert 1998 ; blackman 1999 ; medvedev 2000 ) . these studies do not agree on a single value of @xmath46 , but generally suggest that @xmath46 is likely to be much larger than @xmath50 . recently , sharma et al . ( 2007 ) considered heating by the dissipation of pressure anisotropy and showed that @xmath51 . it should be noted that , even if electrons and ions receive equal amounts of the dissipated energy ( i.e. , @xmath52 ) , the plasma can still be two - temperature . this is because a large part of the heating in an adaf is by compression ( since the density increases inward ) . once @xmath53 , which is the case for @xmath54 a few hundred , the electrons become relativistic and have an adiabatic index @xmath55 , whereas the ions continue to be non - relativistic with @xmath56 . since adiabatic heating by compression causes the temperature to scale as @xmath57 , the electrons heat up only as @xmath58 whereas the ions heat up as @xmath59 . thus , even in the limiting case of @xmath60 , adafs naturally become two - temperature at small radii . for instance , in the adaf model of sagittarius a@xmath3 ( sgr a@xmath3 ) proposed by yuan et al . ( 2003 ) , the authors obtain @xmath61 close to the bh even though they assumed @xmath62 . an important property of the adaf solution is that it is thermally stable ( ny95b ; wu & li 1996 ; kato et al . 1997 ; wu 1997 ) . the demonstration of this property was a crucial advance . two decades earlier , in a seminal paper , shapiro , lightman & eardley ( 1976 ) , and after them rees et al . ( 1982 ; ion tori ) , introduced the idea of a two - temperature accretion flow and derived a hot two - temperature accretion flow solution , the sle solution . however , that solution turned out to be thermally unstable ( piran 1978 ) . until the development of the adaf solution , and the recognition that it is different from the sle solution , no stable , hot , optically thin solution was available to model the many accretion systems whose spectra ( especially in the hard state , see below ) demand such a flow . chen et al . ( 1995 ) and yuan ( 2003 ) have explored the relationships among the adaf / riaf , sle , adaf / slim disk and thin disk solutions . the adaf is a full accretion solution which incorporates consistent dynamics , thermal balance , radiation physics , etc . therefore , the radial profiles of all gas properties can be calculated self - consistently once we know the values of certain parameters : bh mass @xmath63 , accretion rate @xmath64 , viscosity parameter @xmath33 , pressure parameter @xmath65 , adiabatic index @xmath34 ( usually @xmath66 ) , viscous heating parameter @xmath46 , advection parameter @xmath67 . actually , apart from the system - specific parameters @xmath63 and @xmath64 ( which may be estimated through observation ) , most of the other parameters are constrained . under the nearly collisionless conditions expected in an adaf , the viscosity parameter is moderately enhanced relative to a collisional gas ( sharma et al . 2006 ) . in the case of dwarf novae in the hot state , smak ( 1999 ) estimates @xmath68 , while numerical simulations of the magneto - rotational instability give @xmath69 ( hawley , gammie & balbus 1996 ) . thus we expect @xmath30 for an adaf ( typical values used in models are @xmath70 ) . numerical simulations further suggest that magnetic fields are generally subthermal , with @xmath71 ( e.g. , hawley et al . 1996 ) , so we expect @xmath72 . by calculating the energy loss via radiation from the hot accretion flow ( synchrotron , bremsstrahlung , compton scattering ) , @xmath67 can be obtained self - consistently ( e.g. , ny95b ; narayan , barret & mcclintock 1997a ; esin et al . 1997 ; yuan et al . thus , we have only one poorly constrained parameter : @xmath46 . the recent work of sharma et al . ( 2007 ) provides a serviceable prescription even for this parameter . at large radii , where the plasma is effectively one - temperature , their formula gives @xmath73 , while in the energetically important inner region , where the plasma is two - temperature , they find @xmath74 , depending on model details . the final two parameters are discussed later : the transition radius @xmath75 , 3.5 , and the wind parameter @xmath48 , 3.6 . . ( from esin et al . 1998),width=316 ] since the earliest days of x - ray astronomy , it has been clear that bh binaries ( bhbs ) have a number of distinct spectral states ( see zdziarski & gierlinski 2004 ; mcclintock & remillard 2006 ; done et al . the most notable among these are the luminous _ high soft state _ , or _ thermal state _ , the slightly less luminous _ low hard state _ , and the very under - luminous _ quiescent state_. the thermal state is well described by the thin disk model , and a multi - color disk ( mcd ) blackbody model ( e.g. , _ diskbb _ , mitsuda et al . 1984 ; _ ezdiskbb _ , zimmerman et al . 2005 ; both available in xspec , arnaud et al . 1996 ) has been successfully used for years to model the x - ray spectra of sources in this state . recently , fully relativistic versions of the mcd model for arbitrary bh spin ( _ kerrbb _ , li et al . 2005 ; _ bhspec _ , davis & hubeny 2006 ) have been developed , based on the relativistic thin disk model of novikov & thorne ( 1973 ) . these models provide excellent fits to the x - ray spectra of bhbs in the thermal state ( e.g. , mcclintock et al . 2006 ; davis , done & blaes 2006 ) . whereas a satisfactory theoretical model , viz . , the thin disk model , was established early on for the thermal state , the hard state was for many years a mystery . figure 1 shows the spectrum of a typical bhb , gro j0422 + 32 , in the hard state ( esin et al . the observations indicate that the accreting gas is very hot , @xmath76 kev . the gas must also be optically thin , since optically thick blackbody emission at a temperature of 100 kev would correspond to a luminosity @xmath77 for any reasonable estimate of the radiating area @xmath78 . the most natural explanation of the emission in j0422 and other bhbs in the hard state is that it is produced by thermal comptonization . until the adaf model was established , no accretion model could reproduce such a spectrum . ( the sle solution could , but it was unstable . ) indeed , astronomers were reduced to using empirical comptonization models ( sunyaev & titarchuk 1980 ) in which they postulated a comptonizing cloud with some arbitrary geometry and parameterized the cloud with an adjustable temperature and an optical depth ( e.g. , zdziarski et al . 1996 , 1998 ; gierlinski et al . 1997 ) . the situation changed with the recognition of the adaf solution . this model turned out to have the precise properties density , electron temperature , stability needed to explain the hard state . the solid line in fig . 1 shows an adaf model of j0422 ( esin et al . 1998 ) in which the accretion rate has been adjusted to fit the spectrum ; the required rate is about a tenth of the eddington mass accretion rate @xmath79 , where @xmath80 , i.e. , it is the mass accretion rate at which a disk with radiative efficiency 0.1 would radiate at the eddington luminosity . it is gratifying that both the temperature ( which determines the position of the peak ) and the compton @xmath81-parameter ( which determines the power - law slope below the peak ) are reproduced well . esin et al . ( 1997 , 1998 , 2001 ) present models of other bhbs in the hard state . -0.2 in limits from asca ( narayan et al . 1997a ) , and the upper limit in the euv is derived from the absence of a heii @xmath82 line . the solid line corresponds to an adaf model with @xmath83 . the dashed line is a thin disk model whose @xmath64 has been adjusted to fit the optical data . this model fits poorly in the x - ray band and is inconsistent with the euv limit . ( from narayan et al . 1997a),title="fig:",width=355 ] -1.5 in . the radio data are from falcke et al . ( 1998 ; open circles ) and zhao et al . ( 2003 ; filled circles ) , the ir data are from serabyn et al . ( 1997 ) and hornstein et al . ( 2002 ) , and the two `` bow - ties '' in the x - ray band correspond to the quiescent ( lower ) and flaring ( higher ) data from baganoff et al . ( 2001 , 2003 ) . the solid line is an adaf model of sgr a@xmath3 in the quiescent state . the mass accretion rate is @xmath84 near the bh . ( from yuan et al . 2003),width=288 ] a typical accreting bh observed in the hard state has a luminosity on the order of a few percent of eddington . at much lower luminosities , we have the quiescent state , where the spectrum becomes noticeably different . figures 2 and 3 show observations of a quiescent bhb , v404 cyg , and a quiescent supermassive bh , the galactic center source sgr a@xmath3 . although these spectra look very different from the one shown in fig . 1 , the adaf model is able to fit these and other observations of quiescent systems ( narayan , mcclintock & yi 1996 ; narayan et al . 1997a ; yuan et al . all it requires is a lower value of @xmath64 , as appropriate for the lower luminosity . the qualitative features of the spectrum , e.g. , a softening of the x - ray power - law index ( see corbel , tomsick & kaaret 2006 ) , follow naturally . however , a caveat is in order : in some cases , the x - ray emission in quiescence may be from a jet lauched from the adaf ( 3.6 ) , rather than from the adaf itself ( e.g. , yuan & cui 2005 ) . of an accretion flow around a bh , as a function of the eddington - scaled mass accretion rate ( hopkins , narayan & hernquist 2006b ) . the horizontal segment between 0.01 and 0.1 of the eddington accretion rate shows a transition regime in which a part of the accretion flow is an adaf , but the radiative efficiency is still large . it might correspond to the intermediate state , and perhaps the upper end of the hard state ( 3.5 , fig . although this plot is based on calculations shown in fig . 11 of ny95b and fig . 13 of esin et al . ( 1997 ) , it is still very approximate.,width=316 ] as expressed in equation ( [ adafdef ] ) , the defining property of an adaf is that it is radiatively inefficient , @xmath85 calculations show that the adaf solution is possible only for low mass accretion rates . specifically , only when @xmath86 is the gas density low enough to permit a two - temperature plasma ( ny95b ; narayan 1996 ; esin et al . 1997 ) . near the critical luminosity @xmath87 or critical mass accretion rate @xmath88 at which the adaf solution first becomes viable , the radiative efficiency is , by continuity , not very different from that of a thin disk : @xmath89 . however , with decreasing @xmath64 , the efficiency decreases . very roughly , we estimate ( fig . 4 ) @xmath90 ( we note that for the different prescription for @xmath46 used by sharma et al . 2006 , @xmath91 falls rapidly only for @xmath92 . ) the extreme inefficiency of an adaf at very low accretion rates is critical for understanding the peculiar properties of accreting bhs in the quiescent state ( as first discussed by narayan et al . 1996 for bhbs and narayan et al . 1995 for supermassive bhs ) . we do not discuss this topic here , but point the reader to other reviews for details ( e.g. , quataert 2001 ; narayan 2002 , 2005 ) . the quiescent state has also played a major role in our efforts to test for the presence of an event horizon in bhbs . this is a key topic of 4 . ( based on esin et al . the adaf is represented by the hatched ellipses , with the intensity of hatching indicating the density of the hot gas . the horizontal lines represent a standard thin disk . the lowest panel shows the quiescent state ( qs ) , which corresponds to a very low mass accretion rate ( say @xmath93 ) , a weak jet ( 3.6 ) , a low radiative efficiency ( fig . 4 ) , and a large transition radius ( fig . 6 ) . the second panel from the bottom shows the hard state ( hs ) , where the mass accretion rate is higher ( @xmath94 ) but still below the critical rate @xmath88 , the jet is stronger , the radiative efficiency is somewhat larger , and the transition radius is smaller . the second panel from the top shows the intermediate state ( is ) , where @xmath95 , the jet is even stronger , the radiative efficiency is high @xmath96 , and the transition radius is fairly close to the isco . finally , the top panel shows the thermal state ( ts ) , where there is no adaf , the thin disk extends down to the isco , @xmath89 , and there is no jet . esin et al . ( 1997 ) included a tentative proposal for the so - called very high state ( see also done et al . 2007 ) , now called the steep power - law state ( spl , mcclintock & remillard 2006 ) , but we do not include this still - mysterious state here.,width=355 ] based on the properties of the adaf solution discussed above , narayan ( 1996 ) proposed a simple model for understanding the spectral states of accreting bhs . this picture was developed in detail by esin et al . the basic idea is illustrated in fig . the key parameter that determines the spectral state of an accreting bh is @xmath64 . when @xmath97 , only the thin disk solution is available , and so the accretion flow is in the form of a thin disk all the way down to the innermost stable circular orbit ( isco ) . the system is then in the thermal state , and its spectrum is well - described by the mcd model . once @xmath64 falls below @xmath88 , both the thin disk and adaf solutions become viable ( at least at small radii ) . now accretion continues as a thin disk at radii larger than a transition radius , @xmath98 , but the flow switches to an adaf at smaller radii . when the transition from a pure thin disk ( thermal state ) to a disk - plus - adaf configuration first occurs , i.e. , when @xmath64 is just below @xmath88 , the adaf is very small in size and we have a more - or - less radiatively efficient flow , with @xmath89 . this corresponds to the so - called _ intermediate state_. then , at a somewhat lower @xmath64 , the adaf expands a bit and @xmath91 is modestly lower and we have the classic hard state . finally , when @xmath64 is much lower than @xmath88 , the adaf becomes much larger , @xmath99 , with @xmath100 ( narayan et al . 1996 , 1997a ; menou , narayan & lasota 1999b ) . this is the quiescent state . the above paradigm has proved durable ( see zdziarski & gierlinski 2004 ; done et al . in particular , considerable evidence has accumulated that the thin disk retreats from the innermost stable circular orbit ( isco ) to a large radius in the quiescent state . the evidence is strongest in transient bhbs and cvs . the spectra of quiescent bhbs show absolutely no sign of any soft blackbody - like x - ray emission from a thin disk at small radii ( narayan et al . 1996 , 1997a ; mcclintock , narayan & rybicki 2004 ) . timing properties also indicate a large a large size for the adaf ( hynes et al . 2003 ; shahbaz et al . 2005 ) . in additional , theoretical arguments indicate that the thermal - viscous disk instability , which causes the transient behavior in these systems , is incompatible with observations unless the disk is severely truncated in the quiescent state ( lasota , narayan & yi 1996b ; hameury , lasota & dubus 1999 ; lasota 2001 , 2008 ; dubus , hameury & lasota 2001 ; yungelson et al . there are also indications from the time delay between the optical and x - ray outbursts in the bhb gro j165540 ( orosz et al . 1997 ; hameury et al . 1997 ) that the cool disk is truncated at a large radius in the quiescent state . in the case of supermassive bhs in the quiescent state , there is no feature in the spectrum that might be associated with a thin disk , suggesting that there is no disk at all ; examples are sgr a@xmath3 ( narayan et al . 1998a ; narayan 2002 ) and m87 ( di matteo et al . 2000 , 2003 ) . intermediate luminosity agn do exhibit optical emission from a disk , but the `` big blue bump '' is much less pronounced than in high - luminosity agn ( ho 1999 ) ; this suggests that the disk is truncated at a radius @xmath101 and the interior is filled with an adaf ( gammie , narayan & blandford 1999 ; quataert et al . incidentally , the thermal - viscous disk instability does not appear to operate in agn disks ( menou & quataert 2001 ; hameury , lasota & viallet 2007 ) . in the more luminous hard state , again , there is considerable spectral evidence that the disk is truncated at a transition radius outside the isco , and that the inside is filled with an adaf - like hot flow . the most spectacular example is the bhb xte j1118 + 480 , for which observations carried out in the hard state had unprecedented spectral coverage . the observations are fit well with an adaf model , with a transition radius at @xmath102 ( esin et al . the model even fits the complicated timing behavior of the source ( yuan , cui & narayan 2005 ) . it is hard to imagine that the same data could be explained with any model in which a cool disk extends down to the isco . done et al . ( 2007 ) review spectral observations of a number of other bhbs where again the data require a truncated disk . nemmen et al . ( 2006 ) show that both the spectrum and the double - peaked balmer line profile of the liner source ngc 1097 are consistent with a disk truncated at a few hundred @xmath103 . in an interesting study of cyg x1 , gilfanov , churazov & revnivtsev ( 1999 ; see cui et al . 1999 ; done et al . 2007 ; for discussions of other sources ) found that , as the characteristic frequency in the variability spectrum of the source increases , the power - law tail in the spectral energy distribution steepens and the amplitude of the reflection component in the spectrum increases . this is exactly what one expects when the transition radius between the outer cool disk and the inner hot adaf varies . with decreasing transition radius , ( i ) the noise frequency ( which is likely related in some fashion to the keplerian frequency at the transition radius ) should increase , ( ii ) the hot medium should be cooled more effectively by soft photons from the disk , giving a steeper power - law tail , and ( iii ) the solid angle subtended by the cool disk at the adaf should increase , and there should be a larger reflection component . zdziarski , lubinski & smith ( 1999 ) have noted that a correlation between spectral slope and reflection is commonly seen in both bhbs and agn . a few bhbs in the hard state have been found to show a soft blackbody - like component in their spectra ( balucinska - church et al . 1995 ; di salvo et al . 2001 ; miller et al . 2006a , b ; ramadevi & seetha 2007 ; rykoff et al . 2007 ) . this could be interpreted as evidence that the thin disk extends all the way down to the isco , not at the isco ] . some authors have noted that it is possible for a thin disk to evaporate to an adaf at a relatively large radius and for the hot gas to then re - condense into a thin disk at small radii ( rozanska & czerny 2000 ; liu et al . 2007 ; mayer & pringle 2007 ) . such models might explain the occurrence of a soft spectral component in some hard state sources . however , we note that the soft component typically has only 10% of the total observed luminosity . it is hard to understand how a radiatively efficient thin disk located at the isco could be such a minor contributor to the emitted radiation . this difficulty is highlighted in the work of dangelo et al . they point out that , in any model of the hard state , there will be considerable interaction between the hot gas which produces the hard x - rays and cool gas that may be present in a conventional disk . the interaction will occur via x - ray irradiation ( unless one has large outward beaming , which seems unlikely , e.g. , narayan & mcclintock 2005 ) as well as particle bombardment . using a prototype model for ion bombardment ( deufel et al . 2002 ; spruit & deufel 2002 ; dullemond & spruit 2005 ) , dangelo et al . ( 2008 ) show that a weak soft component in the x - ray spectrum arises quite naturally when the thin disk is truncated at @xmath104 . the model fits the observations surprisingly well ; in fact , the same model would predict a much stronger soft component , and would strongly disagree with the observations , if the cool disk were to extend down to the isco . , plotted as a function of the eddington - scaled accretion luminosity @xmath105 , for a sample of bhbs and low - luminosity agn ( yuan & narayan 2004 ) . the individual estimates of @xmath75 are obtained by fitting spectral observations and are very uncertain . nevertheless , there seems to be a trend of increasing transition radius with decreasing luminosity.,width=297 ] one of the most vexing problems in the theory of adafs is that we do not have a robust method of estimating the location of the transition radius @xmath75 . from the earliest studies ( e.g. , ny95b ; meyer & meyer - hofmeister 1994 ) , it has been plausibly argued that @xmath75 increases with decreasing @xmath64 . however , reliable predictions of the exact dependence have proved difficult , although a number of studies have come up with semi - quantitative results ( meyer & meyer - hofmeister 1994 ; honma 1996 ; rozanska & czerny 2000 ; meyer , liu & meyer - hofmeister 2000 ; spruit & deufel 2002 ; mayer & pringle 2007 ) . yuan & narayan ( 2004 ) tried to use observations to deduce the run of @xmath75 with accretion luminosity . figure 6 shows their results . another issue that has been discussed recently is hysteresis in the transition between the thermal state and the hard state . specifically , with increasing @xmath64 , the hard state survives up to fairly high luminosities @xmath106 , whereas with decreasing @xmath64 , the transition from the thermal state to the hard state occurs at a much lower luminosity @xmath107 ( miyamoto et al . 1995 ; maccarone & coppi 2003 ; zdziarski et al . 2004 ; done et al . meyer - hofmeister , liu & meyer ( 2005 ) have provided a plausible explanation . their disk evaporation model , coupled with compton - cooling of the hot electrons in the adaf , naturally produces a hysteresis in the location of the transition radius . the brightest systems in the hard state ( @xmath108 ) are difficult to model with the standard adaf model . a variant of the adaf solution a natural extension of the model called the luminous hot flow ( lhaf ; yuan 2001 , 2003 ; yuan & zdziarski 2004 ; yuan et al . 2007 ; see also machida , nakamura & matsumoto 2006 ) , looks promising for modeling these sources . ny94 , ny95a discovered an unexpected property of the adaf solution : _ the accreting gas in an adaf has a positive bernoulli parameter _ , which is defined as the sum of the kinetic energy , potential energy and enthalpy , @xmath109 a positive bernoulli constant means that the gas is not bound to the bh . ( this is not surprising , since the gas is not losing energy through radiation . ) therefore , the above authors suggested that adafs should be associated with strong winds and jets ( see also meier 2001 ) . strong outflows have been seen in numerical simulations of adafs . the first indications came from 2d and 3d hydrodynamic simulations ( stone , pringle & begelman 1999 ; igumenshchev & abramowicz 2000 ; igumenshchev , abramowicz & narayan 2000 ) , but it was soon confirmed in mhd simulations as well ( stone & pringle 2001 ; hawley & balbus 2002 ; igumenshchev , narayan & abramowicz 2003 ; pen , matzner & wong 2003 ; machida , nakamura & matsumoto 2004 ; mckinney & gammie 2004 ; igumenshchev 2004 ) . apart from producing a gas - dominated , large - scale outflow , mhd simulations of adafs also have a second distinct outflow component along the axis in the form of a collimated , poynting - dominated , relativistic jet ( mckinney 2005 , 2006 ) . the original suggestion of ny94 that adafs would have outflows and jets has thus been confirmed by these simulations . nevertheless , some authors have disputed a connection between a positive bernoulli parameter and an outflow since it is possible to come up with explicit models that have a positive bernoulli parameter but no outflow ( paczyski 1998 ; abramowicz et al . 2000 ) . observational evidence for the association of nonthermal relativistic jets with adafs has accumulated in recent years with the discovery of radio emission in virtually every bhb in the hard state ( corbel et al . 2000 ; fender 2001 ; fender , belloni & gallo 2004 ; fender & belloni 2004 ) . the radio emission is generally too bright to be produced by thermal electrons in the accretion flow . it is therefore very likely to come from nonthermal electrons in a jet ; in fact , radio vlbi imaging has revealed a resolved jet in a few sources . thus , it is now observationally well - established that the hard state / adaf is associated with relativistic jets . according to the discussion in 3.5 , the quiescent state also has an adaf and should have a ( weaker ) jet . indeed , radio emission has been seen from two quiescent systems , v404 cyg ( hjellming et al . 2000 ; gallo , fender & hynes 2005 ) and a062000 ( gallo et al . 2006 ) , confirming this expectation . how much of the x - ray emission in an adaf system comes from the accretion flow and how much from the jet ? a strong correlation has been seen between radio and x - ray luminosity in the hard state and quiescent state ( corbel et al . 2003 ; gallo , fender & pooley 2003 ) . at first sight this might suggest that the x - ray must also come from the jet ( e.g. , falcke , krding & markoff 2004 ) . however , since the jet flows out of the adaf and is thus highly coupled to it , any model in which the x - ray emission is from the adaf and radio is from the jet is equally compatible with the observations . in recent times , several authors have come down in favor of an adaf origin for the x - rays ( e.g. , heinz & sunyaev 2003 ; merloni , heinz & di matteo 2003 ; heinz 2004 ; heinz et al . 2005 ; yuan et al . 2005 ) . additional evidence on this issue comes from hard state spectra indicating thermal emission ( e.g. , see fig . 1 ) , which is very different from the nonthermal power - law emission one expects from a jet . markoff , falcke & fender ( 2001 ) , among others , have argued that the x - ray emission is due to synchrotron emission from a carefully tuned distrubution of nonthermal electrons . however , zdziarski et al . ( 2003 ) and zdziarski & gierlinski ( 2004 ) showed that the high - energy cutoff in the spectra of bhbs in the hard state , which arises naturally in a thermal adaf - like model , is very difficult to reproduce in a nonthermal synchrotron model . a variant of the jet model assigns the x - ray emission to thermal electrons in the `` base of the jet '' ( markoff , nowak & wilm 2005 ) , but the discussion then becomes semantic . the base of the jet is surely embedded in the underlying adaf and it is not clear that one gains anything by simply relabeling the radiation from the adaf as jet emission . the issue is discussed in greater depth in narayan ( 2005 ) , who advocates a ` jet - adaf ' model ( as described in yuan et al . 2005 ; malzac , merloni & fabian 2004 ) , in which the high energy x - ray emission in the hard state comes from the adaf and the low - energy radio ( and infrared ) emission comes from a jet . the situation is less clear - cut in the quiescent state , where nonthermal emission from the jet might dominate even in x - rays ( yuan & cui 2005 ; wu , yuan & cao 2007 ) . it is interesting , however , that the most quiescent system we know , sgr a@xmath3 , shows no sign of a jet . radio images with a resolution of @xmath110 appear to be jet - free ( shen et al . 2005 ) , while the quiescent x - ray emission is spatially resolved and appears to be from thermal gas near the bondi radius at @xmath111 , not from a jet ( baganoff et al . 2001 , 2003 ) . turning our attention now to the extended wind that is predicted from an adaf , and seen in numerical simulations , one consequence of the wind is that , at each radius , a fraction of the accreting gas is lost from the system . thus , the accretion rate itself varies with radius . this is usually parameterized with an index @xmath48 such that @xmath112 . operationally , this means that , in the self - similar regime , the gas density varies with radius as @xmath113 rather than as @xmath114 in the ny94 model . unfortunately , apart from the rather general constraint that @xmath48 should lie between 0 ( no mass loss ) and 1 ( the limit of a cdaf , 3.7 ) , there is no good theoretical estimate of the value of @xmath48 . we have to resort to numerical simulations or observations . the former tends to give somewhat larger values , e.g. , @xmath115 ( pen et al . 2003 ) , while the one case where the latter approach has been tried , sgr a@xmath3 , yields a smaller value , @xmath116 ( yuan et al . 2003 ) . another general property of adafs was highlighted by ny94 ( see also begelman & meier 1982 ) : adafs have entropy increasing inward and should be convectively unstable by the schwarzschild criterion . numerical hydrodynamic simulations of adafs confirmed the presence of strong convection ( igumenshchev et al . 2000 ; narayan , igumenshchev & abramowicz 2000 ) and led to the development of an analytical self - similar model called the convection - dominated accretion flow ( cdaf ; narayan et al . 2000 ; quataert & gruzinov 2000 ) . the cdaf model employs a simplified one - dimensional treatment of the accretion flow , in which all fluxes are assumed to be in the radial direction . the resulting density varies as @xmath117 . in the language of 3.6 , a cdaf corresponds to @xmath118 . however , technically , an idealized cdaf has no mass outflow , only an outward flow of energy by convection ; the energy is assumed to flow out into a surrounding medium . in practice , the flow is never perfectly one - dimensional ; 2d effects intrude and one expects substantial winds and mass loss as well . a real adaf thus involves a complicated interplay between convection and winds . magnetic fields introduce further complexity , and there has been some discussion in the literature on whether mhd adafs do or do not have real convection ( machida , matsumoto & mineshige 2001 ; balbus & hawley 2002 ; narayan et al . 2002b ; pen et al . 2003 ; igumenshchev 2004 ) . the question has no practical significance , however , since there is general agreement that adafs have both unstable entropy gradients and strong magnetic stresses . in the particular case of a spherical , non - rotating , mhd flow ( the magnetic analog of the bondi problem ) , some complications are absent and the problem becomes relatively clean . here one finds that convection and magnetic fields strongly influence the accretion physics , and the resulting flow is very different from the standard bondi solution ( igumenshchev & narayan 2002 ; igumenshchev 2004 , 2006 ) . this result is likely to have significant implications for astrophysics ; for instance , isolated neutron stars accreting from the interstellar medium will be very much dimmer than one might expect based on the bondi solution ( perna et al . 2003 ) . the adaf solution is essentially independent of the mass @xmath63 of the central bh . that is , if length and time are scaled by @xmath63 and the accretion rate is scaled by the eddington rate ( also proportional to @xmath63 ) , then the same solution is valid for any bh mass . therefore , any successful application of the adaf model to a bhb system has immediate consequences for supermassive bhs ( smbhs ) in an equivalent state , and vice versa . this close connection between adafs in bhbs and adafs in agn was highlighted in narayan ( 1996 ) ; it is also empirically obvious from fig . 6 , which combines observations and models of bhbs and agn . the first smbh to be modeled as an adaf was the galactic center source sgr a@xmath3 ( narayan et al . this was soon followed by fabian & rees ( 1995 ) , who suggested that quiescent nuclei in nearby giant ellipticals ( e.g. , m87 ) must be accreting via adafs , and by lasota et al . ( 1996a ) , who argued that low ionization nuclear emission - line region sources ( liners , e.g. , ngc 4258 ) and low - luminosity active galactic nuclei ( llagn ) must have adafs . both suggestions have turned out to be correct ( e.g. , reynolds et al . 1996 ; mahadevan 1997 ; gammie et al . 1999 ; quataert et al . 1999 ; di matteo et al . 2000 , 2003 ; loewenstein et al . 2001 ; ulvestad & ho 2001 ; nemmen et al . 2006 ) . other authors have suggested that all of the following systems have adafs : fri sources ( baum , zirbel & odea 1995 ; reynolds et al . 1996 ; begelman & celotti 2004 ) , bl lac sources ( maraschi & tavecchio 2003 ) , x - ray bright optically normal galaxies ( xbongs ; yuan & narayan 2004 ) , and even some seyferts ( chiang & blaes 2003 ) . all of these sources are relatively low - luminosity agn , where an adaf is likely to be present ( figs . 5 , 6 ) . as discussed in 3.6 , a feature of the adaf model is the presence of outflows and jets . this implies that adaf systems should generally be radio - loud . this is certainly the case for the fri and bl lac sources mentioned above . ho ( 2002 ; see also nagar et al . 2000 ; falcke et al . 2000 ) presents in fig . 5b of his paper a very interesting plot of radio loudness , defined as the ratio of the flux at 6 cm to the flux in the optical b band versus eddington - scaled luminosity , for a sample of galactic nuclei . virtually every source in the plot with an eddington - ratio below about 0.01 is radio loud , which is perfectly consistent with our expectation that all such sources should have adafs . moreover , the degree of radio loudness increases with decreasing eddington ratio , again consistent with our expectation that the adaf should become more and more dominant with decreasing @xmath64 . the majority of sources in ho s plot that have @xmath119 are radio - quiet , as we would expect from fig . 5 if these systems have cool disks . however , a small minority of these bright sources do show powerful jets ( they are generally frii sources ) . the exact nature of the accretion in these sources is unclear . a more complete plot , with updated data , can be found in sikora , stawarz & lasota ( 2007 ) . the jet and the extended wind from an adaf carry with them substantial kinetic energy . this energy will be dumped into the external medium and will have important consequences . in the context of galaxy formation , there has been discussion recently of the so - called `` radio mode '' of accretion ( croton et al . 2006 ) in which outflowing energy from an accreting smbh in the galactic nucleus heats up the surrounding medium . this kind of agn feedback can lead to various effects such as shutting off accretion and/or star formation ( di matteo et al . 2005 ; hopkins et al . 2006a ) . it is worth noting that this radio mode is nothing other than the adaf mode of accretion reviewed in this article . investigators of agn feedback may find it profitable to study the considerable work that has been done on adafs over the last fifteen years . the first bh , cygnus x-1 , was identified and established in 1972 via a measurement of its mass , which was shown to be too large for a neutron star ( ns ) . the surest evidence for the existence of bhs continues to be through dynamical mass measurements . we now know of 20 additional compact binary x - ray sources ( mcclintock & remillard 2006 ; orosz et al . 2007 ) with primaries that are too massive to be a ns or any stable assembly of cold degenerate matter , assuming that gr is valid . similarly , dynamical data have established the existence of supermassive bhs , most notably in the nucleus of our milky way galacy ( schdel et al . 2002 ; ghez et al . 2005a ) and in ngc 4258 ( miyoshi et al . 1995 ) . are these compact objects genuine bhs pockets of fully collapsed matter that are walled off from sight by self gravity and that , like a shadow , reveal no detail or are they exotic objects that have no event horizons but manage to masquerade as bhs ? most astrophysicists believe that they are genuine bhs . there are several reasons for this confidence , the most important being that bhs are an almost inevitable prediction of gr . however , this argument is circular because it presumes that gr is the correct theory of gravity . furthermore , it ignores the many exotic alternatives to bhs that have been suggested . thus , the collapsed objects that we refer to throughout as `` black holes '' are strictly speaking dynamical bh candidates . the current evidence for bhs is not decisive , nor can dynamical measurements be expected to make it so . we now consider some approaches aimed at establishing that these dynamical bhs are genuine . the defining property of a bh is its event horizon . demonstrating the existence of this immaterial surface would be the certain way to prove the reality of bhs . unfortunately , unlike any ordinary astronomical body such as a planet or a star of any kind , it is quite impossible to detect radiation from the event horizon s surface of infinite redshift . ( hawking radiation is negligibly weak for massive astrophysical bhs . ) nevertheless , despite the complete absence of any emitted radiation , it is possible to marshal strong _ circumstantial evidence _ for the reality of the event horizon . the fruitful approaches described below are based on comparing x - ray binary systems that contain bh primaries with very similar systems that contain ns primaries . such investigations are motivated by the simple fact that the termination of an accretion flow at the hard surface of a ns has observational consequences . the earliest and strongest evidence for the event horizon is based on the faintness in quiescence of bh transient systems relative to comparable ns systems . in 4.1 , we discuss the physical arguments that underpin this evidence , and in 4.2 we present the comparative luminosity data for bhs and nss . alternative explanations for the lower luminosities of the bh systems , which do not involve the event horizon , are considered in 4.3 . in 4.4 , we present three independent and additional arguments for the existence of the event horizon , which are again rooted in comparing bh and ns x - ray binaries . in 4.5 , we discuss the extreme faintness and properties of sgr a * and the evidence that this supermassive bh has an event horizon , and in 4.6 we argue that even very exotic objects with enormously strong surface gravity ( e.g. , gravastars ) can not escape our arguments for the event horizon . a test particle in a circular orbit at radius @xmath28 around a mass @xmath63 has , in the newtonian limit , kinetic energy per unit mass equal to @xmath120 , and potential energy equal to @xmath121 . in the context of a gaseous accretion disk , this means that a gas blob at radius @xmath28 retains @xmath122 of its potential energy as kinetic energy . the remaining 50% was transformed into thermal energy during the viscous accretion of the blob to its current radius ( frank et al . 2002 ) . we now turn to consider accretion on to the material surface of a ns . if the mass @xmath63 at the center of an accretion disk has a surface , then the kinetic energy in the accreting gas will be converted to thermal energy in a viscous boundary layer and radiated ( frank et al . in addition , the residual thermal energy that the gas possesses when it reaches the inner edge of the disk will also be radiated from the surface . if accretion occurs via an adaf , the luminosity from the accretion disk @xmath123 and that from the stellar surface @xmath124 will satisfy @xmath125 where @xmath126 is the radius of the stellar surface . the accretion luminosity is small because the flow is radiatively inefficient . therefore , essentially all the potential energy of the accreting gas remains in the gas in the form of thermal and kinetic energy . if the central object has a surface , e.g. , it is a ns , the total luminosity we observe will be equal to @xmath124 . however , if the object has an event horizon , i.e. , it is a bh , there will be no radiation from a stellar surface and the luminosity we observe will only be equal to @xmath123 . thus , we expect @xmath127 , regardless of whether the accretion flow is radiatively efficient or not . in the case of a bh , however , the radiative efficiency @xmath91 of the accretion flow is of paramount importance , and the observed luminosity becomes @xmath128 when the accretion rate is highly sub - eddington.,width=316 ] figure 7 shows schematically the luminosity difference we predict between a ns and a bh . this plot is based on the accretion efficiency estimate shown in fig . 4 . especially at low mass accretion rates , say @xmath129 ( deep quiescent state ) , we expect a huge luminosity difference between nss and bhs . efforts to test this prediction are described in 4.2 . if the accretion disk is radiatively efficient , then the gas has little thermal energy when it reaches the inner edge of the disk . in this case , we expect . moreover , if the central mass spins too close to the `` break - up '' limit , the extra luminosity due to the surface can actually exceed @xmath130 by a large factor ( popham & narayan 1991 ) . ] @xmath131 the above result is based on a newtonian analysis and is okay so long as the central object has a radius @xmath132 , the radius of the isco . for more compact objects , we must allow for the fact that , inside the isco , the accreting gas no longer spirals in by viscosity but free - falls in the gravitational potential of the central mass . now we have @xmath133 the accretion luminosity is limited by the binding energy of the gas at the isco , which gives a radiative efficiency @xmath89 . on the other hand , the total energy budget is @xmath134 , which means that a larger fraction of the luminosity is released at the surface . ( the free - falling gas inside the isco , crashes on the surface and releases its energy in a shock . ) in the limit when @xmath126 is arbitrarily close to @xmath103 ( cf . , the discussion of gravastars in 4.6 ) , the total luminosity is equal to @xmath135 , i.e. , the entire rest mass energy of the accreting gas is converted to radiation ( 100% radiative efficiency ) . note that all luminosities discussed here refer to measurements by an observer at infinity . combining the results in equations ( [ adafsurf ] ) , ( [ surf2 ] ) and ( [ surf3 ] ) , we see that , regardless of the radius of the accretor and whether the accretion flow is radiatively efficient or inefficient , we expect @xmath136 although this relation is weaker than ( [ adafsurf ] ) , it can still be used in favorable cases to test for the presence of an event horizon ( 4.5 ) . bh and ns transient binary systems are very similar in many respects , and it is reasonable to expect that their mass accretion rates and luminosities would be comparable under similar conditions . there is , however , one important qualitative difference between the two kinds of object nss have surfaces whereas bhs do not . often , this difference is not important . however , as discussed in 4.1 ( see eq . [ adaflum ] ) , when accretion occurs via an adaf , a ns binary should be much more luminous than a bh binary ( ny95b ) . the difference will be especially large in the quiescent state , when the accretion flow is radiatively extremely inefficient ( fig . 7 ) . narayan , garcia & mcclintock ( 1997b ) and garcia , mcclintock & narayan ( 1998 ) collected available x - ray data on quiescent ns and bh transients and showed that bh systems are consistently fainter than ns systems . this was the first indication that the objects that astronomers call `` black holes '' are indeed genuine bhs with event horizons . fainter than ns systems with similar orbital periods . , width=316 ] lasota & hameury ( 1998 ) made the important point that it is necessary to compare quiescent ns and bh transients at similar mass accretion rates , and the surest way to achieve this is to plot luminosities as a function of the binary orbital period @xmath137 ( the relevant arguments are outlined in menou et al . 1999b ; lasota 2000 , 2008 ) . since 1999 , this has been the standard way of plotting the data ( menou et al . 1999a ; garcia et al . 2001 ; narayan , garcia & mcclintock 2002a ; hameury et al . 2003 ; mcclintock et al . figure 8 shows the current status of the comparison , with the luminosities of quiescent bh and ns transients plotted in ( most - appropriate ) eddington units . 9 shows the same data without the luminosities being scaled . it is clear that , for comparable orbital periods , the bh systems are 2 to 3 orders of magnitude fainter than their ns cousins . as discussed in 4.2 , a radiatively inefficient adaf provides a natural explanation for the large luminosity deficit of the bh systems . in this model , the gas approaches the center with a large amount of thermal energy . a bh is dim because the bulk of this thermal energy is trapped in the advective flow , passes through the event horizon , and is lost from sight . on the other hand , a ns is bright because the thermal energy is radiated from its surface . a number of alternative models have been put forward in an attempt to rationalize the luminosity differences between quiescent ns and bh systems . notice below that some of these models generate x - ray emission without invoking accretion at all . we first consider a recent challenge centered on the discovery of the extremely low - luminosity ns transient 1h 1905 + 00 . next , we discuss the possibility that the bulk of the accretion power is channeled into a steady jet rather than into x - ray emission . we then consider several diverse models that are discussed in further detail in narayan et al . ( 2002a ) . _ the case of 1h 1905 + 00 : _ jonker et al . ( 2007 ) claim that the extremely low x - ray luminosity of the ns transient and type i burst source 1h 1905 + 00 ( hereafter , h1905 ) undermines the evidence summarized in 4.2 for the existence of the event horizon . the luminosity of this ns ( @xmath138 ; @xmath139 kpc ) , is the same as the luminosity of a062000 and several other short - period ( @xmath140 hr ) bh systems . based on this result , jonker et al . assert that the evidence for event horizons is `` unproven . '' however , they ignore a key point of our argument : namely , as discussed above and illustrated in figures 8 and 9 , the case for event horizons depends critically on comparing bh and ns systems with _ similar orbital periods_. we first note that the orbital period of h1905 is unknown . more importantly , this unknown period is believed to be very short so short that there are no bh systems with comparable periods . this conclusion is based on deep optical imaging data and the lack of a counterpart . jonker et al . ( 2007 ) conclude that the secondary `` can only be a brown or a white dwarf , '' and that the system is probably an ultracompact binary . hence the orbital period is likely to be tens of minutes , far less than the shortest bh binary period of 4.1 hr . thus , there is no bh system comparable to h1905 . and there is no way to usefully predict the mass accretion rate , as jonker et al . conclude in the paper s final sentence . the rough trend of luminosity with period indicated in figure 9 for nss might imply a very low luminosity for h1905 , as observed , but such an extrapolation is unwarranted . see lasota ( 2007 , 2008 ) for a more detailed discussion of this and other issues . jonker et al . ( 2007 ) additionally argue that an unknown amount of mass transfered from the secondary might be lost from the system in winds or jets . their one example of a wind , viz . , an extraordinary mass - loss episode observed in gro j1655 - 40 ( miller et al . 2006c ) when the source was in outburst and had a luminosity @xmath141 times the quiescent level , is quite inappropriate . also , their mention of possible mass expulsion for quiescent nss via the propeller mechanism ( lasota & hameury 1998 ; menou et al . 1999a ) , if effective , would reduce the luminosities of nss and would only strengthen our argument . _ jet outflows : _ fender et al . ( 2003 ) argue that transient bhs at low accretion rates ( @xmath142 ) `` should enter ` jet - dominated ' states , '' in which the majority of the accretion power drives a radiatively - inefficient jet . as we have seen in 3.6 , there is good evidence for this . for instance , the presence of a radio jet has been reasonably well established in quiescence for a062000 ( gallo et al . 2006 ) , the closest and one of the least luminous of the bhs in question . fender et al . ( 2003 ) appeal to the empirical result that bhs are @xmath143 times as ` radio loud ' as nss at similar accretion rates to deduce that quiescent bhs should be @xmath144 times _ less _ luminous in x - rays than quiescent nss . their argument is a little convoluted since , if the radio and x - ray emission in quiescence come from the jet ( see 3.6 ) , one would think the bhs would be @xmath143 times _ more _ luminous , not 100 times less luminous . therefore , naively , the jet argument only makes the discrepancy in figs . 8 and 9 more severe . fender et al . ( 2003 ) get around this difficulty by postulating , in addition to different jet efficiencies , also different origins for the x - rays seen in quiescent bhs and nss . it is unsatisfying that the reason for the difference in jet activity between bhs and nss `` remains unclear . '' at bottom , the authors are suggesting that in quiescence the bhs are in the ` jet - dominated ' regime and the nss are , `` if not jet - dominated , close to the transition to this regime . '' a key uncertainty in this suggestion is whether the scaling relation that has been established for bhs between radio luminosity and jet luminosity ( @xmath145 ) is also valid for nss , as is assumed . in addition , occam s razor suggests that the relative faintness of bhs is unlikely to be attributable to their stronger jets : in 4.5 , we show that sagittarius a@xmath3 , which is radiating in quiescence at the same level of eddington - scaled luminosity as a0620 - 00 , does not possess an energetically - important jet . _ coronal emission from bh secondaries : _ the quiescent x - ray luminosity of the bh systems has been attributed to a rotationally - enhanced stellar corona in the secondary star ( bildsten and rutledge 2000 , but see lasota 2000 ) . however , as discussed in detail by narayan et al . ( 2002a ) , the luminosities of three of the systems plotted in figures 8 and 9 exceed by a factor of 660 the maximum predicted luminosity of the coronal model ; likewise the three systems with adequate data quality show x - ray spectra that are harder / hotter than that typically seen in stellar coronae . finally , if stellar coronae do contribute at some level , then the accretion luminosities of the bhs are even lower than our estimates , which would further strengthen the evidence for event horizons . _ incandescent neutron stars : _ the quiescent luminosity of ns transients has been attributed to heating of the star s crust during outburst followed by cooling in quiescence . this model likewise has problems . ( for details and references , see narayan et al . briefly , the rapid variability of the prototypical ns transient system cen x-4 is not expected in a cooling model and implies that no more than about a third of the quiescent luminosity is due to crustal cooling . furthermore , power - law tails that carry about half the total luminosity are observed for many ns transients ( e.g. , cen x-4 and aql x-1 ) . these are unlikely to arise from a cooling ns surface , but could easily be produced by accretion . finally , strong evidence for continued accretion in quiescence comes from optical variability , which is widely observed for these quiescent systems . _ pulsar wind / shock emission : _ in another accretionless model , the ns transient switches to a radio pulsar - like mode in quiescence ( campana & stella 2000 ) . the x - ray luminosity is expected to be of order the `` pulsar shock '' luminosity @xmath146 , which is close to the observed level . both the observed power - law and thermal components of emission ( see the previous paragraph ) are naturally explained by this model : the former component is produced by the shock and the latter is radiated from the ns surface . the lack of any significant periodicity in the quiescent emission ( in any electromagnetic band ) could be a problem for this model . millisecond x - ray pulsars such as sax j1808.4 - 3658 do show periodicities in outburst , but that emission is clearly the result of accretion , not a pulsar wind / shock . _ optical / uv luminosity : _ campana & stella ( 2000 ) note that the comparison shown in figs . 8 and 9 assumes the x - ray luminosity is an accurate measure of the accretion rate near the bh or ns . they argue that the optical and uv luminosity also originates near the central object and should therefore be included in the comparison . the non - stellar optical / uv luminosity is much greater than the x - ray luminosity . when it is included , the difference between the bh and ns systems largely disappears . however , as detailed in narayan et al . ( 2002a ) , there are some problems with this argument . the level of optical / uv emission generated in the inner region of adafs is strongly suppressed by winds and convection , as evidenced by a comparison of the luminosities of nss and white dwarfs ( loeb , narayan & raymond 2001 ) . it thus appears unlikely that the optical and uv emission is generated in the hot gas close to the accretor ( see shahbaz et al . 2005 ) . in the case of cvs , it has been established that a large fraction of the optical / uv emission comes from the `` hot spot , '' and it is quite reasonable to expect that this is true for bh and ns systems as well ( there is some evidence to support this ; narayan et al . further observations are needed to determine the origin of optical / uv emission in these systems . in addition to the arguments discussed above , we briefly summarize three additional lines of evidence for the existence of event horizons . all three are based on comparisons between bh and ns x - ray binaries . as in the examples above , the first argument considers quiescent systems , whereas the latter two consider active states of accretion . in quiescence , a soft component of thermal emission is very commonly observed from the surfaces of accreting nss , which is widely attributed to either deep crustal heating ( brown , bildsten & rutledge 1998 ) or to accretion . no such component is present in the spectrum of the quiescent bhb xte j1118 + 480 ( hereafter j1118 ) , as one would expect if the compact x - ray source is a bona fide bh that possesses an event horizon ( mcclintock et al . 2004 ) . because of the remarkably low column density to j1118 ( @xmath147 @xmath148 ) the limit on a hypothetical thermal source is very strong ( @xmath149 0.011 kev ) ; it is in fact a factor of @xmath150 lower in flux than the emission predicted by the theory of deep crustal heating , assuming that j1118 has a material surface analogous to that of nss . likewise , there is no evidence that accretion is occurring in quiescence onto the surface of j1118 , which is the mechanism often invoked to explain the far greater thermal luminosities of nss . the simplest explanation for the absence of any thermal emission is that j1118 lacks a material surface and possesses an event horizon . type i x - ray bursts are very common in ns x - ray binaries , but no type i burst has been seen among the bh systems . a model developed by narayan & heyl ( 2002 , 2003 ) , which reproduces the gross observational trends of bursts in ns systems , shows that , if the dynamical bhs have surfaces , they should exhibit instabilities similar to those that lead to type i bursts on nss . remillard et al . ( 2006 ) , following earlier work by tournear et al . ( 2003 ) , measured the rates of type i x - ray bursts from a sample of 37 nonpulsing x - ray transients observed with _ rxte _ during 19962004 . among the ns sources , they found 135 type i bursts in 3.7 ms of pca exposures ( 13 sources ) with a burst rate function consistent with the narayan & heyl model . however , for the bh group ( 18 sources ) , they found no confirmed type i bursts in 6.5 ms of exposure . their upper limit on the incidence of burst activity in these sources is inconsistent with the model predictions at a high statistical significance if the accretors in bhbs have solid surfaces . the results provide strong indirect evidence for bh event horizons , and it would appear that the evidence can be refuted only by invoking rather exotic physics . likewise drawing on the extensive archive of _ rxte _ data , done and gierlinski ( 2003 ) have examined the patterns of x - ray spectral evolution of active bh and ns sources and identified a distinct type of soft spectrum that is occasionally observed only in the bh sources . they attribute this spectrum to thermal emission from the inner accretion disk ( corresponding to the high state discussed earlier ) . they then argue that nss with a similar accretion rate can not exhibit such a simple , low - temperature spectrum because they would have a second component in the emission from the boundary layer where accreting matter impacts the stellar surface . they present a thermal / nonthermal comptonization model for the boundary layer emission which has significant uncertainty , given that so many details of the accretion physics are complex and poorly understood . nevertheless , done & gierlinski ( 2003 ) do appear to have identified a systematic difference in the x - ray spectra of accreting bhs and nss , and it is surely worth pursuing this signature in the effort to amass evidence for the reality of event horizons . the supermassive bh in sgr a@xmath3 has a mass @xmath151 ( schdel et al . 2002 ; ghez et al . 2005a ) but an accretion luminosity of only @xmath152 ; thus the source is highly sub - eddington , @xmath153 . from our earlier discussion , the accretion flow must be in the form of an adaf , i.e. , the radiating gas must be extremely hot and optically thin . this expectation is confirmed by the @xmath154 k brightness temperature of the radio / millimeter emission ( shen et al . 2005 ) and the fact that most of the emission is in this band rather than at frequencies @xmath155 ( x - ray/@xmath34-ray band ) . the adaf model has been successfully applied to sgr a@xmath3 ( narayan et al . 1995 , 1998a ; manmoto et al . 1997 ; mahadevan 1998 ; yuan et al . 2003 ; to mention a few ) . if we assume that the accretion flow is radiatively very inefficient , then it is easy to make a strong case for sgr a@xmath3 not having a surface ( narayan et al . however , as broderick & narayan ( 2006 , 2007 ) showed , we can argue for the presence of an event horizon even without assuming an adaf . if sgr a@xmath3 does not have an event horizon , but has a surface , then any emission from the surface will be blackbody - like . this is because we expect the surface to be optically thick . however , the radio / millimeter emission mentioned above can not be from this surface because of its incredibly high @xmath154 k brightness temperature . thus , the radio / millimeter radiation is from the accretion flow , and its luminosity gives a strict lower bound on @xmath123 . by equation ( [ lsurfacc ] ) then , we expect a luminosity of at least @xmath156 from the surface . the surface radiation would be thermal and blackbody - like and should come out in the infrared ( as easily shown , given the luminosity and the likely area of the surface ) . however , there is no sign of it ! ( assuming the source has a surface ) as a function of the surface radius @xmath28 . each limiting curve is derived from a limit on the quiescent flux of sgr a@xmath3 in an infrared band . the hatched area at the top labeled `` typical riaf range '' corresponds to the mass accretion rate in typical adaf models of sgr a@xmath3 ( e.g. , yuan et al . the horizontal dashed line represents the minimum accretion rate needed to power the bolometric luminosity of sgr a@xmath3 . ( from broderick & narayan 2006),width=316 ] figure 10 shows constraints on the mass accretion rate in sgr a@xmath3 . the horizontal dashed line is the minimum accretion rate @xmath157 in sgr a@xmath3 ; this is the rate needed , even with a radiatively efficient flow ( @xmath89 ) , just to power the observed radio / millimeter radiation . the four solid lines show the maximum mass accretion rate @xmath158 allowed if sgr a@xmath3 has a spherical surface of radius @xmath28 . each curve corresponds to a measured limit on the steady quiescent infrared flux in a particular band ( stolovy et al . 2003 ; clenet et al . 2004 ; ghez et al . 2005b ) and provides an independent upper limit on @xmath158 . for radii less than @xmath159 ( the upper limit on the size of sgr a@xmath3 ; shen et al . 2005 ) , we see that all four bands give upper limits on @xmath158 that are far below the minimum @xmath157 ; in fact , the discrepancy is larger than a factor of 100 in the @xmath160 band . note that the discrepancy would be much larger if we assumed that the accretion flow is radiatively inefficient ( corresponding to the hatched region in fig . the only way to avoid the large discrepancy illustrated in fig . 10 is to give up the assumption that sgr a@xmath3 has a surface . obviously , if the source has an event horizon , then we do not expect any surface radiation , and there is no problem . what if sgr a@xmath3 ejects all the accreting mass via a jet before the gas reaches the surface ? is this a viable explanation for the lack of surface emission ? we can easily rule out this possibility . recall that the source of energy in an accretion system is gravity . at a bare minimum , we know that matter is being accreted at a rate @xmath161 ( shown by the horizontal dashed line in fig . 10 ) in order to produce the observed radiation . all of this mass _ has _ to fall into the potential well in order to release its energy . if sgr a@xmath3 has a jet with a kinetic luminosity @xmath162 , the energy for this must also come from accretion . we will then require an even larger @xmath161 , and hence a larger @xmath163 , since the accreting gas now has to power both the radiation and the jet . the discrepancy with the observed limits on the quiescent infrared flux would then be even larger . the arguments for the event horizon presented in 4.2 , 4.5 are based on newtonian ideas . some authors ( e.g. , abramowicz , kluzniak & lasota 2002 ) have questioned whether the arguments might be substantially modified by strong gravity in the vicinity of the compact accretor . buchdahl ( 1959 ) showed , for a wide class of reasonable equations of state , that the smallest radius allowed for a compact non - rotating object is @xmath164 . an object with this limiting radius has a gravitational redshift from its surface @xmath165 . at such modest redshifts , we do not expect the effects of strong gravity to be particularly large , and so the arguments presented in 4.2 , 4.5 will continue to hold . recently , however , a new class of solutions has been discussed , which goes variously under the name of `` gravastar '' and `` dark energy star '' ( mazur & mottola 2001 ; chapline et al . 2003 ; visser & wiltshire 2004 ; carter 2005 ; lobo 2006 ) . in this model , the radius @xmath28 of an object of mass @xmath63 is allowed to be arbitrarily close to @xmath103 : @xmath166 . the surface redshift can then be arbitrarily large : @xmath167 although the gravastar model is very artificial , it is nevertheless interesting to ask whether such a model , which has no event horizon , can explain the observations described in 4.2 , 4.5 ( abramowicz et al . 2002 ) . broderick & narayan ( 2006 , 2007 ) show that it can not . we briefly summarize the arguments here . a large value of @xmath168 means that any radiation emitted from the surface is redshifted greatly before it reaches the observer at infinity . this looks like an easy way of hiding the surface luminosity . however , a simple energy conservation argument shows otherwise . assuming steady state , in the limit as @xmath169 , the total luminosity observed at infinity , @xmath170 , _ must _ be equal to @xmath135 ( the rate of accretion of rest mass energy ) . moreover , the effective radius of the source as viewed by a distant observer is @xmath171 , and so the surface radiation will be in the x - ray band for accreting bhbs ( 4.2 ) and in the infrared for sgr a@xmath3 ( 4.5 ) , exactly as in the newtonian analysis . could the large gravitational redshift cause a large delay in the signals from the surface of the star , and could this be why we do not see the radiation ? the extra delay due to relativity is easily estimated by considering null geodesics in the schwarzschild metric : @xmath172 since the dependence is only logarithmic , the extra delay is not significant . for instance , even if we take @xmath173 to be comparable to the planck length @xmath174 cm ( the smallest length we can legitimately consider ) , the delay is only @xmath175 ms for a bhb and @xmath176 s for sgr a@xmath3 . we assumed that the radiation from the surface would have a blackbody - like spectrum . is this likely ? actually , it is virtually guaranteed as @xmath177 . when the radius of an object is very close to the schwarzschild radius , most rays emitted from the surface are bent back on to the surface , and only a tiny fraction of rays escapes to infinity . the solid angle corresponding to escape is @xmath178 . in this limit , the surface behaves just like a furnace with a pinhole ( the textbook example of a blackbody ! ) , and so the escaping radiation is certain to have a nearly perfect blackbody spectrum . could the surface luminosity escape in the form of particles rather than radiation ? by the blackbody argument given above , whatever escapes must be in thermodynamic equilibrium at the temperature @xmath179 observed at infinity . for the sources of interest to us , @xmath180 ev ( sgr a@xmath3 ) and kev ( bhbs ) . in thermodynamic equilibrium , the only particles with any significant number density at these temperatures are neutrinos . even if we include all three species of neutrinos , the radiation flux is reduced by a factor of only 8/29 ( broderick & narayan 2007 ) . this is a small correction compared to the large discrepancies @xmath181 that we described in 4.2 , 4.5 . finally , we note that our arguments for the event horizon are based on a steady state assumption . specifically , we assume that the surface luminosity is proportional to the average mass accretion rate on the surface . in the gravastar model , in particular , one can imagine scenarios in which steady state is not reached . however , broderick & narayan ( 2007 ) show that this explanation can be ruled out , at least with current gravastar models . we thus conclude that strong gravity effects are unable to weaken our arguments for the presence of an event horizon in quiescent bhbs and in sgr a@xmath3 . we must look to more conventional astrophysical explanations . the only idea that we find somewhat plausible is that the radiation we observe from quiescent x - ray binaries and sgr a@xmath3 is not produced by accretion at all , but by some other process . this would undercut all our arguments since we assume , as an article of faith , that the radiation we observe is powered by gravity through accretion . given the success of the accretion paradigm in explaining a vast body of observations on a variety of bh systems in many different spectral states , it seems rather extreme to abandon the idea of accretion in just those particular sources where we find evidence for the presence of an event horizon . the aim of this article is two - fold : to give an updated and current account of the adaf model , with an emphasis on applications to accreting bhs ( 3 ) , and to review the considerable body of evidence presently available for the presence of event horizons in astrophysical bhs ( 4 ) . observations of bh binaries ( bhbs ) and active galactic nuclei ( agn ) indicate that , at luminosities of a few percent or less of eddington , accretion occurs via a very different mode than the standard thin accretion disk . at these luminosities , sources have hard x - ray spectra , quite unlike the soft blackbody - like spectra seen in more luminous sources . prior to the establishment of the adaf model , observations in the hard state were modeled in an ad hoc way using empirical thermal comptonization models . in the mid-1990s , the adaf solution was shown to have precisely the densities , temperatures , radiative ( in)efficiencies and stability required to provide a physical description of the observations . moreover , since the adaf model is essentially mass - independent , observations of bhbs and agn are both explained with more - or - less the same adaf model . the model gives satisfactory results for a wide range of luminosity , from about @xmath182 down to @xmath183 of eddington ( below which there are no observations ) . the defining characteristic of an adaf is that the gas is not radiatively efficient . a significant fraction of the energy released by viscous dissipation is retained in the accreting gas and advected with the flow . the trapped thermal energy causes the accreting gas to be weakly bound to the bh . based on this property , it was predicted already in the earliest adaf papers that sources in the adaf state would have strong winds and jets . nonthermal radio emission has been detected in recent years from bhbs in the hard state and quiescent state , as well as from all agn with luminosities below @xmath184 of eddington . these sources have spectra consistent with the adaf model , thus confirming a strong connection between adafs and jets / outflows . a currently active topic of research is the role of agn feedback on galaxy formation . research in this area can now draw on the extensive literature on adafs . of special interest in this work are those accreting objects for which there exist estimates or constraints on the two system specific parameters : bh mass @xmath63 and mass accretion rate @xmath185 . specifically , we are referring here to saggitarius a@xmath3 and a selected sample of quiescent bh and ns x - ray binaries . we have featured these two examples because they provide strong observational evidence for the existence of the event horizon . in the case of the binaries , one can make a plausible argument that , for comparable orbital periods , the mass accretion rates and luminosities of both types of systems should be comparable . remarkably , however , the bh systems are observed to be dimmer by a factor of @xmath186 . we review a wide range of unsatisfactory attempts to explain this large luminosity difference . in contrast , the adaf model provides an entirely straightforward explanation for the faintness of quiescent bhs : a ns must radiate the trapped thermal radiation that is advected with the accretion flow and rains down on its surface , whereas the bh hides the energy behind its event horizon . additional evidence for the event horizon is provided in further comparisons of nss and bhs in outburst : the bhs lack type i x - ray bursts , and they lack a distinctive boundary - layer component of emission . both of these properties are expected if the bhs possess event horizons , but hard to explain otherwise . the supermassive bh in sgr a@xmath3 is extraordinarily quiescent . the observations strongly support the existence of an adaf , and it is consequently easy to provide a compelling argument for an event horizon . however , for sgr a@xmath3 , one can make an even stronger argument for the lack of a hypothetical material surface without at all invoking an adaf . the large radio / millimeter accretion luminosity of sgr a@xmath3 , which has a brightness temperature @xmath187 k , obviously can not be emitted by an optically - thick surface , and so it must be radiated from the accretion flow . this establishes a hard lower limit on the accretion luminosity and @xmath64 . meanwhile , high angular resolution radio observations constrain the radius of the surface to be @xmath188 . this constraint and the lower limit on @xmath185 predict a near - ir flux of thermal , blackbody - like surface emission that is far above the observed limits . the obvious explanation is that there is no material surface , only an event horizon . as an added bonus , it is energetically impossible to explain away the lack of surface emission by appealing to mass loss in a jet . the case for event horizons in both sgr a@xmath3 and in the stellar - mass bhs is robust against appeals to strong gravity or the leading models of exotic stars . first , gr effects will be mild for nearly all conventional models of degenerate stars , whose radii are restricted to be @xmath189 and whose surface redshifts are therefore @xmath190 . secondly , even for an exotic star ( a gravastar ) with an extraordinary surface redshift of a million or more , the full accretion luminosity from its surface will be delivered as x - rays ( in binaries ) or infrared ( in sgr a@xmath3 ) to a distant observer . thus , extreme redshifts have practically no effect on the argument for the event horizon . to play on carl sagan s famous comment , there will always be an absence of _ direct _ evidence for the event horizon , but this surely can not be taken as evidence of its absence in nature . on the contrary , and with a pun in mind , many indicators show that the event horizon is an inescapable reality .

as the luminosity of an accreting black hole drops to a few percent of eddington , the spectrum switches from the familiar soft state to a hard state that is well - described by a distended and tenuous advection - dominated accretion flow ( adaf ) . an adaf is a poor radiator , and the ion temperature can approach @xmath0 k near the center , although the electrons are cooler , with their temperature typically capped at @xmath1 k. the foundational papers predicted that the large thermal energy in an adaf would drive strong winds and jets , as later observed and also confirmed in computer simulations . of chief interest , however , is the accreting gas that races inward . it carries the bulk of the accretion energy as stored thermal energy , which vanishes without a trace as the gas passes through the hole s event horizon . one thus expects black holes in the adaf regime to be unusually faint . indeed , this is confirmed by a comparison of accreting stellar - mass black holes and neutron stars , which reside in very similar transient x - ray binary systems . the black holes are on average observed to be fainter by a factor of @xmath2 . the natural explanation is that a neutron star must radiate the advected thermal energy from its surface , whereas a black hole can hide the energy behind its event horizon . the case for an event horizon in sagittarius a@xmath3 , which is immune to caveats on jet outflows and is furthermore independent of the adaf model , is especially compelling . these two lines of evidence for event horizons are impervious to counterarguments that invoke strong gravity or exotic stars . ,

You are an expert at summarizing long articles. Proceed to summarize the following text: @xcite proposed that neutron star mergers will be accompanied by macronovae ( kilonovae ) , which are optical infrared transients powered by radioactive decay of the merger s debris . these macronovae are among the most promising electromagnetic counterparts to gravitational - wave merger events ( e.g. @xcite ) . recently , macronova candidates have been discovered in the afterglows of several short gamma - ray bursts @xcite . @xcite re - analyzed the afterglow light curves of historical nearby short gamma - ray bursts and suggested that macronovae are ubiquitous in short grbs afterglows . the radioactive heat generated by @xmath0-process nuclei play an essential role in powering macronovae . due to strong adiabatic cooling the ejecta s initial internal energy is practically negligible at the time that the ejecta become optically thin , i.e. when @xmath5 , where @xmath6 is the optical depth and @xmath7 is the velocity of the ejecta . as we discuss later , with typical parameters the peak emission time is around a few days and hence this is the important timescale to focus on . detailed computations using nuclear database and numerical simulations have been widely used to obtain the radioactive heating rates @xcite . another approach to calculate the heating rate is to consider the @xmath0-process material as a statistical assembly of radioactive nuclei . incorporating fermi s theory of beta decay with such an approach provides a clear physical understanding of the nuclear heating rate ( see , e.g. , @xcite for a discussion on the energy generation by fission products ) . we follow this approach and use the fermi theory to estimate the heating rate in neutron star mergers ejecta . * see also @xcite ) considered this approach to estimate the radioactive luminosity of supernovae . but they used only the relativistic regime of fermi s theory , which is not relevant on the macronova peak timescale as we discuss later . furthermore , they assumed that each element decays to a stable one rather than following a decay chain , which we consider in this paper . in this paper , we analytically derive the nuclear heating rate of macronovae based on fermi s theory . in [ sec : summary ] , we begin with a brief summary of the basic concepts and the outcome of this work . in [ sec : fermi ] , we introduce the key ingredients of the theory needed to calculate the heating rate . we derive the heating rate of the beta decay chains of allowed transitions in [ sec : heat ] . we discuss , in [ sec : discussion ] , the role of forbidden transitions and other effects that we ignore and we estimate their possible effect . we summarize the results and discuss the implication to macronova studies in [ sec : conclusion ] . radioactive nuclei that are far from the stability valley are produced in @xmath0-process nucleosynthesis . these nuclei undergo beta decay without changing their mass number . a series of beta decays in each mass number is considered as a decay chain . because the mean lives of radioactive nuclides typically become longer when approaching the stability valley , the nuclei in a decay chain at given time @xmath8 stay at some specific nuclide with a mean life @xmath9 . this means that the number of decaying nuclei in a logarithmic time interval is constant , i.e , the decay rate is @xmath10 , where @xmath11 is the total number of nuclei in the chain . then the beta decay heating rate per nucleus is given by @xmath12 , where @xmath13 is the disintegration energy of the beta decay as a function of the mean life . as we will see later , two important concepts in beta decay theory enable us to determine @xmath13 . first , there are four physical constants in the problem , the fermi s constant @xmath14 , the electron mass @xmath15 , the speed of light @xmath16 , and the planck constant @xmath17 . the fundamental timescale of beta decay @xmath18s can be obtained from these physical constants . second , there is a well known relation between the disintegration energy and mean life as @xmath19 . therefore , the heating rate per nucleus can be roughly estimated as @xmath20 this gives a correct order of magnitude and a reasonable estimate of the time dependence of the beta decay heating rate of @xmath0-process material . in the following we refine these ideas . the nature of beta decay was successfully described by @xcite . here we briefly describe key ingredients of fermi s theory needed for obtaining the macronova heating rate . in a beta decay one of the neutrons in a nucleus disintegrated to a proton , an electron , and an anti - neutrino . using fermi s golden rule , the beta - disintegration probability of a beta - unstable nucleus per unit time in a unit momentum interval of the electron is written as @xmath21 where @xmath22 and @xmath23 are momentum and kinetic energy of the electron , @xmath24 is the matrix element of the interaction hamiltonian responsible for the beta disintegration , and @xmath25 is the number density of final states of the light particles between @xmath23 and @xmath26 . the number of final states is assumed to be proportional to the volume of the accessible phase space of the light particles : @xmath27 where @xmath28 is the total disintegration energy , @xmath29 is momentum of the neutrino , and energy conservation @xmath30 has been used . we use the fact that the neutrino mass is sufficiently small compared to @xmath31 and assume that there is no angular correlation between the electron and neutrino . here we imagine that the whole system is enclosed in a large box with a volume @xmath32 . hereafter we change the notation as @xmath33 and @xmath34 . in fermi s theory , the four particles interact at a single point with a coupling constant @xmath14 so that the matrix element is written as @xmath35 where @xmath36 is the wave function of each particle involved in the beta disintegration , @xmath37 and @xmath38 are operators acting on the light particle s spin , nucleon s spin and isospin ( see , e.g. , @xcite for a discussions on beta interaction ) . the wave function of the light particles can be evaluated at @xmath39 because their de broglie wavelengths are much larger than the nuclear size . when the light particles do not carry off orbital angular momentum with respect to the central nucleus , the wave function of each light particle at @xmath40 is just a normalization factor of @xmath41 with a coulomb correction for the electron s wave function . thus the square of the matrix element can be written as @xmath42 is the coulomb correction factor , @xmath43 is the proton number of the daughter nucleus , and @xmath44 is the nuclear matrix element . the transitions described here are _ allowed _ transition . more specifically , allowed transitions are transitions which satisfy both conditions that the light particles do nt carry off orbital angular momentum and the parity of the nucleus does not change via its disintegration . otherwise the transition is a _ forbidden _ transition . because the population of allowed transitions is larger and because of their simplicity , we focus on allowed transitions in this and the next sections . we will discuss the role of forbidden transitions in 4 . integrating eq . ( [ fermi ] ) over the accessible phase space , the mean - life of a beta - unstable nuclide with the disintegration energy of @xmath31 is obtained as @xmath45 where the variables in the integral are in units of @xmath46 and @xmath16 and @xmath47 is the fundamental timescale of beta decay : @xmath48 note that , although this fundamental timescale is a characteristic timescale of allowed beta decay , the lifetime of beta unstable nuclides spreads over many orders of magnitude because of the phase space factor of eq . ( [ mean ] ) . the coulomb correction factor in the matrix element is obtained by evaluating the electron s wave function at the nuclear radius @xmath49 @xcite : @xmath50 } ( 2p\rho)^{2s-2 } e^{\pi\eta } \left|(s-1+i\eta ) ! \right|^2 , \end{aligned}\ ] ] where @xmath51 , @xmath7 is the velocity of the electron , @xmath52 , @xmath53 , @xmath54 is the electron charge , and @xmath55 is the fine - structure constant . for @xmath56 , the coulomb correction factor slowly increases with @xmath57 as @xmath58 . a simple form of the coulomb correction factor is obtained in the non - relativistic limit of eq . ( [ cc ] ) , @xmath59 and @xmath60 : @xmath61 the coulomb correction factor is unity for @xmath62 and approaches to @xmath63 for @xmath64 . this enhances the transition probability at lower energies . at these energies the electron is pulled by the nucleus due to the coulomb force and the amplitude of the electron s wave function is larger near the nucleus . as a result , the lifetime of beta unstable nuclei becomes shorter than the one estimated without the coulomb correction and the dependence of the lifetime on @xmath31 is weakened . note that one can also obtain an identical form to eq . ( [ cn ] ) by solving the schr@xmath65dinger equation and evaluating the electron s wave function at @xmath40 . as the integral in eq . ( [ mean ] ) is easily calculated for given @xmath28 and @xmath43 , comparative half - lives @xmath66 are often used for comparison with the experimental data : @xmath67 although @xmath68 of each beta transition can not be calculated within fermi s theory , @xmath69 can be determined from the measurements of the lifetime and the electron s spectrum . it is sufficient for our purpose to know the statistical distribution of this quantity . for allowed transitions , the distribution of @xmath66 is known to have a peak around @xmath70s corresponding to @xmath71 ( e.g. @xcite ) , which we take as a reference value in this paper.h , the comparative half - lives are @xmath72 s corresponding to @xmath73 . such transitions are called as _ superallowed _ transitions . these transitions are , however , absent in @xmath0-process material . ] one can show that @xmath74 attains simple forms in the following three regimes : @xmath75 where @xmath76 . the non - relativistic regime exists only for @xmath77 , and thus , there is no such a regime in @xmath0-process material . in previous a work @xcite applied only the relativistic regime @xmath19 . however , as we will see later , the mean - lives of the nuclei are rather proportional to @xmath78 or @xmath79 on the relevant timescale of macronovae , i.e. , a few days . , where @xmath80 is euler number.,width=302 ] in the context of macronovae , we are interested in the relation between the lifetime and the mean electron s energy since the neutrinos do nt contribute to the heat deposition in the merger ejecta . the fraction of energy of the electrons to the total energy is : @xmath81 in the three regimes discussed earlier @xmath82 satisfies : @xmath83 where we assumed @xmath84 . neutron - rich nuclei produced via the @xmath0-process undergo beta decay towards the beta - stable valley without changing their mass number . a series of beta decays of nuclei in each mass number can be considered as a decay chain . here we consider ideal - chains of radioactive nuclei with a series of mean lives ( @xmath85 ) , in which each chain conserves the total number of nuclei throughout the decay process and sufficiently many chains exist . within this approximation , the number of decaying nuclei in a logarithmic time interval is constant and the beta decays at a given time @xmath8 are dominated by nuclides with mean - lives of @xmath9 ( see fig . [ fig : dn ] ) . this is , of course , valid for @xmath86 , where @xmath87 is the mean life of the first nuclide in a decay chain . the heating rate per unit mass is then @xmath88 where @xmath89 is the mean mass number of the @xmath0-process material , and @xmath90 is the atomic mass unit . note that @xmath80 is the euler number , which arises from the fact that the decay rate of each nuclide is proportional to @xmath91 . one can obtain @xmath92 by using eq . ( [ mean ] ) and ( 11 ) . in the relativistic and non - relativistic coulomb regimes , we can derive simple explicit forms of eq . ( [ heat ] ) . as the lifetime of beta - unstable nuclides monotonically increases with decreasing @xmath31 , the relativistic regime is valid at early times and the non - relativistic coulomb regime is valid at late times . more specifically , the relativistic regime is valid until @xmath93 and the non - relativistic coulomb regime is valid after @xmath94 . using eqs . ( [ f ] ) and ( [ e ] ) , we obtain the heating rate in these regimes : @xmath95 where @xmath96 is time in units of a day , @xmath97 is the mean mass number normalized by @xmath98 , and @xmath99 is the mean proton number normalized by @xmath100 . note that the overall magnitude of the heating rate is determined by the mean values of the nuclear quantities , @xmath101 , @xmath43 , and @xmath44 . these values should be constant within an order of magnitude , and thus , the magnitude of the heating rate does not depend significantly on the details of the abundance pattern of the @xmath0-process nuclei . furthermore , we emphasize that the formula of eq . ( [ heat2 ] ) is independent of the distribution of the nuclear decay energy . figure [ fig : heat ] depicts the heating rate obtained from eq . ( [ heat ] ) and the one derived using a nuclear database ( @xcite ; see also similar heating rates in @xcite ) . we find that the heating rate based on the simple analytic formula reproduces the one based on the database remarkably well . in order to see more details , the right panel of fig . [ fig : heat ] shows the heating rates normalized to the values obtained for the relativistic regime ( eq . [ heat2 ] ) . the normalized analytic heating rate ( blue solid line ) is flat at early times and it approaches the non - relativistic coulomb regime ( magenta dotted line ) at late times . it is worthy noting that the formula with the non - relativistic coulomb limit reproduces the full heating rate after @xmath102s , even though it should be valid only after @xmath103s . this can be understood as follows . the mean life is approximately proportional to @xmath104 between the relativistic and the non - relativistic regimes , and thus , the energy generation rate evolves as @xmath105 . in addition , in this stage , @xmath106 changes from @xmath107 to @xmath108 , which approximately corresponds to @xmath109 . as a result , the electron heating rate is @xmath110 , which is quite similar to the one in the non - relativistic coulomb regime . note that @xcite and @xcite assume that a nucleus that undergoes a radioactive decay reaches the valley of stability in a single step . in this case the total number of radioactive nuclei decreases with time . this assumption is valid if the radioactive nuclei are distributed just next to the stable nuclei , i.e. , at late times . under such an assumption , the resulting heating rate declines more steeply as @xmath111 in the relativistic regime . as we will discuss in the next section , the actual situation is in between these two assumptions . the analytic formula derived in the previous section reproduces remarkably well the result based on the nuclear database . however , there are two important effects that have not been taken into account . here we discuss the role of these effects . _ higher orbital - angular momentum transitions ( unique forbidden ) _ : the light particles wave function in the matrix element eq . ( [ matrix ] ) can be expanded in a series of spherical harmonics , of which the @xmath112th term is proportional to @xmath113 , where @xmath114 is the total momentum of the light particles . the @xmath112th transition corresponds to the transition in which the light particles carry off orbital angular momentum of @xmath115 . this expansion converges rapidly on the energy scale of beta decay on the length scale of nucleus @xmath116 . as a result , the @xmath112th transition probability is suppressed by a factor of @xmath117 . for first unique forbidden transitions , an additional shape factor @xmath118 should be multiplied in the electron spectrum of eq . ( [ mean ] ) . this shape factor results in @xmath119 in the non - relativistic coulomb regime , which can be seen in fig . [ fig : half ] ( a blue dotted line ) . even though the number of beta unstable nuclides that disintegrate mainly via unique forbidden transitions is small , they may play a role by increasing the heating rate after a few hours . _ relativistic transitions ( parity forbidden ) _ : some interactions mix the large and small components of dirac spinor of the nucleon in the matrix element eq . ( [ matrix ] ) . a transition due to such an interaction is a _ parity forbidden _ transition as it changes the nucleus parity without removing the orbital angular momentum ( see magenta crosses in fig . [ fig : half ] ) . the corresponding amplitude is suppressed by a factor of @xmath120 or @xmath121 compared to allowed transitions . here the velocity of nucleons @xmath122 is typically @xmath123 . as a result , the probability of these transitions is lower than the allowed ones by a factor of @xmath124 or @xmath125 . the theoretical curves of the first order parity forbidden transitions are shown as the dashed and the dot - dashed lines in fig . [ fig : half ] . here we use a suppression factor of @xmath126 and @xmath127 , respectively . the first order parity forbidden transitions have an electron spectral shape that is similar to the allowed transitions . as one can see in this figure , the curves of these transitions have the same shapes to the allowed one with a constant shift in the half - life . the existence of these transitions in addition to the allowed ones increases the heating rate . the lifetimes of second order parity forbidden transitions , in which angular momentum of @xmath17 is carried off by the light particles , are too long to be relevant for the macronova heating rates ( see green points in fig . [ fig : half ] ) . -process nuclides . open circles , closes , filled squares , and filled circles are allowed , first parity forbidden , first unique forbidden , and second parity forbidden transitions respectively . here the data points are taken from evaluated nuclear data file endf / b - vii.1 library @xcite . each curve depicts the theoretical expectation with a constant nuclear matrix element of each type of transitions : the allowed ( red solid ) , the first parity forbidden ( magenta dashed and dot - dashed ) , the first unique forbidden ( blue dotted ) , and the second parity forbidden ( green ) . here we adopt @xmath128 and @xmath129 for all analytic models but @xmath130 for the first unique transition . , width=302 ] beta decay chains terminate when they reach stable nuclides . once this happens these terminated chains do nt contribute to the heating rate any more . the overall lifetime of a chain , @xmath131 , can be estimated from the sum of the half - lives of nuclides in the chain . the cumulative distribution of the chains for @xmath132@xmath133 as a function of the chains lifetime is shown in fig . [ fig : chain ] . the number of the chains begins to decrease slowly as @xmath134 at @xmath135s . after about @xmath136 days it decreases slightly faster as @xmath137 . this steep decline at late times due to the termination of the decay chains is consistent with the assumption made by @xcite and @xcite . in summary , the contribution of forbidden transitions to the heating rate slightly increases the heat generation at late times . on the contrary , at the same time , the termination of the beta decay chains slightly decreases it . as a result , the combined effects on the heating rate somehow cancel out . note that these corrections to the heating rate depend on the actual abundance distribution of the chains . we derive an analytic form of the macronova heating rate by considering statistical assembly of radioactive @xmath0-process nuclides and fermi s theory of beta decay . the resulting analytic formula reproduces the heating rate derived from the nuclear database remarkably well . within the assumption that the ideal decay chains of allowed beta transitions generate radioactive heats , we show that the heating rate evolves as @xmath138 at early times and @xmath139 at late times . the overall magnitude of heating rate is determined by the mean value of the nuclear matrix elements , mass and atomic number of beta unstable nuclides involved in the decay chains . is the sum of the half - lives of beta unstable nuclides of a decay chain.,width=302 ] we discuss the role of forbidden transitions and the deviation from the ideal - chains approximation . the former slightly increases the heating rate at late times and the latter slightly decreases it . as a result , these corrections somehow cancel out with each other . the robust and simple form of the heating rate suggests that observations of the late - time bolometric macronova light curve can provide and observational evidence that is is driven by a radioactive decay of @xmath0-process material . furthermore , determination of the bolometric luminosity will enable us to estimate the total amount of @xmath0-process nuclei produced in a merger . using the non - relativistic coulomb regime of eq . ( [ heat2 ] ) , the late - time bolometric light curve is written as @xmath140 where @xmath141 is the ejecta mass . this expression is valid after the peak time given by @xmath142 where @xmath143 is the ejecta velocity and @xmath144 is the bound - bound opacity of @xmath0-process elements @xcite . confirming this behavior by observation may be difficult because the light curve may have large fluctuation due to the temperature and density dependent opacity @xcite . however , as suggested in the context of supernovae ( @xcite ) , the time - weighted integral of the bolometric luminosity after the peak provides a more robust estimate the radioactive power in the ejecta . in this method , the time - weighted integral of the bolometric luminosity should behave as @xmath145 . the bolometric luminosity that we derived here is the total radioactive power emitted in the electrons . at late times , this power is not necessarily thermalized in the ejecta ( see @xcite for a detailed study ) . the inefficiency of the electron thermalization may reduce the bolometric luminosity by a factor of @xmath146 on the macronova timescale . at the same time , we have ignored , additional heating due to @xmath147-rays , @xmath148-particles and fission fragments . the role of these decay products in the macronova heating is still under debate . for instance , it has been suggested that the heat generation by spontaneous fission and @xmath148-decay can be comparable to or even larger than the beta decay heating ( see @xcite ) . we explore the role of these effects in the estimating the total amount of @xmath0-process material ejected in a macronva from the integrated bolometric light curve in a separate work . we thank michael paul , kohsaku tobioka , and shinya wanajo for useful discussions . this research was supported by an erc advanced grant ( trex ) and by the i - core program of the planning and budgeting committee and the israel science foundation ( grant no 1829/12 ) , and an isf grant .

macronovae ( kilonovae ) that arise in binary neutron star mergers are powered by radioactive beta decay of hundreds of @xmath0-process nuclides . we derive , using fermi s theory of beta decay , an analytic estimate of the nuclear heating rate . we show that the heating rate evolves as a power law ranging between @xmath1 to @xmath2 . the overall magnitude of the heating rate is determined by the mean values of nuclear quantities , e.g. , the nuclear matrix elements of beta decay . these values are specified by using nuclear experimental data . we discuss the role of higher order beta transitions and the robustness of the power law . the robust and simple form of the heating rate suggests that observations of the late - time bolometric light curve @xmath3 would be a direct evidence of a @xmath0-process driven macronova . such observations could also enable us to estimate the total amount of @xmath0-process nuclei produced in the merger . [ firstpage ] stars : neutron@xmath4gamma - ray burst : general

You are an expert at summarizing long articles. Proceed to summarize the following text: the atlas central detector @xcite consists of an inner tracking detector ( @xmath0 ) , electromagnetic and hadronic calorimeters ( @xmath1 ) and the muon spectrometer ( @xmath2 ) . in addition , atlas is also equipped with the lucid @xcite , zdc @xcite and alfa @xcite detectors which partially cover the forward rapidity region . the cross - sections for elastic and diffractive production are large . at the centre - of - mass energy of 14 tev the elastic cross - section is estimated to be 25 - 30 mb . the cross - section for single and double diffraction are estimated 10 - 15 mb . thus , elastic and diffractive processes account for roughly half of the @xmath3 total cross - section of @xmath4 mb . thus only modest luminosity is required to study these processes . this is fortuitous since event pile - up resulting from higher luminosity running will tend to destroy the rapidity gap signature of these forward physics processes . lucid is composed of two modules located at @xmath5 m from the interaction point that provide a coverage @xmath6 for charged particles . each lucid detector is a symmetric array of 1.5 m long polished aluminium tubes that surrounds the beam - pipe and points toward the atlas interaction point ( ip ) . this results in a maximum of cerenkov emission from charged particles from the ip that traverse the full length of the tube . each tube is 15 mm in diameter and filled with c4f10 gas maintained at a pressure of 1.2 - 1.4 bar giving a cerenkov threshold of 2.8 gev for pions and 10 mev for electrons . the cerenkov light emitted by the particle traversing the tube has a half - angle of @xmath7 and is reflected an average 3 - 4 times before the light is measured by photomultiplier tubes which match the size of cerenkov tubes . the fast timing response ( a few ns ) provides the unambiguous measurements of individual bunch - crossings . lucid is sitting in the high radiation area that is estimated to receive a radiation dose of @xmath8 mrad per year at maximum luminosity ( @xmath9 ) . lucid is a relative luminosity detector and during the initial period of lhc operation , the absolute calibration would come from the lhc machine parameters allowing the luminosity to be determined to a precision of @xmath10% . after an initial period of lhc running @xmath11 boson counting can be used , as the production cross sections are known well enough to allow and absolute luminosity calibration to 5 - 8% accuracy . qed processes such as exclusive muon pair production via two photon exchange can be calculated to be better than 1% providing another physics based calibration . however , the rates of such processes are quite low and their experimental acceptance and detection efficiency are difficult to estimate accurately . the final absolute luminosity calibration will be determined to a precision of a few percent using elastic proton - proton scattering in the coulomb nuclear interference ( cni ) region covered by the alfa detector . this method requires special low luminosity high beta runs and consequently it is unlikely that this source of calibration will be available in initial lhc running . the zero degree calorimeters ( zdcs ) provide coverage of the region @xmath12 for neutral particles . they reside in a slot in the tan ( target absorber neutral ) absorber , which would otherwise contain copper shielding . the zdc is located at @xmath13 m from the interaction point , at a place where the straight section of the beam - pipe divides into two independent beam - pipes . there will be four zdc modules installed per arm : one electromagnetic ( em ) module and three hadronic modules . each em module consists of 11 tungsten plates , with their faces perpendicular to the beam direction . the height of these plates is extended in the vertical direction with 290 mm long steel plates . two types of quartz radiator are used : vertical quartz strips for energy measurement and horizontal quartz rods which provide position information . at present only hadronic modules are installed . the em module will be installed once the lhcf project has completed data taking . the roman - pot spectrometers are located @xmath14 m away from the interaction point ( ip ) . there will be two roman pot stations separated by four meters on either side of the ip . the main requirements on the alfa scintillating fibre detectors that will be housed in the roman pots are : a spatial resolution of about @xmath15 m ; no significant inactive region ; minimal sensitivity to the radio frequency noise from the lhc beams ; and , ability to operate in the vacuum maintained in the roman pots . at the beginning of the run , the alfa detectors are in withdrawn position far from the beam . after the beam has stabilized , the detectors are moved back to within 1.5 mm of the beam . elastic and diffractive protons deflected from the beam pass through arrays of scintillating fibre trackers ( 20 @xmath16 64 fibres in each array ) , which measure the distance of the proton to the beam . traditionally , the absolute luminosity at hadron colliders has been determined via elastic scattering at small angles . atlas also pursues this approach with the alfa detector . the extremely small angles ( @xmath17 ) needed to make these measurements are smaller than the nominal beam divergence . so special beam conditions e.g. high - beta ( @xmath18 ) optics in combination with reduced beam emittance , are required . alfa will be used to determine the absolute luminosity via elastic scattering at small angles in the coulomb - nuclear interference region . single diffractive ( sd ) can be tagged by identifying the rapidity gap , by requiring that the forward detector system register little hadronic activity . the atlas forward calorimeter ( fcal ) , lucid and the zdc can be utilized as part of a rapidity gap requirement for the sd analysis . di - jet production by sd should be measurable with @xmath19 of the data , corresponding to around 1.5 years of data acquisition at @xmath20 . the cross - section for sd di - jet production is predicted by the pomwig event generator @xcite to be 3.6 ( 0.20)@xmath21b for @xmath22 for jet transverse energy ( @xmath23 ) greater than 20 ( 40)gev , where @xmath24 is the fractional momentum lost by proton during the interaction . di - jet production permits a study of factorization breaking in diffractive events . additional soft interactions and multiple parton - parton scattering during the @xmath25-interaction reduce the observed cross - section for diffractive processes at hadron colliders , with respect to the predicted cross - section obtained from diffractive parton distribution function measured at hera . the central exclusive production ( cep ) is defined as the process @xmath26 , where all of the energy lost by proton goes into the production of a hard central system , @xmath27 . thus the final state consists of two outgoing protons , a hard central system and no other activity . the cep allows direct access to quantum numbers of @xmath27 and has the direct relation of the scattered protons energy loss to the central mass @xmath28 with clean azimuthal correlation of the both scattered protons . at certain scenario it could provide a clean higgs discovery channel @xcite . the di - jet cross section is predicted by the exhume event generator @xcite to be approximately 8nb for a minimum jet transverse energy of 20gev . it should be noted that the current measurements of cep by the cdf collaboration @xcite are in good agreement with the theoretical predictions that form the physics basis of the exhume generator . given the large pre - scale on low @xmath23 jets , one would expect approximately 100 events in 100pb@xmath29 of data . it is necessary to reduce the level 1 ( l1 ) pre - scaling to obtain the good measurement of the cep . it may be possible to do this in atlas by exploiting the clean nature of the exclusive event by requiring a rapidity gap in the l1 trigger ( using lucid , zdc ) , in conjunction with a triggered jet . the single diffractive ( sd ) is characterized by a centrally produced system separated by a rapidity gap , or lack of hadronic activity , from an outgoing proton . in sd exchange the outgoing proton can be tagged and measured during special lhc runs by the alfa detectors . however , the low luminosity means that only soft sd processes can be studied , in particular the forward proton spectrum at low @xmath24 . the acceptance is @xmath30 50% ( 10% ) for @xmath31 0.01 ( 0.1 ) . it is expected that at a luminosity of @xmath32 there would be 1.2 to 1.8 million events recorded in 100 hours of data acquisition . the @xmath24 measurement resolution is approximately 8% for @xmath33 , falling to 2% for @xmath34 . the aim of the atlas forward proton ( afp ) @xcite project is to install spectrometers at @xmath35 m and @xmath36 m from the interaction point of atlas . in exclusive central production processes where the incident protons remain intact , the precise measurement of fractional momentum loses ( @xmath37 ) can be used to determine the mass of the central system with great accuracy , using the relation @xmath38 where @xmath39 is square of the center of mass energy . the @xmath24 acceptance of the afp detectors at 220 m and 420 m is @xmath40 and @xmath41 , respectively . this implies a mass acceptance of the central system spanning the range from 80 gev to masses in excess of 1 tev . afp project opens up a possibility of searching for new physics in cep processes , such as higgs boson production in sm , mssm and nmssm @xcite . because of the limited available space at 420 m , the traditional roman pot technique can not be used . instead , the afp group have opted to employ so called a hamburg movable beam pipe system to deploy the afp forward spectrometers . in order to achieve a good acceptance and mass resolution , 3d silicon edgeless technology has been chosen for the spectrometers , where the 3d silicon sensors have rectangular pixels of dimensions 50 microns by 400 microns . it is envisaged that the pile - up background can be handled by using ultra precise time - of - flight ( tof ) detectors to differentiate the vertex of interest from the vertices of the pile - up events by measuring , the arrival time of the two deflected protons in the tof detectors with a precision of @xmath4210ps . there are two approaches to tof measurement currently being studied . both of these approaches utilize cerenkov detectors readout by microchannel plate ( mcp ) photomultipliers . one of these approaches utilizes a gas cerenkov radiator ( gastof ) , whilst the other employs a fused - silica radiator ( quartic ) . the luminosity monitor lucid , calibrated by the alfa detector , will allow the luminosity delivered to atlas to be determined to better than 5% accuracy . the zdc will measure forward spectators for heavy ion collisions and provide trigger and centrality measurements . it will also provide a luminosity measurement and , measure forward particle production for mc tuning . low luminosity forward physics topics include : elastic scattering using alfa ; sd forward proton spectrum ( alfa ) ; single diffractive di - jet and @xmath43 production and di - jets from double pomeron exchange ( dpe ) and cep ( with rapidity gap vetoes from fcal , lucid , zdc ) . at high luminosities the atlas forward proton ( afp ) project aims to deploy proton taggers at @xmath35 m and @xmath36 m in order to obtain access to a rich new vein of cep physics , that includes sm / mssm / nmssm higgs boson studies , @xmath43 pair production , slepton production and gluino pair production , etc . 9 g. aad _ et al . _ , ( the atlas collaboration ) , jinst 3 s08003 ( 2008 ) . et al . _ , ( the atlas collaboration ) , jinst 3 s08003 , * 206 * ( 2008 ) . et al . _ , ( the atlas collaboration ) , jinst 3 s08003 , * 214 * ( 2008 ) . et al . _ , ( the atlas collaboration ) , jinst 3 s08003 , * 211 * ( 2008 ) . cox and j.r . forshaw , comput . 144 , 104 ( 2002 ) . khoze , a.d . martin and m.g . ryskin , eur . j. c. * 34 * ( 2004 ) 327 ; j.r . forshaw , arxiv:0901.3040 ( 2009 ) . t. aaltonen _ et al . _ [ cdf coll . ( 2007 ) 242002 ; t. aaltonen _ et al . _ [ cdf coll . d 77 ( 2008 ) 052004 . j. monk and a. pilkington , comput . communication 175 ( 2006 ) 232 . s. heinemeyer _ et . j. c 53 ( 2008 ) . b. cox _ et . al . _ , jhep 0710:090 , ( 2007 ) .

in this communication i describe the atlas forward physics program and the detectors , lucid , zdc and alfa that have been designed to meet this experimental challenge . in addition to their primary role in the determination of atlas luminosity these detectors - in conjunction with the main atlas detector - will be used to study soft qcd and diffractive physics in the initial low luminosity phase of atlas running . finally , i will briefly describe the atlas forward proton ( afp ) project that currently represents the future of the atlas forward physics program .

You are an expert at summarizing long articles. Proceed to summarize the following text: kullback - leibler ( kl ) divergence ( relative entropy ) can be considered as a measure of the difference / dissimilarity between sources . estimating kl divergence from finite realizations of a stochastic process with unknown memory is a long - standing problem , with interesting mathematical aspects and useful applications to automatic categorization of symbolic sequences . namely , an empirical estimation of the divergence can be used to classify sequences ( for approaches to this problem using other methods , in particular true metric distances , see @xcite , @xcite ; see also @xcite ) . in @xcite ziv and merhav showed how to estimate the kl divergence between two sources , using the parsing scheme of lz77 algorithm @xcite on two finite length realizations . they proved the consistence of the method by showing that the estimate of the divergence for two markovian sources converges to their relative entropy when the length of the sequences diverges . furthermore they proposed this estimator as a tool for an `` universal classification '' of sequences . a procedure based on the implementations of lz77 algorithm ( gzip , winzip ) is proposed in @xcite . the estimate obtained of the relative entropy is then used to construct phylogenetic trees for languages and is proposed as a tool to solve authorship attribution problems . moreover , the relation between the relative entropy and the estimate given by this procedure is analyzed in @xcite . two different algorithms are proposed and analyzed in @xcite , see also @xcite . the first one is based on the burrows - wheeler block sorting transform @xcite , while the other uses the context tree weighting method . the authors proved the consistence of these approximation methods and show that these methods outperform the others in experiments . in @xcite it is shown how to construct an entropy estimator for stationary ergodic stochastic sources using non - sequential recursive pairs substitutions method , introduced in @xcite ( see also @xcite and references therein for similar approaches ) . in this paper we want to discuss the use of similar techniques to construct an estimator of relative ( and cross ) entropy between a pair of stochastic sources . in particular we investigate how the asymptotic properties of concurrent pair substitutions might be used to construct an optimal ( in the sense of convergence ) relative entropy estimator . a second relevant question arises about the computational efficiency of the derived indicator . while here we address the first , mostly mathematical , question , we leave the computational and applicative aspects for forthcoming research . the paper is structured as follows : in section [ sec : notations ] we state the notations , in section [ sec : nsrps ] we describe the details of the non - sequential recursive pair substitutions ( nsrps ) method , in section [ sec : scaling ] we prove that nsrps preserve the cross and the relative entropy , in section [ sec : convergence ] we prove the main result : we can obtain an estimate of the relative entropy by calculating the 1-block relative entropy of the sequences we obtain using the nsrps method . we introduce here the main definitions and notations , often following the formalism used in @xcite . given a finite alphabet @xmath0 , we denote with @xmath1 the set of finite words . given a word @xmath2 , we denote by @xmath3 its length and if @xmath4 and @xmath5 , we use @xmath6 to indicate the subword @xmath7 . we use similar notations for one - sided infinite ( elements of @xmath8 ) or double infinite words ( elements of @xmath9 ) . often sequences will be seen as finite or infinite realizations of discrete - time stochastic stationary , ergodic processes of a random variable @xmath10 with values in @xmath0 . the @xmath11-th order joint distributions @xmath12 identify the process and its elements follow the consistency conditions : @xmath13 when no confusion will arise , the subscript @xmath11 will be omitted , and we will just use @xmath14 to denote both the measure of the cylinder and the probability of the finite word . equivalently , a distribution of a process can also be defined by specifying the initial one - character distribution @xmath15 and the successive conditional distributions : @xmath16 given an ergodic , stationary stochastic source we define as usual : @xmath17 @xmath18 where @xmath19 denotes the concatenated word @xmath20 and @xmath21 is just the process average . @xmath22 the following properties and results are very well known @xcite , but at the same time quite important for the proofs and the techniques developed here ( and also in @xcite ) : * @xmath23 * a process @xmath24 is @xmath25-markov if and only if @xmath26 . * _ entropy theorem _ : for almost all realizations of the process , we have @xmath27 in this paper we focus on properties involving pairs of stochastic sources on the same alphabet with distributions @xmath24 and @xmath28 , namely _ cross entropy _ and the related _ relative entropy _ ( or _ kullback leibler divergence _ ) : _ n - conditional cross entropy _ @xmath29 _ cross entropy _ @xmath30 _ relative entropy ( kullback - leibler divergence ) _ @xmath31 note that @xmath32 moreover we stress that , if @xmath28 is k - markov then , for any @xmath24 @xmath33 namely @xmath34 for any @xmath35 : @xmath36 & = - \sum_{\omega \in a^{l - k},\,b\in a^k,\,a\in a } \mu ( \omega ba ) \log \nu(a\vert b ) \\ & = - \sum_{b\in a^k,\,a\in a } \mu(ba ) \log \nu(a\vert b)= h_k(\mu\|\nu ) \end{array}\ ] ] note that @xmath37 depends only on the two - symbol distribution of @xmath24 . entropy and cross entropy can be related to the asymptotic behavior of properly defined _ returning times _ and _ waiting times _ , respectively . more precisely , given an ergodic , stationary process @xmath24 , a sample sequence @xmath38 and @xmath39 , we define the returning time of the first @xmath11 characters as : @xmath40 similarly , given two realizations @xmath41 and @xmath42 of @xmath24 and @xmath28 respectively , we define the @xmath43 obviously @xmath44 . we now have the following two important results : [ returning ] if @xmath24 is a stationary , ergodic process , then @xmath45 [ waiting ] if @xmath24 is stationary and ergodic , @xmath28 is k - markov and the marginals @xmath12 of @xmath24 are dominated by the corresponding marginals @xmath46 of @xmath28 , i.e. @xmath47 , then @xmath48 we now introduce a family of transformations on sequences and the corresponding operators on distributions : given @xmath49 ( including @xmath50 ) , @xmath51 and @xmath52 , a _ pair substitution _ is a map @xmath53 which substitutes sequentially , from left to right , the occurrences of @xmath54 with @xmath55 . for example @xmath56 or : @xmath57 @xmath58 is always an injective but not surjective map that can be immediately extended also to infinite sequences @xmath59 . the action of @xmath60 shorten the original sequence : we denote by @xmath61 the inverse of the contraction rate : @xmath62 for @xmath24-_typical _ sequences we can pass to the limit and define : @xmath63 an important remark is that if we start from a source where admissible words are described by constraints on consecutive symbols , this property will remain true even after an arbitrary pair substitution . in other words ( see theorem 2.1 in @xcite ) : a pair substitution maps pair constraints in pair constraints . a pair substitution @xmath64 naturally induces a map on the set of ergodic stationary measures on @xmath65 by mapping typical sequences w.r.t . the original measure @xmath24 in typical sequences w.r.t . the transformed measure @xmath66 : given @xmath67 then ( theorem 2.2 in @xcite ) @xmath68 exists and is constant @xmath24 almost everywhere in @xmath69 , moreover @xmath70 are the marginals of an ergodic measure on @xmath71 . again in @xcite , the following results are proved showing how entropies transform under the action of @xmath72 , with expanding factor @xmath73 : _ invariance of entropy _ @xmath74 _ decreasing of the 1-conditional entropy _ @xmath75 moreover , @xmath76 maps 1-markov measures in 1-markov measures . in fact : @xmath77 _ decreasing of the k - conditional entropy _ @xmath78 moreover @xmath76 maps @xmath25-markov measures in @xmath25-markov measures . while later on we will give another proof of the first fact , we remark that this property , together with the decrease of the 1-conditional entropy , reflect , roughly speaking , the fact that the amount of information of @xmath79 , which is equal to that of @xmath80 , is more concentrated on the pairs of consecutive symbols . as we are interested in sequences of recursive pair substitutions , we assume to start with an initial alphabet @xmath0 and define an increasing alphabet sequence @xmath81 , @xmath82 , @xmath83 , . given @xmath84 and chosen @xmath85 ( not necessarily different ) : * we indicate with @xmath86 a new symbol and define the new alphabet as @xmath87 ; * we denote with @xmath88 the substitution map @xmath89 which substitutes whit @xmath90 the occurrences of the pair @xmath91 in the strings on the alphabet @xmath92 ; * we denote with @xmath93 the corresponding map from the measures on @xmath94 to the measures on @xmath95 ; * we define by @xmath96 the corresponding normalization factor @xmath97 . we use the over - line to denote iterated quantities : @xmath98 and also @xmath99 the asymptotic properties of @xmath100 clearly depend on the pairs chosen in the substitutions . in particular , if at any step @xmath84 the chosen pair @xmath91 is the pair of maximum of frequency of @xmath101 then ( theorem 4.1 in @xcite ) : @xmath102 regarding the asymptotic properties of the entropy we have the following theorem that rigorously show that @xmath103 becomes asymptotically 1-markov : if @xmath102 then @xmath104 the main results of this paper is the generalization of this theorem to the cross and relative entropy . before entering in the details of our construction let us sketch here the main steps . in particular let us consider the cross entropy ( the same argument will apply to the relative entropy ) of the measure @xmath24 with respect to the measure @xmath28 : i.e. @xmath105 . as we will show , but for the normalization factor @xmath106 , this is equal to the cross entropy of the measure @xmath107 w.r.t the measure @xmath108 : @xmath109 moreover , as we have seen above , if we choose the substitution in a suitable way ( for instance if at any step we substitute the pair with maximum frequency ) then @xmath110 and the measure @xmath108 becomes asymptotically 1-markov as @xmath111 . interestingly , we do not know if @xmath112 also diverges ( we will discuss this point in the sequel ) . nevertheless , noticing that the cross entropy of a 1-markov source w.r.t a generic ergodic source is equal to the 1-markov cross entropy between the two sources , it is reasonable to expect that the cross entropy @xmath105 can be obtained as the following limit : @xmath113 this is exactly what we will prove in the two next sections . we first show how the relative entropy between two stochastic process @xmath24 and @xmath28 scales after acting with the _ same _ pair substitution on both sources to have @xmath66 and @xmath114 . more precisely we make use of theorem [ waiting ] and have the following : [ main1 ] if @xmath24 is ergodic , @xmath28 is a markov chain and @xmath47 , then if @xmath60 is a pair substitution @xmath115 _ proof . _ to fix the notations , let us denote by @xmath116 and @xmath117 the infinite realizations of the process of measure @xmath24 and @xmath28 respectively , and by @xmath118 and @xmath119 the corresponding finite substrings . let us denote by @xmath49 the characters involved in the pair substitution @xmath58 . moreover let us denote the waiting time with the shorter notation : @xmath120 we now explore how the waiting time rescale with respect to the transformation @xmath60 : we consider the first time we see the sequence @xmath121 inside the sequence @xmath122 . to start with , we assume that @xmath123 as we can always consider th . [ waiting ] for realizations with a fixed prefix of positive probability . moreover we choose a subsequence @xmath124 such that @xmath125 is the smallest @xmath126 such that @xmath127 . of course @xmath128 as @xmath129 . in this case , it is easy to observe that @xmath130 then , using theorem [ waiting ] @xmath131 = \nonumber\\ & = & \lim_{i\to + \infty } \frac{n_i}{|g(w_1^{n_i})|}\frac{\log|g(w_1^{t_{n_i}})|}{n_i}= \nonumber\\ & = & \lim_{i\to + \infty } \frac{n_i}{|g(w_1^{n_i})|}\left[\frac{1}{n_i}\log ( t_{n_i } ) + \frac{1}{n_i}\log\left(\frac{|g(w_1^{t_{n_i}})|}{t_{n_i}}\right)\right]=\nonumber\\ & = & z^{\mu } h(\mu\|\nu ) \label{kl1}\end{aligned}\ ] ] where in the last step we used the fact that @xmath132 as @xmath129 , the definition of @xmath133 and theorem [ waiting ] for @xmath24 and @xmath28 . note that for @xmath134 , equation ( [ kl1 ] ) reproduces the content of theorem 3.1 of @xcite : @xmath135 that thus implies @xmath136 note that the limit in th . [ waiting ] is almost surely unique and then the initial restrictive assumption @xmath137 and the use of the subsequence @xmath125 have no consequences on the thesis ; this concludes the proof . @xmath138 before discussing the convergence of relative entropy under successive substitutions we go thorough a simple explicit example of the theorem [ main1 ] , in order to show the difficulties we deal with , when we try to use the explicit expressions of the transformed measures we find in @xcite . _ example . _ we treat here the most simple case : @xmath24 and @xmath28 are bernoulli binary processes with parameters @xmath139 and @xmath140 respectively . we consider the substitution @xmath141 given by @xmath142 . it is long but easy to verify that @xmath66 is a stationary , ergodic , 1-markov with equilibrium state @xmath143 where @xmath144 . for example , given a @xmath66-generic sequence @xmath145 , corresponding to a @xmath24-generic sequence @xmath146 ( @xmath147 ) : @xmath148 clearly : @xmath149 using the same argument as before , it is now possible to write down the probability distribution of pair of characters for @xmath66 . again the following holds for a generic process : @xmath150 \frac{{\mathcal g}\mu(10)}z= \mu(10)-\mu(010 ) -\mu(101)+\mu(0101 ) & \frac{{\mathcal g}\mu(11)}z= \mu(11)-\mu(011 ) & \frac{{\mathcal g}\mu(12)}z= \mu(101)-\mu(0101)\\[4pt ] \frac{{\mathcal g}\mu(20)}z= \mu(010)-\mu(0101 ) & \frac{{\mathcal g}\mu(21)}z= \mu(011 ) & \frac{{\mathcal g}\mu(22)}z= \mu(0101 ) \end{array}\ ] ] it is easy to see that @xmath151 . now we can write the transition matrix @xmath152 for the process @xmath66 as @xmath153 : @xmath154 for bernoulli processes : @xmath155 we now denote with @xmath156 the transition matrix for @xmath157 . for the two 1-markov processes , we have @xmath158 via straightforward calculations , using the product structure of the measure @xmath24 : @xmath159\\ + z\mu(11)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ + z\mu(01)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ = z\mu(00 ) d(\mu\vert\vert\nu)+ z\mu(1)\left[\mu(00)\log\frac{\mu(00)}{\nu(00)}+\mu(1)\log\frac{\mu(1)}{\nu(1)}+\mu(01)\log\frac{\mu(01)}{\nu(01)}\right]\\ = z\mu(00 ) d(\mu\vert\vert\nu)+z\mu(1)\left[\mu(0)d(\mu\vert\vert\nu)+ d(\mu\vert\vert\nu)\right]\\ = z d(\mu\vert\vert\nu ) ( \mu(00)+\mu(10)+\mu(1))\\ = z d(\mu\vert\vert\nu)\end{aligned}\ ] ] we now prove that the renormalized 1-markov cross entropy between @xmath12 and @xmath46 converges to the cross - entropy between @xmath160 and @xmath161 as the number of pair substitution @xmath11 goes to @xmath162 . more precisely : [ main2 ] if @xmath163 as @xmath164 , @xmath113 _ proof . _ let us define , as in @xcite the following operators on the ergodic measures : @xmath165 is the projection operator that maps a measure to its 1-markov approximation , whereas @xmath166 is the operator such that for any arbitrary @xmath28 @xmath167 we notice ( see @xcite for the details ) that the normalization constant for @xmath168 is the same of that for @xmath28 : @xmath169 the measure @xmath168 is not @xmath170-markov , but we know that it becomes 1-markov after @xmath84 steps of substitutions , in fact it becomes @xmath171 . moreover , as discussed in @xcite , it is an approximation of @xmath28 if @xmath172 diverges : for any @xmath80 of length @xmath25 , @xmath173 now it is easy to establish the following chain of equalities : @xmath174 where we have used the conservation of the cross entropy @xmath175 and the fact that @xmath176 if @xmath177 are 1-markov , as shown in eq . [ h - k - markov ] . to conclude the proof we have to show that @xmath178 this is an easy consequence of eq . [ convergenza ] the definition [ hk ] and eq . [ hktoh ] . it is important to remark that we are assuming the divergence of @xmath179 too , as not being necessary for the convergence to the ( rescaled ) two - characters relative entropy . nevertheless , it would be interesting to understand both the topological and statistical constraints that prevent or permit the divergence of the expanding factor @xmath179 . experimentally , it seems that if we start with two measures with finite relative entropy ( i.e. with absolutely continuous marginals ) , then if we choose the standard strategy ( most frequent pair substitution ) for the sequence of pair substitutions that yields the divergence of @xmath180 , we also simultaneously obtain the divergence of @xmath181 ( see for instance fig . [ fig : z ] ) . on the other hand , it seems possible to consider particular sources and particular strategies of pairs substitutions withdiverging @xmath180 , that prevent the divergence of @xmath181 . at this moment we do not have conclusive rigorous mathematical results on this subject . finally , let us note that th . [ main2 ] do not give directly an algorithm to estimate the relative entropy : in any implementation we would have to specify the `` optimal '' number of pairs substitutions , with respect to the length of initial sequences and also with respect to the dimension of the initial alphabet . namely , in the estimate we have to take into account at least two correction terms , which diverges with @xmath84 : the entropy cost of writing the substitutions and the entropy cost of writing the frequencies of the pairs of characters in the alphabet we obtain after the substitutions ( or equivalent quantities if we use , for instance , arithmetic codings modeling the two character frequencies ) . for what concerns possible implementations of the method it is important to notice that the nsrps procedure can be implemented in linear time @xcite . therefore it seems reasonable that reasonably fast algorithms to compute relative entropy via nsrps can be designed . anyway , preliminary numerical experiments show that for sources of finite memory this method seems to have the same limitations of that based on parsing procedures , with respect to the methods based on the analysis of context introduced in @xcite . in fig . [ fig : h ] we show the convergence of the estimates of the entropies of the two sources and of the cross entropy , given th . [ main2 ] , for two markov process of memory 5 . in this case , the numbers of substitutions @xmath182 is small with respect to the length of the sequences @xmath183 , then the correction terms are negligible . let us finally note that the cross entropy estimate might show large variations for particular values of @xmath84 . this could be interpreted by the fact that for these values of @xmath84 pairs with particular relevance for one source with respect to the other have been substituted . this example suggest that the nsrps method for the estimation of the cross entropy should be useful in sequences analysis , for example in order to detect strings with a peculiar statistical role . 99 d. benedetto , e. caglioti , d. gabrielli : non - sequential recursive pair substitution : some rigorous results . _ issn : 1742 - 5468 ( on line ) * 09 * pp . 121 doi:10.1088/1742.-5468/2006/09/p09011 ( 2006 )

the entropy of an ergodic source is the limit of properly rescaled 1-block entropies of sources obtained applying successive non - sequential recursive pairs substitutions @xcite,@xcite . in this paper we prove that the cross entropy and the kullback - leibler divergence can be obtained in a similar way . _ keywords _ : information theory , source and channel coding , relative entropy .

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Proceed to summarize the following text: on 2008 january 9.56 ut , while observing the supernova ( sn ) 2007uy in the nearby spiral galaxy ngc2770 ( @xmath0 ) , the x - ray telescope onboard _ swift _ detected a bright x - ray transient ( xt ) , with a peak luminosity of @xmath1 erg s@xmath2 and a duration of about 10 minutes @xcite . its power - law spectrum and light curve shape were reminiscent of gamma - ray bursts ( grbs ) and x - ray flashes , but the energy release was at least 2 orders of magnitude lower than for typical and even subluminous grbs , also allowing for beaming ( e.g. , @xcite ) . the discovery of the xt prompted the search for , and discovery of , an optical counterpart @xcite . we performed spectroscopy of the source as soon as possible , starting 1.75 days after the xt , using the fors2 spectrograph on the eso very large telescope ( vlt ) . subsequent spectroscopic monitoring of the object was carried out at the nordic optical telescope ( not ) and the william herschel telescope ( wht ) . all spectra have been reduced using standard techniques . on january 18.22 ut ( 8.65 days after the xt ) we secured a high - resolution spectrum using the uves instrument on the vlt . for this observation , we adopted the eso cpl pipeline ( v3.3.1 ) , and flux calibration was performed using the master response curves . the observing log of the spectra is reported in table [ speclog ] . imaging observations were conducted using the not , the vlt , the liverpool telescope ( lt ) and the united kingdom infrared telescope ( ukirt ) . image reduction was carried out using standard techniques . for photometric calibration , we observed optical standard star fields on five different nights , and defined a local sequence in the ngc2770 field . in the near - infrared we used two micron all sky survey stars as calibrators magnitudes were computed using small apertures , and subtracting the background as measured in an annulus around the sn position . the contribution from the underlying host galaxy light was always negligible , as also apparent from archival sloan digital sky survey images . our photometric results are listed in table [ tab : photo ] . one of our spectra covers the nucleus of ngc2770 , allowing a precise measurement of its redshift : @xmath3 . this is slightly larger than the value listed in the nasa extragalactic database ( @xmath4 ) . for @xmath5 km s@xmath2 mpc@xmath2 , the luminosity distance is 29.9 mpc . k , reddened assuming @xmath6 mag ( section [ sec : av ] ) . the d narrow absorption from the interstellar medium in ngc2770 is also noted , as well as the two strong telluric features ( marked with ` @xmath7').[fg : earlyspec ] ] our first optical spectrum of the transient source ( figure [ fg : earlyspec ] ) exhibits d absorption lines at @xmath8 , thus establishing its extragalactic nature . broad features are also apparent across the whole spectrum ( @xmath9 km s@xmath2 ) , which led us to identify the object as a core - collapse sn @xcite . @xcite describe nearly simultaneous spectra as featureless , probably due to their smaller covered wavelength range ( @xmath10 ) . @xcite report features consistent with those in our data . we initially classified the sn as a very young type ib / c , based on the absence of conspicuous si and h lines @xcite . as the spectrum is among the earliest observed for any sn , comparable only to the very first spectrum of @xcite , there is no obvious resemblance with known sn spectra . it is notable , however , that the earliest spectrum of the type - ic was essentially flat with broad , low - amplitude undulations ( though the covered wavelength range was limited ; @xcite ) . early spectra , also mostly featureless , are available for the h - rich type - iip @xcite . @xcite interpret them in terms of high temperature and ionization . a striking feature in the spectrum is a conspicuous w - shaped absorption with minima at 3980 and 4190 ( rest frame ) . it was detected using two different instrument setups ( figure [ fg : earlyspec ] ) , and also reported by @xcite . if interpreted as due to p cyg profiles , the inferred expansion velocity is @xmath11 km s@xmath2 , computed from the position of the bluest part compared to the peak . its origin is unclear , although , following @xcite , @xcite propose that it is due to a combination of , , and . interpreting the broad absorption at @xmath12 as @xmath13 6347 , 6371 , some ejecta reached @xmath14 km s@xmath2 . such large velocities have been seen only in broad - lined ( bl ) type - ic sne , at significantly later stages @xcite . ] figure [ fg : lc ] shows the optical and near - infrared light curves of . in the first days after the xt , the flux dropped faster in the bluer bands , with the color becoming progressively redder . this can be interpreted as due to the stellar envelope cooling after the shock breakout @xcite . our first spectra were taken during this stage , before energy deposition by radioactive nuclei became dominant , hence the physical conditions of the emitting material might be different than later . we note that the w - shaped absorption discussed in section [ sec : earlyspec ] was no longer visible from 3.5 days after the xt onward ( fig . [ fg : specevo ] ) . the later spectra , acquired during the radioactivity - powered phase and extending over more than two months in time , established as a type - ib sn @xcite . from january 17 and onward unambiguous he lines are observed ( figure [ fg : specevo ] ) , consistent with other reports @xcite . in figure [ fg : photov ] , we plot the velocities at maximum absorption of a few transitions determined using the synow code @xcite . for comparison , we also plot the velocity of the bl @xcite and of the normal type - ic @xcite , showing that the velocities of are lower than those of bl sne . stands for `` high - velocity h@xmath15 '' ) . the narrow emission line at 6560 is residual h@xmath15 from the sn host galaxy . the feature around 6800 is affected by the b - band atmospheric absorption.[fg : specevo ] ] it follows from the detection of strong d with an equivalent width ( ew ) of 1.3 that the extinction toward is substantial in ngc2770 . our best estimate of the reddening comes from comparing the colors of with those of stripped - envelope sne , which have @xmath16 around maximum ( e.g. , @xcite ) . the resulting reddening is @xmath6 mag , corresponding to an extinction @xmath17 mag ( using the extinction law by @xcite with @xmath18 ) . a large dust content is supported by absorption features in our high - resolution spectrum ( see also @xcite ) . the d1 absorption line indicates a multicomponent system , spanning a velocity range of 43 km s@xmath2 , which sets a lower limit @xmath19 mag ( * ? ? ? * their figure 4 ) . the d versus @xmath20 relation for sne @xcite suggests @xmath21 mag . diffuse interstellar bands ( dibs ) are also detected at 5781.2 , 5797.8 , and 6283.9 ( rest frame ) . their ews suggest @xmath22 mag @xcite . a dusty environment has been directly revealed through millimeter imaging of ngc2770 @xcite . last , the hydrogen column density in the x - ray spectrum of the xt is @xmath23 @xmath24 ( assuming solar abundances ; @xcite ) . the gas - to - dust ratio is @xmath25 @xmath24 mag@xmath2 , close to the galactic value @xmath26 @xmath24 mag@xmath2 @xcite . ( open symbols ) . the latter interpretation is unlikely due to the lack of corresponding h@xmath27 ( see also @xcite ) . shows velocities lower than the prototypical hypernova @xcite and comparable to @xcite.[fg : photov ] ] the precise explosion epoch is so far only known for a few type - ii sne , thanks to either the detection of the neutrino signal (; @xcite ) or of the uv flash by _ galaxy evolution explorer _ @xcite , and for bl type - ic sne associated with grbs ( e.g. , @xcite ) . is the first type - ib sn with a precisely constrained explosion epoch , since the xt is expected to occur less than 1 hr after the stellar collapse @xcite . the nature of the xt shock breakout versus relativistic ejecta is still debated @xcite , thus it is unclear whether this phenomenon is common . the light - curve evolution of is very similar to that of @xcite . the initial fading can be interpreted as due to the envelope cooling through expansion following the initial x - ray / uv flash @xcite . the subsequent rebrightening is due to the energy released by radioactive material in the inner layers and gradually reaching the optically thin photosphere . from our @xmath28 data , we constructed the bolometric light curve of ( figure [ fg : lcbolo ] ) . for comparison we also show the two other he - rich sne caught during the early cooling phase : @xcite and @xcite . strikingly , the three sne had very similar light curves during the photospheric phase . given its peak luminosity , synthesized about @xmath29 of @xmath30ni based on arnett s rule @xcite . the emission during the early cooling phase , however , varied substantially for the three sne ( seen only in @xmath31 for ) . furthermore , significant radiation may be emitted blueward of the @xmath31 band during this early phase , so that the bolometric values may be underestimated . has different properties from the sne associated with grbs , namely the presence of he in the ejecta , a lower peak luminosity ( @xmath32 mag ) , lower expansion velocities ( a factor of @xmath33 ) , and lower @xmath30ni mass ( a factor of @xmath34 ) . the sn environment is also unlike that of grbs @xcite . whether these two kinds of high - energy transients are separate phenomena or form a continuum is unclear . addressing this issue will require theoretical modeling and an enlarged sample . the discovery of a short - lived xt associated with an ordinary type - ib sn opens the possibility of accessing the very early phases of ordinary sne , which will provide new insights into sn physics . future x - ray sky - scanning experiments , such as lobster or erosita , may turn out , rather unexpectedly , ideally suited to examine this issue , alerting us to the onset of many core collapse sne . the dark cosmology centre is supported by the dnrf . is supported by the spanish research programs aya2004 - 01515 and esp2005 - 07714-c03 - 03 and p.m.v . by the eu under a marie curie intra - european fellowship , contract meif - ct-2006 - 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18.22 & 8.66 & vlt / uves+dic#1 & @xmath39 + jan 26.15 & 16.59 & wht / isis+r300b / r316r & @xmath40 + jan 29.01 & 19.45 & not / alfosc+g4 & @xmath37 + feb 01.02 & 53.46 & not / alfosc+g4 & @xmath41 + feb 04.05 & 56.49 & not / alfosc+g4 & @xmath41 + feb 18.15 & 70.59 & not / alfosc+g4 & @xmath37 + feb 25.88 & 78.32 & not / alfosc+g4 & @xmath42 + mar 02.18 & 112.62 & not / alfosc+g4 & @xmath37 + mar 18.87 & 129.31 & not / alfosc+g4 & @xmath43 @llccl@ jan 11.26528 & 01.70082 & @xmath31 & 18.60@xmath440.02 & not+stancam + jan 13.01426 & 03.44980 & @xmath31 & 19.16@xmath440.05 & not+alfosc + jan 15.22931 & 05.66485 & @xmath31 & 19.40@xmath440.05 & not+alfosc + jan 16.03116 & 06.46670 & @xmath31 & 19.29@xmath440.07 & not+alfosc + jan 17.25019 & 07.68573 & @xmath31 & 19.19@xmath440.07 & not+alfosc + jan 18.00103 & 08.43657 & @xmath31 & 19.08@xmath440.04 & not+stancam + feb 01.06841 & 22.50395 & @xmath31 & 19.11@xmath440.04 & not+alfosc + feb 02.00892 & 23.44446 & @xmath31 & 19.26@xmath440.05 & not+alfosc + feb 04.03992 & 25.47546 & @xmath31 & 19.45@xmath440.04 & not+alfosc + feb 11.03650 & 32.47204 & @xmath31 & 20.54@xmath440.08 & not+alfosc + jan 11.04261 & 01.47815 & @xmath45 & 19.00@xmath440.01 & not+stancam + jan 11.26873 & 01.70427 & @xmath45 & 19.03@xmath440.02 & not+stancam + jan 13.01980 & 03.45534 & @xmath45 & 19.21@xmath440.02 & not+alfosc + jan 15.23748 & 05.67302 & @xmath45 & 19.27@xmath440.04 & not+alfosc + jan 16.03701 & 06.47255 & @xmath45 & 19.11@xmath440.01 & not+alfosc + jan 17.25617 & 07.69171 & @xmath45 & 18.99@xmath440.04 & not+alfosc + jan 18.00862 & 08.44416 & @xmath45 & 18.99@xmath440.05 & not+stancam + jan 25.90236 & 16.33790 & @xmath45 & 18.38@xmath440.08 & lt+ratcam + jan 29.05104 & 19.48658 & @xmath45 & 18.45@xmath440.01 & not+alfosc + jan 30.98094 & 21.41648 & @xmath45 & 18.48@xmath440.07 & not+alfosc + feb 01.07844 & 22.51398 & @xmath45 & 18.59@xmath440.01 & not+alfosc + feb 02.01195 & 23.44749 & @xmath45 & 18.67@xmath440.01 & not+alfosc + feb 04.04304 & 25.47858 & @xmath45 & 18.81@xmath440.03 & not+alfosc + feb 11.04021 & 32.47575 & @xmath45 & 19.55@xmath440.03 & not+alfosc + feb 18.98034 & 40.41588 & @xmath45 & 20.18@xmath440.09 & not+alfosc + mar 01.09311 & 51.52865 & @xmath45 & 20.51@xmath440.03 & not+alfosc + mar 10.88157 & 61.31711 & @xmath45 & 20.74@xmath440.04 & not+mosca + apr 02.93574 & 84.37128 & @xmath45 & 21.14@xmath440.05 & not+mosca + jan 11.02120 & 01.45674 & @xmath46 & 18.33@xmath440.01 & not+stancam + jan 11.27136 & 01.70690 & @xmath46 & 18.35@xmath440.06 & not+stancam + jan 13.02744 & 03.46298 & @xmath46 & 18.45@xmath440.01 & not+alfosc + jan 15.24242 & 05.67796 & @xmath46 & 18.13@xmath440.05 & not+alfosc + jan 16.04183 & 06.47737 & @xmath46 & 18.14@xmath440.01 & not+alfosc + jan 17.26096 & 07.69650 & @xmath46 & 17.99@xmath440.02 & not+alfosc + jan 18.01366 & 08.44920 & @xmath46 & 17.86@xmath440.13 & not+stancam + jan 20.02994 & 10.46548 & @xmath46 & 17.75@xmath440.03 & not+stancam + jan 25.90873 & 16.34427 & @xmath46 & 17.27@xmath440.10 & lt+ratcam + jan 29.05856 & 19.49410 & @xmath46 & 17.39@xmath440.01 & not+alfosc + jan 30.19193 & 20.62747 & @xmath46 & 17.39@xmath440.02 & not+alfosc + jan 30.97634 & 21.41188 & @xmath46 & 17.40@xmath440.08 & not+alfosc + feb 01.08387 & 22.51941 & @xmath46 & 17.45@xmath440.01 & not+alfosc + feb 02.01399 & 23.44953 & @xmath46 & 17.49@xmath440.01 & not+alfosc + feb 04.04479 & 25.48033 & @xmath46 & 17.59@xmath440.01 & not+alfosc + feb 11.04238 & 32.47792 & @xmath46 & 18.02@xmath440.01 & not+alfosc + feb 18.98778 & 40.42332 & @xmath46 & 18.53@xmath440.04 & not+alfosc + mar 01.09818 & 51.53372 & @xmath46 & 18.87@xmath440.02 & not+alfosc + mar 10.88568 & 61.32122 & @xmath46 & 19.06@xmath440.05 & not+mosca + mar 17.86086 & 68.29640 & @xmath46 & 19.24@xmath440.08 & not+alfosc + apr 02.94851 & 84.38405 & @xmath46 & 19.58@xmath440.03 & not+mosca + apr 16.95106 & 98.38660 & @xmath46 & 19.90@xmath440.07 & not+stancam @llccl@ jan 11.00922 & 01.44476 & @xmath47 & 17.94@xmath440.01 & not+stancam + jan 11.26190 & 01.69744 & @xmath47 & 17.91@xmath440.02 & not+stancam + jan 11.30417 & 01.73971 & @xmath47 & 17.94@xmath440.02 & vlt+fors2 + jan 13.02367 & 03.45921 & @xmath47 & 18.00@xmath440.01 & not+alfosc + jan 15.24730 & 05.68284 & @xmath47 & 17.80@xmath440.01 & not+alfosc + jan 16.04659 & 06.48213 & @xmath47 & 17.69@xmath440.01 & not+alfosc + jan 17.26593 & 07.70147 & @xmath47 & 17.54@xmath440.01 & not+alfosc + jan 18.01774 & 08.45328 & @xmath47 & 17.42@xmath440.01 & not+stancam + jan 20.04708 & 10.48262 & @xmath47 & 17.16@xmath440.04 & not+stancam + jan 29.06847 & 19.50401 & @xmath47 & 16.82@xmath440.01 & not+alfosc + jan 30.21722 & 20.65276 & @xmath47 & 16.80@xmath440.02 & not+alfosc + jan 30.97162 & 21.40716 & @xmath47 & 16.88@xmath440.05 & not+alfosc + feb 01.08694 & 22.52248 & @xmath47 & 16.84@xmath440.01 & not+alfosc + feb 02.01603 & 23.45157 & @xmath47 & 16.87@xmath440.01 & not+alfosc + feb 04.04619 & 25.48173 & @xmath47 & 16.92@xmath440.01 & not+alfosc + feb 11.04420 & 32.47974 & @xmath47 & 17.25@xmath440.01 & not+alfosc + feb 18.99276 & 40.42830 & @xmath47 & 17.62@xmath440.01 & not+alfosc + mar 01.10190 & 51.53744 & @xmath47 & 18.04@xmath440.01 & not+alfosc + mar 02.12923 & 52.56477 & @xmath47 & 18.07@xmath440.02 & not+alfosc + mar 10.88805 & 61.32359 & @xmath47 & 18.20@xmath440.02 & not+mosca + mar 17.86648 & 68.30202 & @xmath47 & 18.37@xmath440.05 & not+alfosc + apr 02.95326 & 84.38880 & @xmath47 & 18.72@xmath440.04 & not+mosca + apr 16.94566 & 98.38120 & @xmath47 & 19.01@xmath440.04 & not+stancam + jan 11.03211 & 01.46765 & @xmath48 & 17.43@xmath440.01 & not+stancam + jan 11.27381 & 01.70935 & @xmath48 & 17.40@xmath440.01 & not+stancam + jan 15.25279 & 05.68833 & @xmath48 & 17.21@xmath440.01 & not+alfosc + jan 16.05135 & 06.48689 & @xmath48 & 17.09@xmath440.01 & not+alfosc + jan 17.27084 & 07.70638 & @xmath48 & 16.95@xmath440.01 & not+alfosc + jan 18.02196 & 08.45750 & @xmath48 & 16.79@xmath440.01 & not+stancam + jan 20.05173 & 10.48727 & @xmath48 & 16.62@xmath440.07 & not+stancam + jan 29.07600 & 19.51154 & @xmath48 & 16.18@xmath440.01 & not+alfosc + jan 30.22517 & 20.66071 & @xmath48 & 16.19@xmath440.05 & not+alfosc + jan 30.99911 & 21.43465 & @xmath48 & 16.16@xmath440.04 & not+alfosc + feb 01.09031 & 22.52585 & @xmath48 & 16.17@xmath440.01 & not+alfosc + feb 02.01851 & 23.45405 & @xmath48 & 16.18@xmath440.01 & not+alfosc + feb 04.04764 & 25.48318 & @xmath48 & 16.23@xmath440.01 & not+alfosc + feb 11.04589 & 32.48143 & @xmath48 & 16.45@xmath440.01 & not+alfosc + feb 18.99647 & 40.43201 & @xmath48 & 16.76@xmath440.01 & not+alfosc + mar 01.07881 & 51.51435 & @xmath48 & 17.09@xmath440.01 & not+alfosc + mar 06.11299 & 56.54853 & @xmath48 & 17.15@xmath440.03 & vlt+fors1 + mar 10.89042 & 61.32596 & @xmath48 & 17.16@xmath440.04 & not+mosca + mar 17.87137 & 68.30691 & @xmath48 & 17.48@xmath440.06 & not+alfosc + apr 02.95712 & 84.39266 & @xmath48 & 17.65@xmath440.07 & not+mosca + apr 16.95661 & 98.39215 & @xmath48 & 17.97@xmath440.02 & not+stancam + jan 12.555 & 02.99100 & @xmath49 & 16.66@xmath440.02 & ukirt+ufti + jan 14.598 & 05.03400 & @xmath49 & 16.40@xmath440.02 & ukirt+ufti + jan 15.457 & 05.89300 & @xmath49 & 16.28@xmath440.02 & ukirt+ufti + jan 17.645 & 08.08100 & @xmath49 & 15.92@xmath440.02 & ukirt+ufti + jan 21.004 & 11.44000 & @xmath49 & 15.52@xmath440.02 & not+notcam + jan 23.576 & 14.01200 & @xmath49 & 15.47@xmath440.02 & ukirt+ufti + feb 16.240 & 37.67600 & @xmath49 & 15.54@xmath440.02 & ukirt+wfcam + feb 25.295 & 46.73100 & @xmath49 & 15.90@xmath440.02 & ukirt+wfcam + mar 24.250 & 74.69000 & @xmath49 & 16.85@xmath440.02 & ukirt+wfcam + jan 12.555 & 02.99100 & @xmath50 & 16.17@xmath440.02 & ukirt+ufti + jan 14.598 & 05.03400 & @xmath50 & 16.05@xmath440.02 & ukirt+ufti + jan 15.457 & 05.89300 & @xmath50 & 15.91@xmath440.02 & ukirt+ufti + jan 17.645 & 08.08100 & @xmath50 & 15.55@xmath440.02 & ukirt+ufti + jan 21.004 & 11.44000 & @xmath50 & 15.13@xmath440.02 & not+notcam + jan 23.576 & 14.01200 & @xmath50 & 15.05@xmath440.02 & ukirt+ufti + feb 16.240 & 37.67600 & @xmath50 & 15.05@xmath440.02 & ukirt+wfcam + feb 25.295 & 46.73100 & @xmath50 & 15.32@xmath440.02 & ukirt+wfcam + mar 24.250 & 74.69000 & @xmath50 & 16.07@xmath440.02 & ukirt+wfcam + jan 12.555 & 02.99100 & @xmath51 & 15.99@xmath440.02 & ukirt+ufti + jan 14.598 & 05.03400 & @xmath51 & 15.71@xmath440.02 & ukirt+ufti + jan 15.457 & 05.89300 & @xmath51 & 15.55@xmath440.02 & ukirt+ufti + jan 17.645 & 08.08100 & @xmath51 & 15.20@xmath440.02 & ukirt+ufti + jan 21.004 & 11.44000 & @xmath51 & 14.83@xmath440.02 & not+notcam + jan 23.576 & 14.01200 & @xmath51 & 14.72@xmath440.02 & ukirt+ufti + feb 16.240 & 37.67600 & @xmath51 & 14.68@xmath440.02 & ukirt+wfcam + feb 25.295 & 46.73100 & @xmath51 & 14.98@xmath440.02 & ukirt+wfcam + mar 24.250 & 74.69000 & @xmath51 & 15.75@xmath440.02 & ukirt+wfcam

was discovered while following up an unusually bright x - ray transient ( xt ) in the nearby spiral galaxy ngc2770 . we present early optical spectra ( obtained 1.75 days after the xt ) which allowed the first identification of the object as a supernova ( sn ) at redshift @xmath0 . these spectra were acquired during the initial declining phase of the light curve , likely produced in the stellar envelope cooling after shock breakout , and rarely observed . they exhibit a rather flat spectral energy distribution with broad undulations , and a strong , w - shaped feature with minima at 3980 and 4190 ( rest frame ) . we also present extensive spectroscopy and photometry of the sn during the subsequent photospheric phase . unlike sne associated with gamma - ray bursts , displayed prominent he features and is therefore of type ib .

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Proceed to summarize the following text: galactic studies have been one of the main topics of the tartu observatory over the whole period of its existence . these studies began with the work by f.g.w . struve on double stars and measurements of stellar parallaxes in early 19th century , when tartu observatory was founded in the kaiserliche universitt zu dorpat . modern era of galactic studies began about 95 years ago when ernst pik determined the dynamical density of matter in the disk of the galaxy in 1915 , the distance to the andromeda nebula in 1922 , and found main principles of stellar structure and evolution in 1938 . his student grigori kuzmin developed principles of galactic modeling and calculated the local density of matter near the sun , suggesting the absence of local dark matter in large quantities . the present generation of astronomers follows these traditions . in the following i give an overview of the development of the ideas of the structure of galaxies and the universe over the whole period of activity of the tartu observatory . the period of last 50 years is described in more detail . it was a tradition in classical universities to have an astronomical observatory . tartu university observatory was built in 1810 , its astronomical instruments were installed by astronomy and mathematics student friedrich georg wilhelm struve ( 1793 - 1864 ) , who after defending his phd was nominated to professor of astronomy and mathematics and director of the observatory . struve understood well the needs of contemporary astronomy and geodesy . the main goal at this time was to understand the nature of stars and stellar systems . struve started to measure stellar positions , which was basis for better description of the universe . he soon realised that conventional astronomical instruments , available in tartu , were not sufficient to solve most interesting problems . thus he applied to get support to buy a new larger telescope . his efforts succeeded and in 1825 a 9 inch fraunhofer refractor was installed in observatory . for about 15 years this was the largest and best telescope of this type in the world . struve made excellent use of the new telescope . first he made a survey of the whole northern sky and published a catalogue of double stars detected . a catalogue followed , which contained exact measurements of positions of double stars@xcite . after repeated observations have been made it is possible to calculate orbits of double stars and to get information on their masses . this is a foundation of the new astronomy astrophysics . as a by - product of the measurements of double stars he also published his determination of the distance of a star vega . with this measurement it was finally demonstrated that stars are distant suns . struve double star catalogue is used even in present days , his achievements have found place in astronomy textbooks . another important scientific - historic achievement of f.g.w . struve was the astronomic - trigonometric measurement of tartu ( struve ) meridian arc ( 1816 - 1852 ) . this measurement was made together with carl friedrich tenner ( 1783 - 1859 ) . they succeeded in determining the almost 3000 km long section of the meridian arc between the mouth of danube and the arctic ocean with the accuracy of @xmath0 m @xcite . the comparison of data of different arc sections indicated that the length of arc , corresponding to one degree of latitude , increases towards the pole , i.e. the earth is flattened . the measurements were used by f.w . bessel for determination of spheroidal earth s new parameters . struve geodetic meridian arc is included to the unesco world heritage list . in 1839 f.g.w . struve was appointed director of the new pulkovo observatory . the next director of tartu observatory johann heinrich mdler ( 1794 - 1874 ) was interested in the dynamics of the milky way . in his book centralsonne he tried to determine the center of the galaxy using proper motions of stars . the accuracy of data was not sufficient for this task , the method itself is correct . the founder of the modern astronomy school in tartu university was ernst pik ( 1893 - 1985 ) . he started his astronomical career as student of the moscow university , and made his principal discoveries as observator of the tartu university observatory . one of his first scientific papers was devoted to the question : what is the density of matter near the plane of the milky way stellar system ? his calculations showed that the gravitating matter density can be fully explained by observable stars , and that there is no evidence for hypothetical matter near the symmetry plane of the galaxy . this work seems to be the first attempt to address the dark matter problem@xcite . next pik devoted his attention to spiral nebulae . their nature was not known at this time : are they gaseous objects within the milky way or distant worlds similar in structure to our galaxy ? immediately when relative velocity measurements of the central part of andromeda nebula m31 had been published , pik developed a method how to use this information to estimate the distance to m31 . his result was 440 kpc ( about 1.5 million light years)@xcite . with this work he solved the problem of the nature of spiral nebulae , and showed that the universe is millions of times larger than our milky way system . another unsolved problem at this time was the source of stellar energy and the evolution of stars . already in early 1920s pik demonstrated , using very simple physical considerations , that gravitational contraction and radioactivity can not be the main sources of stellar energy . he concluded , that some unknown subatomic processes in central regions of stars must be responsible for stellar energy . when data on atomic structure were available , he developed a detailed theory of stellar evolution , based on nuclear reactions in stellar interiors . here under very high temperature hydrogen burns to helium , and huge amounts of energy will be released @xcite . the efficiency of these reactions has just been estimated based on atomic theory , thus pik was able to calculate ages of stars . he demonstrated , that hot giant stars are so luminous that their energy sources will be exhausted within several tens millions years . their presence shows that they have been recently formed in other words , star formation is a process which takes place even today . these results revolutionised our understanding of stars and their evolution . presently they are accepted by the astronomical community . professor of astronomy in tartu university rootsme ( 1885 - 1959 ) applied these ideas to kinematics of stars to find the sequence of formation of different stellar populations@xcite . similar ideas were developed independently by eggen , lynden - bell and sandage@xcite . pik s student grigori kuzmin ( 1917 - 1988 ) continued his mentors work on studying the structure of the galaxy . in his phd thesis he developed further pik s method to derive the density of matter in the galaxy and constructed a mathematical model of the galaxy , much more advanced than previous ones@xcite . in contrast to earlier models by oort and schmidt he used ellipsoids of variable density , and applied the theory first to the andromeda galaxy and then to our own galaxy . one of central problems in modeling the galaxy was the density of matter near the sun . in 1930s famous dutch astronomer jan oort@xcite studied this problem and found , that in addition to ordinary stars there must be in the galaxy an unknown population , so that the total local density exceeds twice the density of visible matter . kuzmin found that the method used by oort is not very accurate and that there is no indication for the presence of local dark matter in the galaxy . later two students of kuzmin , heino eelsalu ( 1930 - 1998 ) @xcite and mihkel jeveer ( 1937 - 2006 ) @xcite , reanalysed the problem , using different data and methods , and confirmed his results . the discrepancy between results of tartu astronomers and the rest of the world continued until 1990s , when finally modern data confirmed that kuzmin was right ( gilmore , wyse & kuijken@xcite ) . thus we came to the conclusion that _ there is no evidence for the presence of large amounts of dark matter in the disk of the galaxy_. if there is some invisible matter near the galactic plane , then it consists probably of low mass stars or jupiters , which have been formed from the flat gas population . kuzmin also developed a method how to describe more accurately the kinematics of stellar populations , using three integrals : the energy and mass conservation integrals , and a third integral , which allows the existence of three - axial velocity ellipsoids of stellar populations@xcite . from his model it follows that orbits of stars in galaxies lie in a toroidal volume . numerical modeling of star orbits in realistic mass models have confirmed kuzmin s prediction . the work by pik , rootsme , kuzmin and their students formed the basis of a concept of the structure and evolution of stellar populations in galaxies , which is rather close to the presently accepted picture . in early 1960s i was interested in the problem too . as new observational data arrived , the need for a better and more accurate model of our galaxy and other galaxies was evident . detailed local structure is known only for our own galaxy , and global information on stellar populations is better known for external galaxies , thus it is reasonable to investigate the structure of our galaxy and other galaxies in parallel . also there was a need for a more detailed method of the construction of composite models of galaxies . this goal was realised in a series of papers in tartu observatory publications@xcite , a summary of the method was published in english@xcite . a natural generalisation of classical galactic models is the use of all available observational data for spiral and elliptical galaxies , both photometric data on the distribution of colour and light , and kinematical data on the rotation and/or velocity dispersion . further , it is natural to apply identical methods for modeling of galaxies of different morphological type ( including our own galaxy ) , and to describe explicitly all major stellar populations . the main principles of model construction were : ( 1 ) galaxies can be considered as sums of physically homogeneous populations ( young flat disk , thick disk , core , bulge , halo ) ; ( 2 ) physical properties of populations ( mass to luminosity ratio @xmath1 , colour ) should be in agreement with models of physical evolution of stellar populations ; ( 3 ) the density of a population can be expressed as ellipsoids of constant flatness and rotational symmetry ; ( 4 ) densities of populations are non negative and finite ; ( 5 ) moments of densities which define the total mass and effective radius of the galaxy are finite . it was found that in a good approximation densities of all stellar populations can be expressed by a generalised exponential law : @xmath2 $ ] , where @xmath3 is the central density , @xmath4 is the distance along the major axis , @xmath5 is the axial ratio of the equidensity ellipsoid , @xmath6 is the core radius ( @xmath7 is the harmonic mean radius ) , @xmath8 and @xmath9 are normalising parameters , depending on the structural parameter @xmath10 , which allows to vary the density behaviour with @xmath11 . the cases @xmath12 and @xmath13 correspond to conventional exponential and de vaucouleurs models , respectively . this density law ( called einasto profile ) was first applied to find a composite model of the galaxy@xcite , based on the new system of galactic constants , using all available data ( einasto and kutuzov@xcite ) . next the method was applied to the andromeda galaxy @xcite . a central problem in galactic modeling is the correct estimation of the mass - to - luminosity ratio of populations . this ratio depends on the evolutionary history of the population and on its chemical composition . in order to bring these ratios for different populations to a coherent system , a model of physical evolution of stellar populations was developed @xcite . the model used as input data the evolutionary tracks of stars of various composition ( metallicity ) and age ; the star formation rate as a function of stellar mass was accepted according to salpeter@xcite , with a low mass limit of star formation of @xmath14 . the model yielded a continuous sequence of population parameters as a function of age ( colour , spectral energy distribution , @xmath1 ) , see fig . [ fig : evolution ] . the results of modeling stellar populations were calibrated using direct dynamical data for star clusters and central regions of galaxies ( velocity dispersions ) by einasto & kaasik@xcite . these data supported relatively high values ( @xmath15 ) for old metal rich stellar populations near centres of galaxies ; moderate values ( @xmath16 ) for discs and bulges ; and low values ( @xmath17 ) for metal poor halo type populations . modern data yield lower values , due to more accurate measurements of velocity dispersions in clusters and central regions of galaxies , and rotation data on bulge dominated s0 galaxies . i had a problem in the modeling of m31 . if rotation data were taken at face value , then it was impossible to represent the rotational velocity with the sum of known stellar populations . the local value of @xmath1 increases towards the periphery of m31 very rapidly , if the mass distribution is calculated directly from rotation velocity . all known old metal poor halo type stellar populations have a low @xmath18 . in contrast , on the basis of rotation data we got @xmath19 on the periphery of the galaxy near the last point with measured rotational velocity . i discussed the problem with my collaborator enn saar . he suggested to abandon the idea , that only stellar populations exist in galaxies . instead it is reasonable to assume the existence of a population of unknown nature and origin , and to look which properties it should have using available data on known stellar populations . so i calculated a new set of models for m31 , our galaxy and several other galaxies of the local group , as well as for the giant elliptical galaxy m87 in the virgo cluster . in most models it was needed to include a new population in order to bring rotation and photometric data into mutual agreement . to avoid confusion with the metal - poor halo the new hypothetical population was called corona . results of these calculations were reported in the first european astronomy meeting in athens in september 1972@xcite . the conclusions were : ( 1 ) there are two dark matter ( dm ) problems : the local dm near the galactic plane , and the global dm forming an extended almost spherical population ( corona ) ; ( 2 ) the local dark matter , if it exists , must be of stellar origin , as it is strongly concentrated to the galactic plane ; ( 3 ) the global dark matter is probably of non - stellar origin . available data were insufficient to determine outer radii and masses of coronas . preliminary estimates indicated that in some galaxies the mass and radius of the corona may exceed considerably the mass and radius of known stellar populations . arguments for the non - stellar origin of galactic coronas were the following . ( 1 ) physical and kinematical properties of the stellar populations depend almost continuously on the age of the population , the oldest have the lowest metallicity and @xmath1-ratio , and there is no place where to put the corona into this sequence@xcite . ( 2 ) since the @xmath1 value and spatial distribution of the corona differ so much of similar properties of known stellar populations , the corona must have been formed much earlier than all known populations ; the total mass of the corona exceeds masses of known populations by an order of magnitude , thus we have a problem : how to transform in an early stage of the evolution of the universe most of gas to the coronal stars ? it is known that star formation is a very inefficient process , as in a star - forming gaseous nebula only about 1 % of matter transforms to stars@xcite . ( 3 ) due to the large size of the corona , coronal stars must have in the vicinity of the sun much higher velocities than all other stars , but no extremely high - velocity stars have been found by jaaniste and saar@xcite . ( 4 ) luminosity decreases in outer regions rapidly , therefore , if the matter is in stars , they must be of very low luminosity . the presence of low - luminosity stars in outer galactic regions without bright ones would require a process of large - scale segregation of stars according to mass ( low - luminosity stars have small masses ) , but this is highly improbable@xcite . the hidden matter can not be in the form of neutral gas , since this gas would be observable@xcite . for these reasons , i assumed that coronas may consist of hot gas . soon it was clear that a fraction of coronal matter is indeed gaseous@xcite , however not all . to find the radii and masses of galactic coronas more distant test objects are needed . one possibility is the use of companion galaxies . if coronas are large enough , then in pairs of galaxies the companion galaxy is moving inside the corona , and it can be considered as a test particle to measure the gravitational attraction of the main galaxy . mean relative velocities , calculated for different distances from the main galaxy , can be used instead of rotation velocities to find the mass distribution of giant galaxies . a collection of 105 pairs of galaxies yield following results : radii and masses of galactic coronas exceed radii and masses of stellar populations of galaxies by an order of magnitude ! together with a. kaasik and e. saar we calculated new models of galaxies including dark coronas . results were reported in the caucasus winter school in january 1974 , and published in nature@xcite . our data suggest that all giant galaxies have massive coronas of some unknown origin ( dark matter ) , the total masses of galaxies including dark coronas exceed masses of known populations by an order of magnitude . it follows , that dark matter is the dominating component in the whole universe . similar results have been obtained by ostriker , peebles and yahil@xcite . additional arguments supporting the physical connection between main galaxies and their companions were found from the morphology of companions@xcite . in january 1975 the first conference on dark matter was held in tallinn , estonia . the rumour on dark matter had spread around the astronomical community and , in contrast to conventional local astronomy conferences , leading soviet astronomers and physicists attended . the main topics was the possible nature of the dark matter . it was evident that a stellar origin is almost excluded@xcite , but a fully gaseous corona also has difficulties , as shown by komberg and novikov@xcite . the problems with baryon nucleosynthesis constraints were discussed by zeldovich . so the nature of dm was not clear . the next dark matter discussion was in july 1975 during the third european astronomical meeting in tbilisi , georgia , where a full session was devoted to the dark matter problem . here the principal discussion was between the supporters of the classical paradigm with conventional mass estimates of galaxies , and of the new one with dark matter . the major arguments supporting the classical paradigm were summarised by gustav tammann@xcite . the most serious arguments were : _ big bang nucleosynthesis suggests a low - density universe with the density parameter @xmath20 ; the smoothness of the hubble flow also favours a low - density universe . _ dark matter problem was also discussed during the iau general assembly in grenoble , 1976 . here arguments for the non stellar nature of dark coronas were again presented@xcite . it was clear that by sole discussion the presence and nature of dark matter can not be solved , new data and more detailed studies were needed . a very strong confirmation of the dark matter hypothesis came from new extended rotation curves of galaxies . vera rubin and her collaborators developed new sensitive detectors to measure optically the rotation curves of galaxies at very large galactocentric distances . their results suggested that practically all spiral galaxies have extended flat rotation curves@xcite . the internal mass of galaxies rises with distance almost linearly , up to the last measured point , see fig . [ fig : bosma_rubin ] . at the same time measurements of a number of spiral galaxies with the westerbork synthesis radio telescope were completed , and mass distribution models were built , all - together for 25 spiral galaxies by bosma@xcite , see fig . [ fig : bosma_rubin ] . observations confirmed the general trend that the mean rotation curves remain flat over the whole observed range of distances from the center , up to @xmath21 kpc for several galaxies . these new observations confirmed the presence of dark coronas of galaxies . the nature of the coronas was still unclear , and the difficulties discussed in tallinn and tbilisi were not clarified . in late 1970s suggestions were made that some sort of non - baryonic elementary particles may serve as candidates for dark matter particles . there were several reasons to search for non - baryonic particles as a dark matter candidate . first of all , no baryonic matter candidate did fit the observational data . second , the total amount of dark matter is of the order of 0.20.3 in units of the critical cosmological density , whereas the nucleosynthesis constraints suggest that the amount of baryonic matter can not be higher than about 0.04 of the critical density . a very important observation was made , which caused doubts to the baryonic matter as the dark matter candidate . in 1964 cosmic microwave background ( cmb ) radiation was detected . initially the universe was very hot and all density and temperature fluctuations of the primordial gas were damped by very intense radiation . at a certain epoch called recombination the gas became neutral , and density fluctuations in the gas had a chance to grow by gravitational instability . but gravitational clustering is a very slow process . in order to have time to build up all observed structures the amplitude of initial density fluctuations at the epoch of recombination must be of the order of @xmath22 of the density itself . density fluctuations are of the same order as temperature fluctuations , and astronomers started to search for temperature fluctuations of the cmb radiation . none were found . as the accuracy of measurement increased , lower and lower upper limits for the amplitude of cmb fluctuations were obtained . in late 1970s it was clear that the upper limits are much lower than the theoretically predicted limit @xmath22 . then astronomers recalled the possible existence of non - baryonic particles , such as heavy neutrinos . this suggestion was made independently by several astronomers ( @xcite ) . if dark matter is non - baryonic , then this helps to explain the paradox of small temperature fluctuations of cosmic microwave background radiation . density perturbations of non - baryonic dark matter start growing already during the radiation - dominated era , whereas the growth of baryonic matter is damped by radiation . if non - baryonic dark matter dominates dynamically , the total density perturbations can have an amplitude of the order @xmath22 at the recombination epoch , which is needed for the formation of the observed structure of the universe . the evolution of perturbations in a neutrino - dominated dark matter medium was discussed in a conference in tallinn in april 1981 ( this conference was probably the first one devoted to the astro particle physics ) . numerical simulations made for a neutrino - dominated universe were made by a number of astronomers . these calculations demonstrated some weak points in the scenario : large - scale structures ( superclusters ) form too late and have no fine structure as observed in the real universe . a new scenario was suggested , among others , by bond , szalay & turner@xcite ; here hypothetical particles like axions , gravitinos or photinos play the role of dark matter . numerical simulations of structure evolution for neutrino and axion dominated universe were made and analysed by melott et al.@xcite . all quantitative characteristics ( connectivity of the structure , multiplicity of galaxy systems , correlation function ) of this new model fit the observational data well . this model was called subsequently the cold dark matter ( cdm ) model , in contrast to the neutrino based hot dark matter model . presently the cdm model with some modifications is the most accepted model of the structure evolution ( blumenthal et al.@xcite ) . after my talk in the caucasus winter school zeldovich turned to me and offered collaboration in the study of the universe . he was developing a theory of the formation of galaxies the pancake theory@xcite ; an alternative whirl theory was suggested by ozernoy , and a third theory of hierarchical clustering by peebles . zeldovich asked for our help in solving the question : can we find some observational evidence which can be used to discriminate between these theories ? initially we had no idea how we can help zeldovich . but soon we remembered our previous experience in the study of galactic populations : kinematical and structural properties of populations hold the memory of their previous evolution and formation ( rootsme@xcite , eggen , lynden - bell & sandage@xcite ) . random velocities of galaxies are of the order of several hundred km / s , thus during the whole lifetime of the universe galaxies have moved from their place of origin only about 1 @xmath23mpc ( we use the hubble constant in units @xmath24 km s@xmath25 mpc@xmath25 ) . in other words if there exist some regularities in the distribution of galaxies , these regularities must reflect the conditions in the universe during the formation of galaxies . in our work to solve the zeldovich question we had a close collaboration with his team . in 1975 doroshkevich , shandarin and novikov@xcite obtained first results of numerical simulations of the evolution of particles according to the theory of gravitational clustering , developed by zeldovich@xcite . this was a 2dimensional simulation with @xmath26 particles ( see fig . [ fig : zeld - nbody ] ) . in this picture a system of high and low density regions was seen : high density regions form a cellular network , which surrounds large under dense regions . one of our challenges was to find out , whether the real distribution of galaxies showed some similarity with the theoretical picture . now we had a guiding idea how to solve the problem of galaxy formation : _ we have to study the distribution of galaxies on larger scales_. both our galactic astronomy and theoretical cosmology teams participated in the effort to find the distribution of galaxies and their systems in space . one approach was the study of the distribution of nearby zwicky clusters . many bright galaxies of nearby zwicky clusters had at this time measured redshifts , so we hoped to determine the distribution of clusters and to find some regularities there . to see the distribution better , we built in the office of saar and jaaniste a 3dimensional model from plastic balls . some regularity was evident : there were several clusters of zwicky clusters superclusters , one of them in the perseus region . but too many clusters had no galaxies with measured redshifts , so it was difficult to get an overall picture . zwicky nearby clusters were used several years later by einasto , jeveer & saar@xcite , when more redshifts were available and for the rest photometric distances were estimated . a different approach was used by mihkel jeveer . he used wedge diagrams . his trick was : he made a number of wedge diagrams in sequence for fixed @xmath27 and @xmath28 intervals , and plotted in the same diagram galaxies , as well as groups and clusters of galaxies , and markarian galaxies . redshift data for clusters , groups and markarian galaxies were almost complete in the northern hemisphere up to a redshift about 15.000 km / s . two wedge diagrams of width @xmath29 , crossing the coma , perseus , hercules and local superclusters , are shown in fig . [ fig : pers ] . in these diagrams a regularity was clearly seen : _ isolated galaxies and galaxy systems populate identical regions , and the space between these regions is empty_. after this success the whole tartu cosmology team continued the study using wedge diagrams and other methods to analyse the distribution of galaxies and their systems . a detailed analysis of the perseus supercluster region was made ; here the number of foreground galaxies is very small . already in 1975 , after the tbilisi meeting , we discussed with zeldovich the possibility to organise a real international conference devoted solely to cosmology . due to soviet bureaucratic system it was extremely difficult for soviet astronomers to attend international conferences in western countries ; thus the only possibility to have a better contact between cosmologists from east and west was to hold the conference within the soviet union . zeldovich suggested to hold it in tallinn . after some discussion we decided to devote it to `` large scale structure of the universe '' . when we started preparations we had no idea what this term could mean . [ fig : pers ] the symposium was held in september 1977 . our main results were presented in the talk by jeveer & einasto@xcite : ( 1 ) galaxies , groups and clusters of galaxies are not randomly distributed , but form chains , concentrated in superclusters ; ( 2 ) the space between galaxy chains contains almost no galaxies and form holes or voids of diameter up to @xmath30 @xmath23mpc ( see fig . [ fig : pers ] ) ; ( 3 ) the whole pattern of the distribution of galaxies and clusters resembles cells of a honeycomb , rather close to the picture predicted by zeldovich . a more detailed analysis was published separately by jeveer , einasto & tago@xcite . the presence of voids in the distribution of galaxies was reported also by other groups : by tully & fisher@xcite , tifft & gregory@xcite , and tarenghi et al.@xcite in the local , coma and hercules superclusters , respectively . theoretical interpretation of the observed cellular structure was discussed by zeldovich@xcite . malcolm longair noted in his concluding remarks : _ the discovery of the filamentary character of the distribution of galaxies , similar to a lace tablecloth , and the overall cellular picture of the large scale distribution was the most exciting result presented at this symposium_. these results demonstrated that the pancake scenario by zeldovich@xcite has many advantages over other rivalling scenarios . the term `` large scale structure of the universe '' got its present meaning . the first problem to solve was to find some explanation for the absence of galaxies in voids . this was done by einasto , jeveer & saar@xcite . saar developed an approximate model of the evolution of density perturbations in under and over dense regions based on zeldovich ideas . he found that the matter flows out of under dense regions and collects in over dense regions until it collapses ( pancake forming ) and forms here galaxies and clusters . in under dense regions the density decreases continuously , but never reaches zero : there must be primordial matter in voids . in these under - dense regions the density is always less than the mean density , thus galaxy formation is not possible . initially we believed that pancakes are 2dimensional surfaces as predicted by zeldovich@xcite . to our surprise we did not find evidence for the presence of wall like structures between voids _ the dominating structural element was a chain ( filament ) of galaxies and clusters_. the absence of wall like pancakes and the dominance of filaments was explained theoretically by bond , kofman & pogosyan@xcite , through the effect of tidal forces . the presence of filaments and voids in the galaxy distribution was met with some scepticism . one objection against the new concept was raised by peebles : the human brain has a tendency to see regularity ( filaments in galaxy distribution ) even in the case if the actual distribution is almost random . zeldovich@xcite addressed this problem in his talk during the iau symposium and suggested that objective criteria should be used to check this aspect of the galaxy distribution . together with the zeldovich team we developed several methods to characterise the filamentary distribution one of these methods was the multiplicity function of galaxy systems , it is sensitive to the richness of galaxy systems . the other test applied was the connectivity or percolation test , which makes difference between galaxy systems consisting of isolated clusters , clusters connected with filaments , and a random distribution of galaxies . applying these tests to real galaxy samples confirmed the presence of galaxy systems of very different richness , and a high connectivity due to galaxy filaments between superclusters . the application of tests to simulated structures showed that the original zeldovich scenario ( where small - scale perturbations were absent ) , as well as the peebles scenario , have some problems . the zeldovich scenario passes the percolation test , but not the multiplicity test there are no fine structures within superclusters . the peebles scenario fails both tests , results of these tests were described by zeldovich , einasto & shandarin@xcite and einasto et al.@xcite . difficulties with both scenarios were solved when the cold dark matter model of structure formation was applied , which unites best properties of both the zeldovich and the peebles scenarios ( melott et al.@xcite , blumenthal et al.@xcite ) . galaxies form by clustering and merging of smaller galaxies as suggested by peebles , but their formation starts in future superclusters , where the density is highest , as it follows from the zeldovich pancake scenario . at the time of the tallinn symposium only relatively small areas of sky were covered by complete magnitude - limited galaxy samples , thus many astronomers were suspicious to the presence of the overall cellular distribution . it was evident that wide area and deeper flux limited galaxy redshift surveys are needed . harvard astronomers made for the whole northern hemisphere a survey up to limiting magnitude 14.5 , later the survey was extended to 15.5 magnitude , cfa1 and cfa2 surveys , respectively . the second cfa surveys shows the filamentary character of the galaxy distribution very clearly , as seen from the first slice of their study by de lapparent , geller & huchra@xcite . the presence of the cosmic web was confirmed . a much deeper redshift survey up to the blue magnitude 19.4 was made using the anglo - australian 4-m telescope . this two degree field galaxy redshift survey ( 2dfgrs ) covers an equatorial strip in the northern galactic hemisphere and a contiguous area in the southern hemisphere@xcite . over 250 thousand redshifts have been measured . presently the largest project to map the universe , the sloan digital sky survey ( sdss ) , has been completed by a number of american , japanese and european universities and observatories@xcite . the goal was to map a quarter of the entire sky : to determine positions and photometric data in 5 spectral bands of galaxies and quasars , and redshifts of all galaxies down to red magnitude r = 17.7 ( about 1 million galaxies ) . all 7 data releases have been made public . this has allowed to map the largest volume of the universe so far . [ fig : sdss ] shows the luminosity density field of a shell of thickness 10 @xmath23mpc at a distance @xmath31 @xmath23mpc from us . the discovery of the non baryonic nature of the dark matter has resolved the first fundamental problem of the dark matter , discussed by tammann in tbilisi in 1975 . the second problem , the smoothness of the hubble flow , was explained only recently with the discovery of dark energy , previously called also cosmological lambda term . two teams , led by riess@xcite ( high - z supernova search team ) and perlmutter @xcite ( supernova cosmology project ) , initiated programs to detect distant type ia supernovae in the early stage of their evolution , and to investigate with large telescopes their properties . these supernovae have an almost constant intrinsic brightness . by comparing the luminosities and redshifts of nearby and distant supernovae it is possible to calculate how fast the universe was expanding at different times . the supernova observations give strong support to the cosmological model with the @xmath32 term . studies of the hubble flow in the nearby space , using observations of type ia supernovae with the hubble space telescope ( hst ) , were carried out by several groups . the major goal of the study was to determine the value of the hubble constant . as a by - product also the smoothness of the hubble flow was investigated . one of these projects was led by allan sandage@xcite . the analysis confirmed earlier results that the hubble flow is very quiet over a range of scales from our local supercluster to the most distant objects observed . this smoothness in spite of the inhomogeneous local mass distribution requires a special agent . sandage emphasises that no viable alternative to dark energy is known at present , thus the quietness of the hubble flow gives strong support for the existence of dark energy . this effect has been investigated in detail by arthur chernin@xcite . * the presence of dark matter in the universe and the cosmic web were established gradually by astronomers from many centers , in several cases tartu astronomers have pioneered in these studies . * initially the dark matter concept had many problems , until its non baryonic nature and the presence of dark energy were found ( @xmath32cdm model ) . also the concept of the web - like distribution of galaxies was initially met with scepticism , which disappeared when wide - area and complete redshift surveys were completed . * the discoveries of dark matter and cosmic web are connected : dark matter as the dominant population in the universe determines properties of the web , and the structure of the web gives information on properties of dm particles . * in the study of dark matter and the structure of the universe our tartu team benefited from the earlier experience in tartu by pik , rootsme and kuzmin , and especially from a close collaboration with the moscow cosmology team led by yakov zeldovich . i thank maret einasto , gert htsi , juhan lauri liivamgi , enn saar , erik tago , elmo tempel , and our colleague volker mller in potsdam for collaboration and for the permission to use our common results in this review talk . my special thank is to late mihkel jeveer for pioneering contributions both in the search for dark matter and of the structure of the universe , as well as to late yakov borissovich zeldovich and his team , who inspired us to study the large scale distribution of galaxies , and participated in the development of the ideas following the discovery of the cosmic web . the present study was supported by the estonian science foundation grant etf 4695 , and by grant to 0060058s98 . i thank icranet , and the astrophysikalisches institut potsdam ( dfg - grant mu 1020/15 - 1 ) where part of this study was performed , for hospitality . w. g. tifft and s. a. gregory , observations of the large scale distribution of galaxies , in _ large scale structures in the universe _ , eds . m. s. longair and j. einasto , iau symposium , vol . 79 , 267 ( 1978 ) . m. tarenghi , w. g. tifft , g. chincarini , h. j. rood and l. a. thompson , the structure of the hercules supercluster , in _ large scale structures in the universe _ , eds . m. s. longair and j. einasto , iau symposium , vol . 79 , 263 ( 1978 ) .

an overview is provided for 200 years of galactic studies at the tartu observatory . galactic studies have been one of the main topics of studies in tartu over the whole period of the history of the observatory , starting from f.g.w . struve and j.h . mdler , followed by ernst pik and grigori kuzmin , and continuing with the present generation of astronomers . our goal was to understand better the structure , origin and evolution of stars , galaxies and the universe .

You are an expert at summarizing long articles. Proceed to summarize the following text: after more than two decades of investigations , black hole thermodynamics is still one of the most puzzling subjects in theoretical physics . one approach to studying the thermodynamical aspects of a black hole involves considering the evolution of quantum matter fields propagating on a classical ( curved ) background spacetime . this gives rise to the phenomenon of black hole radiation that was discovered by hawking in 1974 @xcite . combining hawking s discovery of black hole radiance with the classical laws of black hole mechanics @xcite , leads to the laws of black hole thermodynamics . the entropy of a black hole obtained from this approach may be interpreted as resulting from averaging over the matter field degrees of freedom lying either inside the black hole @xcite or , equivalently , outside the black hole @xcite , as was first anticipated by bekenstein @xcite even before hawking s discovery . the above approach was further developed in the following years @xcite . a second route to black hole thermodynamics involves using the path - integral approach to quantum gravity to study _ vacuum _ spacetimes ( i.e. , spacetimes without matter fields ) . in this method , the thermodynamical partition function is computed from the propagator in the saddle point approximation @xcite and it leads to the same laws of black hole thermodynamics as obtained by the first method . the second approach was further developed in the following years @xcite . the fact that the laws of black hole thermodynamics can be derived without considering matter fields , suggests that there may be a purely geometrical ( spacetime ) origin of these laws . however , a complete geometrical understanding of black hole thermodynamics is not yet present . in general , a basic understanding of the thermodynamical properties of a system requires a specification of the system s ( dynamical ) degrees of freedom ( d.o.f . ) . obtaining such a specification is a nontrivial matter in quantum gravity . in the path - integral approach one avoids the discussion of the dynamical d.o.f .. there , the dominant contribution to the partition function comes from a saddle point , which is a classical euclidean solution @xcite . calculating the contribution of such a solution to the partition function does not require an identification of what the dynamical d.o.f.s of this solution are . though providing us with an elegant way of getting the laws of black hole thermodynamics , the path - integral approach does not give us the basic ( dynamical ) d.o.f . from which we can have a better geometrical understanding of the origin of black hole thermodynamics . it was only recently that the dynamical geometric d.o.f . for a spherically symmetric vacuum schwarzschild black hole were found @xcite under certain boundary conditions . in particular , by considering general foliations of the complete kruskal extension of the schrawzschild spacetime , kucha @xcite finds a reduced system of _ one _ pair of canonical variables that can be viewed as global geometric d.o.f .. one of these is the schwarzschild mass , while the other one , its conjugate momentum , is the difference between the parametrization times at right and left spatial infinities . using the approach of kucha , recently louko and whiting @xcite ( henceforth referred to as lw ) studied black hole thermodynamics in the hamiltonian formulation . as shown in fig . 2 , they considered a foliation in which the spatial hypersurfaces are restricted to lie in the right exterior region of the kruskal diagram and found the corresponding reduced phase space system . this enabled them to find the unconstrained hamiltonian ( which evolves these spatial hypersurfaces ) and canonically quantize this reduced theory . they then obtain the schrdinger time - evolution operator in terms of the reduced hamiltonian . the partition function @xmath0 is defined as the trace of the euclideanised time - evolution operator @xmath1 , namely , @xmath2 , where the hat denotes a quantum operator . this partition function has the same expression as the one obtained from the path - integral approach and expectedly yields the laws of black hole thermodynamics . in a standard thermodynamical system it is not essential to consider _ euclidean_-time action in order to study the thermodynamics . if @xmath3 is the lorentzian time - independent hamiltonian of the system , then the partition function is defined as @xmath4 where @xmath5 is the inverse temperature of the system in equilibrium . however , in many cases ( especially , in time- independent systems ) the euclidean time - evolution operator turns out to be the same as @xmath6 . nevertheless , there are cases where , as we will see in section [ subsec : lwham ] , the euclidean time - evolution operator is not the same as @xmath7 . this is the case for example in the lw approach , i.e. , @xmath8 , where @xmath9 is the reduced hamiltonian of the quantized lw system . there is a geometrical reason for this inequality and in this work we discuss it in detail . in this paper , we ask if there exists a hamiltonian @xmath3 ( which is associated with certain foliations of the schwarzschild spacetime ) appropriate for finding the partition function of a schwarzschild black hole enclosed inside a finite - sized box using ( [ partition - trace ] ) . such a procedure will not resort to euclideanization . in our quest to obtain the hamiltonian that is appropriate for defining the partition function for ( [ partition - trace ] ) , we also clarify the physical significance of the lw hamiltonian . by doing so we hope to achieve a better understanding of the geometrical origin of the thermodynamical aspects of a black hole spacetime . in a previous work @xcite , brown and york ( henceforth referred to as by ) found a general expression for the quasilocal energy on a timelike two - surface that bounds a spatial three - surface located in a spacetime region that can be decomposed as a product of a spatial three - surface and a real line interval representing time . from this expression they obtained the quasilocal energy inside a spherical box centered at the origin of a four - dimensional spherically symmetric spacetime . they argued that this expression also gives the correct quasilocal energy on a box in the schwarzschild spacetime . in this paper we show that , although their expression for the quasilocal energy on a box in the schwarzschild spacetime is correct , the analysis they use to obtain it requires to be extended when applied to the case of schwarzschild spacetime . in this case , one needs to impose extra boundary conditions at the timelike boundary inside the hole ( see fig . 3 ) . as mentioned above , in principle , one can use the hamiltonian @xmath10 so obtained to evaluate the partition function , @xmath11 . this partition function corresponds to the canonical ensemble and describes the thermodynamics of a system whose volume and temperature are fixed but whose energy content is permitted to vary . such a hamiltonian , @xmath10 would then lead to a description of black hole thermodynamics without any sort of euclideanisation . the only obstacle to this route to the partition function is that the trace can be evaluated only if one knows the density of the energy eigenstates . unfortunately , without knowing what the thermodynamical entropy of the system is , it is not clear how to find this density in terms of the reduced phase - space variables of kucha @xcite . so how can one derive the thermodynamical laws of the schwarzschild black hole using a lorentzian hamiltonian without knowing the density of states ? based on an observation that identifies the thermodynamical roles of the by and the lw hamiltonians we succeed in studying black hole thermodynamics within the hamiltonian formulation but without euclideanization . in section [ sec : thermoconsi ] we describe the thermodynamical roles of the by and the lw hamiltonians . identifying these roles allows us to immediately calculate the partition function and recover the thermodynamical properties of the schwarzschild black hole . in section [ sec : geoconsi ] we study the geometrical significance of these hamiltonians . in particular , we extend the work of brown and york @xcite to find the nature of the spatial slices that are evolved by the by hamiltonian in the full kruskal extension of the schwarzschild spacetime . in section [ geothermo ] we use the observations made in sections [ sec : thermoconsi ] and [ sec : geoconsi ] to ascribe geometrical basis to the thermodynamical parameters of the system , thus gaining insight into the geometrical nature of black hole thermodynamics . we conclude the paper in section [ sec : conclu ] by summarising our results and discussing the connection between the foliation geometry and equilibrium black hole thermodynamics . in appendix a we extend our results to the case of two - dimensional dilatonic black holes . in appendix b we discuss an alternative foliation ( see fig . 4 ) , in which the spatial slices are again evolved by the by hamiltonian @xmath12 . this illustrates the non - uniqueness of the foliation associated with the by hamiltonian . we shall work throughout in `` geometrized - units '' in which @xmath13 . it was shown by brown and york @xcite that in 4d spherically symmetric einstein gravity , the quasilocal energy of a system that is enclosed inside a spherical box of finite surface area and which can be embedded in an asymptotically flat space is @xcite @xmath14 where @xmath15 is the adm mass of the spacetime and @xmath16 is the fixed curvature radius of the box with its origin at the center of symmetry . we will call @xmath12 the brown - york hamiltonian . the brown and york derivation of the quasilocal energy can be summarised as follows . the system they consider is a spatial three - surface @xmath17 bounded by a two - surface @xmath18 in a spacetime region that can be decomposed as a product of a spatial three - surface and a real line interval representing time ( see fig . the time evolution of the two - surface boundary @xmath18 is the timelike three - surface boundary @xmath19 . they then obtain a surface stress - tensor on the boundary by taking the functional derivative of the action with respect to the three - metric on @xmath19 . the energy surface density is the projection of the surface stress tensor normal to a family of spacelike two - surfaces like @xmath18 that foliate @xmath19 . the integral of the energy surface density over such a two - surface @xmath18 is the quasilocal energy associated with a spacelike three - surface @xmath17 whose _ orthogonal _ intersection with @xmath19 is the two - boundary @xmath18 . as argued by by , eq . ( [ quasih ] ) also describes the total energy content of a box enclosing a schwarzschild hole . one would thus expect to obtain the corresponding partition function from it by the prescription @xmath11 . as mentioned above , the only obstacle to this calculation is the lack of knowledge about the density of states of the system , which is needed to evaluate the trace . however , as we discuss in the next subsection , there is another hamiltonian associated with the schwarzschild spacetime that allows us to obtain the relevant partition function without euclideanization . this is the louko - whiting hamiltonian . in their quest to obtain the partition function for the schwarzschild black hole in the hamiltonian formulation , lw found a hamiltonian that time - evolves spatial hypersurfaces in a schwarzschild spacetime of mass @xmath15 such that the hypersurfaces extend from the bifurcation 2-sphere to a timelike box - trajectory placed at a constant curvature radius of @xmath20 ( see fig . as we will show in the next subsection , the lw hamiltonian describes the correct free energy of a schwarzschild black hole enclosed inside a box in the thermodynamical picture . the lw hamiltonian is @xmath21 where generically @xmath22 and @xmath23 are functions of time @xmath24 , which labels the spatial hypersurfaces . physically , @xmath25 , where @xmath26 is the time - time component of the spacetime metric on the box . on the other hand , the physical meaning of @xmath23 is as follows . on a classical solution , consider the future timelike unit normal @xmath27 to a constant @xmath24 hypersurface at the bifurcation two - sphere ( see fig . then @xmath23 is the rate at which the constant @xmath24 hypersurfaces are boosted at the bifurcation 2-sphere : @xmath28 where @xmath29 is the initial hypersurface and @xmath30 is the boosted hypersurface . if one is restricted to a foliation in which the spatial hypersurfaces approach the box along surfaces of constant proper time on the box , then @xmath31 . on classical solutions , the spatial hypersurfaces approach the bifurcation 2-sphere along constant killing - time hypersurfaces ( see the paragraph containing eqs . ( [ ls - r ] ) in section [ subsec : lwgeoham ] ) . the lw fall - off conditions ( [ ls - r ] ) , which are imposed on the adm variables at the bifurcation 2-sphere , can be used to show that on solutions , @xmath32 , where @xmath33 is the killing time and @xmath34 is the surface gravity of a schwarzschild black hole . in the particular case where the label time @xmath24 is taken to be the proper time on the box , we have @xmath35^{-1 } \ \ .\ ] ] with such an identification of the label time @xmath24 , eq . ( [ lwht ] ) shows that on classical solutions the time - evolution of these spatial hypersurfaces is given by the hamiltonian @xmath36 where @xmath23 is given by ( [ n0sol ] ) . one may now ask if one can use the lw hamiltonian ( [ lwh ] ) to obtain a partition function for the system and study its thermodynamical properties . unfortunately , one can not do so in a straightforward manner . first , one can not replace @xmath3 in ( [ partition - trace ] ) by @xmath9 , the quantum counterpart of ( [ lwh ] ) , to obtain the partition function . the reason is that classically @xmath37 does not give the correct energy of the system ; the by hamiltonian @xmath12 of ( [ quasih ] ) does . to avoid this problem , lw first construct the schrdinger time - evolution operator @xmath38 . they then euclideanize this operator and use it to obtain the partition function @xmath39 . the partition function so obtained does not equal @xmath40 , but rather it turns out to be the same as that obtained via the path integral approach of gibbons and hawking @xcite . however , apart from this end result , a justification at some fundamental level has been lacking as to why the lw hamiltonian ( [ lwh ] ) and not any other hamiltonian ( eg . , ( [ quasih ] ) ) should be used to obtain the partition function using the lw procedure . in the next subsection , we will find the thermodynamical roles played by the by and lw hamiltonians . we will also show how this helps us in obtaining the partition function without euclideanization . this way we will avoid the ambiguity mentioned above that arises in the lw - method of constructing the partition function . as argued by brown and york @xcite , on solutions , the by hamiltonian @xmath12 in eq . ( [ quasih ] ) denotes the internal energy @xmath41 residing within the box : @xmath42 in fact eq . ( [ e ] ) can be shown to yield the first law of black hole thermodynamics @xmath43 where @xmath44 is the surface pressure on the box - wall @xcite @xmath45 the first term on the rhs of ( [ de ] ) is negative of the amount of work done by the system on its surroundings and , with hindsight , the second term is the product @xmath46 , where @xmath47 is the temperature and @xmath48 is the entropy of the system . we will not assume the latter in the following analysis ; rather we will deduce the form of @xmath47 and @xmath48 from first principles . we next show that the lw hamiltonian @xmath37 of eq . ( [ lwh ] ) plays the role of helmholtz free energy of the system . recall that the helmholtz free energy @xmath49 is defined as @xmath50 where @xmath41 is the internal energy . thus in an isothermal and reversible process , the first law of thermodynamics implies that the amount of mechanical work done by a system , @xmath51 , is equal to the decrease in its free energy , i.e. , @xmath52 as a corollary to this statement it follows that for a mechanically isolated system at a constant temperature , the state of equilibrium is the state of minimum free energy . we now show that under certain conditions on the foliation of the spacetime with spatial hypersurfaces , the lw hamiltonian @xmath37 in eq . ( [ lwh ] ) plays the role of _ free energy_. we choose a foliation such that on solutions the lapse @xmath23 obeys ( [ n0sol ] ) . using the expression ( [ e ] ) for @xmath41 , the hamiltonian @xmath37 in eq . ( [ lwh ] ) can be rewritten as @xmath53 now let us perturb @xmath37 about a solution by perturbing @xmath15 and @xmath16 such that @xmath23 itself is held fixed . then @xmath54 note that keeping @xmath23 fixed , i.e. , @xmath55 , does not necessarily imply through ( [ n0sol ] ) that @xmath56 and @xmath57 are not independent perturbations . this is because , in general , the perturbed @xmath37 may not correspond to a solution and hence the perturbed @xmath23 need not have the form ( [ n0sol ] ) . however , here we will assume that the perturbations do not take us off the space of static solutions and , therefore , the perturbed @xmath23 has the form ( [ n0sol ] ) . hence in our case @xmath58 and @xmath57 are not independent perturbations . using ( [ de ] ) and ( [ n0sol ] ) in ( [ dhitode ] ) yields @xmath59 finally , from ( [ dhdw ] ) and ( [ fw ] ) we get @xmath60 where @xmath61 is a constant independent of @xmath15 . to find @xmath61 , we take the limit @xmath62 . in this limit both @xmath37 and @xmath49 vanish and , therefore , @xmath61 has to be zero . another way to see that @xmath61 should vanish is to identify the geometric quantity @xmath23 with the temperature , @xmath47 , of the system ( up to a multiplicative constant ) . then the perturbation ( [ dhitode ] ) in @xmath37 , keeping @xmath23 ( and , therefore , @xmath47 ) fixed , describes an isothermal process . but eq . ( [ hfc ] ) shows that @xmath61 has to be an extensive function of thermodynamic invariants of the isothermal process since @xmath37 and @xmath49 are both extensive . the only thermodynamic quantity that we assume to be invariant in this isothermal process is the temperature @xmath47 . but since @xmath47 is not extensive , @xmath61 has to be zero . the fact that @xmath23 indeed determines the temperature of the system will be discussed in detail in a later section . the above proof of the lw hamiltonian being the helmholtz free energy immediately allows us to calculate the partition function for a canonical ensemble of such systems , @xmath63 by simply putting @xmath64 . in this way we recover the thermodynamical properties of a schwarzschild black hole without euclideanization . we will do so in detail in section [ geothermo ] but first we establish the geometrical significance of by and lw hamiltonians in the next section . a study of the geometrical roles of the by and lw hamiltonians provides the geometrical basis for the thermodynamical parameters associated with a black hole that were discussed in the preceeding section . in this section we begin by setting up the hamiltonian formulation appropriate for the two sets of boundary conditions that lead to the by and lw hamiltonians as being the unconstrained hamiltonians that generate time - evolution of foliations in schwarzschild spacetime . the notation follows that of kucha @xcite and lw . a general spherically symmetric spacetime metric on the manifold @xmath65 can be written in the adm form as @xmath66 where @xmath67 , @xmath68 , @xmath69 and @xmath70 are functions of @xmath24 and @xmath71 only , and @xmath72 is the metric on the unit two - sphere . we will choose our boundary conditions in such a way that the radial proper distance @xmath73 on the constant @xmath24 surfaces is finite . this implies that the radial coordinate @xmath71 have a finite range , which we take to be @xmath74 $ ] , without any loss of generality . the spatial metric and the spacetime metric will be assumed to be nondegenerate , in particular , @xmath69 , @xmath70 , and @xmath67 are taken to be positive . for the metric ( [ 4-metric ] ) , the einstein - hilbert action is @xmath75 & & \nonumber \\ = \int dt \int_0 ^ 1 d r \ , \bigg [ & & -n^{-1 } \left ( r \bigl ( - { \dot \lambda } + ( \lambda n^r ) ' \bigr ) ( - { \dot r } + r ' n^r ) + \case{1}{2 } \lambda { ( - { \dot r } + r ' n^r ) } ^2 \right ) \nonumber \\ & & + n \left ( - \lambda^{-1 } r r '' + \lambda^{-2 } r r ' \lambda ' - \case{1}{2 } \lambda^{-1 } { r'}^{2 } + \case{1}{2 } \lambda \right ) \bigg ] \ \ , \label{s - lag}\end{aligned}\ ] ] where the subscript @xmath17 denotes that @xmath76 is a hypersurface action that is defined only up to the possible addition of boundary terms . above , the overdot and the prime denote @xmath77 and @xmath78 , respectively . the equations of motion derived from ( [ s - lag ] ) are the full einstein equations for the metric ( [ 4-metric ] ) , and they imply that every classical solution is part of a maximally extended schwarzschild spacetime , where the value of the schwarzschild mass @xmath79 may be positive , negative , or zero . in what follows , we will choose our boundary conditions such that @xmath80 . we shall discuss the boundary conditions and the boundary terms after passing to the hamiltonian formulation . the momenta conjugate to @xmath69 and @xmath70 are found from the lagrangian action ( [ s - lag ] ) to be @xmath81 a dual - legendre transformation then yields the hamiltonian action @xmath82 = \int dt \int_0 ^ 1 dr \left ( p_\lambda { \dot \lambda } + p_r { \dot r } - nh - n^r h_r \right ) \ \ , \label{s - ham}\ ] ] where the super - hamiltonian @xmath83 and the radial supermomentum @xmath84 are @xmath85 it can be verified that the poisson brackets of the constraints close according to the radial version of the dirac algebra @xcite . we next consider the boundary terms that must be added to the hypersurface action ( [ canhypaction ] ) for the total action to yield , upon variation , only a volume term corresponding to the equations of motion . however , the boundary terms depend intricately on the choice of the spacetime foliation . different foliations require different boundary conditions on the geometric variables in the variational analysis , thus requiring the addition of different boundary terms to ( [ canhypaction ] ) . as is well known in general relativity , it is these boundary terms that determine the true hamiltonian of the system . hence , as we show below explicitly , this implies that different foliations correspond to different hamiltonians , which are the generators of time - evolution of the spatial slices in the foliations . in general , the analysis of brown and york ( see subsection [ subsec : byham ] ) breaks down in cases where spacetime regions of non - trivial topologies are enclosed inside the spherical box , particularly so in the case of a schwarzschild black hole enclosed inside the box . as we show below , in this case one is forced to introduce an `` inner''-boundary where the spatial hypersurfaces of brown and york must extend to . this fact becomes transparent when one looks at the full kruskal extension of the schwarzschild metric . there , one begins by performing a @xmath86 decomposition of the spacetime in terms of a one - parameter family of spatial hypersurfaces . the kruskal diagram ( see figs . 2 and 3 ) shows that any such foliation would necessarily require two boundaries : an outer boundary and an inner boundary . the hamiltonian that evolves these spatial hypersurfaces in time will in general depend on the boundary conditions specified on these 2-boundaries . in this section we show that the spatial hypersurfaces that are evolved by the hamiltonian corresponding to the quasilocal energy , given in ( [ quasih ] ) , are ones that extend from the box ( the outer timelike boundary ) , on the right end , to an inner timelike boundary located completely inside the dynamical region of the kruskal diagram , on the left end . we first find the boundary conditions and the foliation that correspond to time evolution by the by hamiltonian and later we compare these with those corresponding to the lw hamiltonian . at @xmath87 , we prescribe the following fall - off conditions [ s - r ] @xmath88 where @xmath89 and @xmath90 are positive . this ensures that on classical solutions @xmath79 is positive . also @xmath91 . here @xmath92 stands for a term whose magnitude at @xmath93 is bounded by @xmath94 times a constant , and whose @xmath95th derivative at @xmath93 is similarly bounded by @xmath96 times a constant for @xmath97 . the fact that the shift @xmath98 vanishes as @xmath99 implies that on solutions , the inner timelike boundary lies along a constant killing - time surface located completely in the past and the future dynamical regions and cutting across the bifurcation two - sphere . also , on solutions , the variable @xmath90 corresponds to the throat - radius . it is straightforward to verify that the conditions ( [ s - r ] ) are consistent with the equations of motion : provided that the constraints obey @xmath100 and the fall - off conditions ( [ s - r - lambda])([s - r - pr ] ) hold for the initial data , and provided that the lapse and shift satisfy ( [ s - r - n ] ) and ( [ s - r - nr ] ) , it then follows that the fall - off conditions ( [ s - r - lambda])([s - r - pr ] ) are preserved in time by the time - evolution equations . on the other hand , at @xmath101 , the boundary conditions are as follows : we fix @xmath70 and @xmath102 to be prescribed positive - valued functions of @xmath24 . this means fixing the metric on the three - surface @xmath101 to be timelike . in the classical solutions , the surface @xmath101 is located in the right exterior region of the kruskal extension of the schwarzschild spacetime . we now give an action principle appropriate for these boundary conditions . to begin , note that the surface action @xmath103 $ ] in ( [ s - ham ] ) is well defined under the above conditions . consider the total action @xmath104 = s_\sigma [ \lambda , r , p_\lambda , p_r ; n , n^r ] + s_{\partial\sigma } [ \lambda , r , p_\lambda , p_r ; n , n^r ] \ \ , \label{s - total}\ ] ] where the boundary action is given by @xmath105 \nonumber \\ & & = \int dt { \biggl [ n r r ' \lambda^{-1 } - n^r \lambda p_\lambda - \case{1}{2 } r { \dot r } \ln \left| { n + \lambda n^r \over n - \lambda n^r } \right| \biggr]}_{r=1 } \ \ , \label{s - boundary}\end{aligned}\ ] ] where @xmath106_a$ ] is value of the _ term _ evaluated at @xmath107 . the variation of the total action ( [ s - total ] ) can be written as a sum of a volume term proportional to the equations of motion , boundary terms from the initial and final spatial surfaces , and boundary terms from @xmath87 and @xmath101 . the boundary terms from the initial and final spatial surfaces take the usual form @xmath108 with the upper ( lower ) sign corresponding to the final ( initial ) surface . these terms vanish provided we fix the initial and final three - metrics . the boundary term from @xmath87 vanishes under the fall - off conditions specified in ( [ s - r ] ) . as will be shown in subsection [ subsec : reduction ] , this is crucial in obtaining a reduced hamiltonian that corresponds to the correct quasilocal energy . the boundary term from @xmath101 reads @xmath109}_{r=1 } \ \ . \label{bt-1}\end{aligned}\ ] ] since @xmath70 and @xmath110 are fixed at @xmath101 , the first three terms in ( [ bt-1 ] ) vanish . the integrand in the last term in ( [ bt-1 ] ) is proportional to the equation of motion ( [ plambda ] ) , which is classically enforced for @xmath111 by the volume term in the variation of the action . therefore , for classical solutions , also the last term in ( [ bt-1 ] ) will vanish by continuity . thus the action ( [ s - total ] ) is appropriate for a variational principle which fixes the initial and final three - metrics , and the three - metric on the timelike boundary at @xmath101 . each classical solution belongs to that region of a kruskal diagram that lies within two timelike boundaries such that the inner boundary lies along a constant killing - time surface located completely in the dynamical regions and the outer boundary is a timelike surface located in the right exterior region ( see fig . the constant @xmath24 slices are spacelike everywhere between the two timelike boundaries . to obtain the reduced action and extract the unconstrained hamiltonian system one needs to first solve the constraints ( [ superham ] ) and ( [ supermom ] ) . in the following , we will follow kucha s way of handling the constraints @xcite . it was shown by kucha that in the context of a vacuum schwarzschild spacetime ( in the absence of timelike boundaries ) there exists a set of new variables , which are related to the adm variables through a canonical transformation , such that in terms of the new variables the constraints are remarkably simple and solvable . this allows one to perform a hamiltonian reduction . in this section we show that the canonical transformation given by kucha from the adm variables @xmath112 to the new variables @xmath113 is readily adapted to our boundary conditions . as mentioned earlier , the boundary conditions ensure that @xmath80 . recall from @xcite that the new variables @xmath113 are defined by [ trans ] @xmath114 where @xmath115 in the classical solution , @xmath79 is the value of the schwarzschild mass and @xmath116 is the derivative of the killing time with respect to @xmath71 . a pair of quantities which will become new lagrange multipliers are defined by [ n - def ] @xmath117 using arguments similar to kucha and lw , it can be shown that under our boundary conditions the transformation ( [ trans ] ) is a canonical transformation , which is also invertible @xcite . the hamiltonian action ( [ s - ham ] ) can now be written in terms of the new variables . using eqs . ( [ n - def ] ) , one sees that the constraint terms @xmath118 in the old surface action ( [ s - ham ] ) take the form @xmath119 . thus the new surface action is @xmath120 = \int dt \int _ { 0}^{1 } dr \left ( p_m { \dot m } + p_{\sf r } \dot{\sf r } + 4 { \sf n } m m ' -n^{\sf r } p_{\sf r } \right ) \ \ , \label{s2-ham}\ ] ] where the quantities to be varied independently are @xmath79 , @xmath121 , @xmath122 , @xmath123 , @xmath124 , and @xmath125 . the complete set of equations of motion is [ eom2 ] @xmath126 we now turn to the boundary conditions and boundary terms . as a preparation for this , let us denote by @xmath127 the quantity @xmath128 when expressed as a function of the new canonical variables and lagrange multipliers . a short calculation using ( [ trans])([n - def ] ) yields @xmath129 in general , @xmath127 need not be positive for all values of @xmath71 , even for classical solutions . however , as in subsection [ subsec : bygeo ] , we shall introduce boundary conditions that fix the intrinsic metric on the three - surface @xmath101 to be timelike , and under such boundary conditions @xmath127 is positive at @xmath101 . consider now the total action @xmath130 = s_\sigma [ m , { \sf r } , p_m , p_{\sf r } ; { \sf n } , n^{\sf r } ] + s_{\partial\sigma } [ m , { \sf r } , p_m , p_{\sf r } ; { \sf n } , n^{\sf r } ] \ \ , \label{s2-total}\ ] ] where the boundary action is given by @xmath131 \nonumber \\ & = & \int dt { \left [ { \sf r } \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \case{1}{2 } { \sf r } \dot{\sf r } \ln \left ( { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right ) \vphantom { { \left| { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right|}^q_q } \right]}_{r=1 } \label{s2-boundary}\end{aligned}\ ] ] with @xmath132 . note that the argument of the logarithm in ( [ s2-boundary ] ) is always positive . the variation of ( [ s2-total ] ) contains a volume term proportional to the equations of motion , as well as several boundary terms . from the initial and final spatial surfaces one gets the usual boundary terms @xmath133 which vanish provided we fix @xmath79 and @xmath121 on these surfaces . similarly one can show that with our choice of boundary conditions ( given in section [ subsec : bygeo ] ) the remaining boundary terms from the timelike surfaces at @xmath87 and @xmath101 also vanish . we now concentrate on the variational principle associated with the action @xmath134 $ ] ( [ s2-total ] ) . we shall reduce the action to the true dynamical degrees of freedom by solving the constraints . the constraint @xmath135 ( [ eom2-mm ] ) implies that @xmath79 is independent of @xmath71 . we can therefore write @xmath136 substituting this and the constraint @xmath137 ( [ eom2-psfr ] ) back into ( [ s2-total ] ) yields the true hamiltonian action @xmath138 = \int dt \left ( { \bf p } { \dot { \bf m } } - { \bf h } \right ) \ \ , \label{s - red}\ ] ] where @xmath139 the reduced hamiltonian @xmath140 in ( [ s - red ] ) takes the form @xmath141 here @xmath142 and @xmath143 are the values of @xmath121 and @xmath127 at the timelike boundary @xmath101 , and @xmath144 . as mentioned before , @xmath142 and @xmath143 are prescribed functions of time , satisfying @xmath145 and @xmath146 . note that @xmath140 is , in general , explicitly time - dependent . the variational principle associated with the reduced action ( [ s - red ] ) fixes the initial and final values of @xmath15 . the equations of motion are [ red - eom ] @xmath147 equation ( [ red - eom1 ] ) is readily understood in terms of the statement that @xmath15 is classically equal to the time - independent value of the schwarzschild mass . to interpret equation ( [ red - eom2 ] ) , recall from sec . [ subsec : transformation ] that @xmath116 equals classically the derivative of the killing time with respect to @xmath71 , and @xmath148 therefore equals by ( [ bfp ] ) the difference of the killing times at the left and right ends of the constant @xmath24 surface . as the constant @xmath24 surface evolves in the schwarzschild spacetime , ( [ red - eom2 ] ) gives the negative of the evolution rate of the killing time at the right end of the spatial surface where it terminates at the outer timelike boundary at @xmath101 . note that @xmath149 gets no contribution from the inner timelike boundary located at @xmath87 in the dynamical region . this is a consequence of the fall - off conditions ( [ s - r ] ) which ensure that on solutions , the rate of evolution of the killing time at @xmath87 is zero . the case of interest is when the radius of the ` outer ' boundary two - sphere does not change in time , i.e. , @xmath150 . in that case the second term in @xmath151 ( [ bfhb ] ) vanishes , and @xmath149 in ( [ red - eom2 ] ) is readily understood in terms of the killing time of a static schwarzschild observer , expressed as a function of the proper time @xmath152 and the blueshift factor @xmath153 . the reduced hamiltonian is given by @xmath154 where @xmath16 is the time - independent value of @xmath142 . unfortunately , the above hamiltonian does not vanish as @xmath15 goes to zero . the situation is remedied by adding the @xmath155 term of gibbons and hawking @xcite to @xmath140 . physically , this added term arises from the extrinsic curvature of the ` outer ' boundary two - sphere when embedded in flat spacetime . with the added term , the hamiltonian becomes @xmath156 this is the quasilocal energy of brown and york when @xmath31 . the choice of @xmath22 determines the choice of time in the above hamiltonian . setting @xmath157 geometrically means choosing a spacetime foliation in which the rate of evolution of the spatial hypersurface on the box is the same as that of the proper time . then the new hamiltonian is @xmath158 namely , the quasilocal energy ( [ quasih ] ) in schwarzschild spacetime . in the next section , we discuss the geometric relevance of the lw hamiltonian that , as we showed earlier , yields the correct free energy of the system . we now summarize the lw choice of the foliation of the schwarzschild spacetime , state the corresponding boundary conditions they imposed , and briefly mention how they obtain their reduced hamiltonian . the main purpose of this section is to facilitate a comparison between the lw boundary conditions and our choice of the boundary conditions ( as discussed in the preceeding subsections ) that yield the by hamiltonian . as shown in fig . 2 , lw considered a foliation in which the spatial hypersurfaces are restricted to lie in the right exterior region of the kruskal diagram . each spatial hypersurface in this region extends from the box at the right end up to the bifurcation 2-sphere on the left end . the boundary conditions imposed by lw are as follows . at @xmath93 , they adopt the fall - off conditions [ ls - r ] @xmath159 where @xmath89 and @xmath90 are positive , and @xmath91 . equations ( [ ls - r - lambda ] ) and ( [ ls - r - r ] ) imply that the classical solutions have a positive value of the schwarzschild mass , and that the constant @xmath24 slices at @xmath93 are asymptotic to surfaces of constant killing time in the right hand side exterior region in the kruskal diagram , all approaching the bifurcation two - sphere as @xmath93 . the spacetime metric has thus a coordinate singularity at @xmath93 , but this singularity is quite precisely controlled . in particular , on a classical solution the future unit normal to a constant @xmath24 surface defines at @xmath93 a future timelike unit vector @xmath160 at the bifurcation two - sphere of the schwarzschild spacetime , and the evolution of the constant @xmath24 surfaces boosts this vector at the rate given by @xmath161 at @xmath101 , we fix @xmath70 and @xmath102 to be prescribed positive - valued functions of @xmath24 . this means fixing the metric on the three - surface @xmath101 , and in particular fixing this metric to be timelike . in the classical solutions , the surface @xmath101 is located in the right hand side exterior region of the kruskal diagram . to obtain an action principle appropriate for these boundary conditions , consider the total action @xmath104 = s_\sigma [ \lambda , r , p_\lambda , p_r ; n , n^r ] + s_{\partial\sigma } [ \lambda , r , p_\lambda , p_r ; n , n^r ] \ \ , \label{ls - total}\ ] ] where the boundary action is given by @xmath105 \nonumber \\ & & = \case{1}{2 } \int dt \ , { \left [ r^2 n ' \lambda^{-1 } \right]}_{r=0 } \ ; \ ; + \int dt { \biggl [ n r r ' \lambda^{-1 } - n^r \lambda p_\lambda - \case{1}{2 } r { \dot r } \ln \left| { n + \lambda n^r \over n - \lambda n^r } \right| \biggr]}_{r=1 } \ \ . \label{ls - boundary}\end{aligned}\ ] ] the variation of the total action ( [ ls - total ] ) can be written as a sum of a volume term proportional to the equations of motion , boundary terms from the initial and final spatial surfaces , and boundary terms from @xmath87 and @xmath101 . to make the action ( [ ls - total ] ) appropriate for a variational principle , one fixes the initial and final three - metrics , the box - radius @xmath70 , and the three - metric on the timelike boundary at @xmath101 . these are similar to the boundary conditions that we imposed to obtain the by hamiltonian . however , for the lw boundary conditions , one has to also fix the quantity @xmath162 at the bifurcation 2-sphere . each classical solution is part of the right hand exterior region of a kruskal diagram , with the constant @xmath24 slices approaching the bifurcation two - sphere as @xmath93 , and @xmath163 giving via ( [ n - boost ] ) the rate of change of the unit normal to the constant @xmath24 surfaces at the bifurcation two - sphere . although we are here using geometrized units , the argument of the @xmath164 in ( [ n - boost ] ) is a truly dimensionless boost parameter " even in physical units . to obtain the ( reduced ) lw hamiltonian , one needs to solve the super - hamiltonian and the supermomentum constraints . just as in the case of the by hamiltonian ( see subsection [ subsec : transformation ] ) , it helps to first make a canonical transformation to the kucha variables ( see lw for details ) . after solving the constraints , one obtains the following reduced action @xmath165 = \int dt \left ( { \bf p } { \dot { \bf m } } - { \bf h } \right ) \ \ , \label{ls - red}\ ] ] where @xmath166 and the reduced hamiltonian @xmath167 in ( [ ls - red ] ) is @xmath168 which is the same as the one given in eq . ( [ lwht ] ) . in obtaining the above reduced form @xmath169 , we have assumed that the box - radius is constant in time , @xmath170 , just as we did in obtaining the by hamiltonian ( [ byhq ] ) . having established the geometrical significance of the by and lw hamiltonians , the basis for their thermodynamical roles becomes apparent . we showed that the by hamiltonian evolves spatial hypersurfaces in such a way that they span the spacetime region both inside and outside the event horizon . this is what one would expect from the fact that it corresponds to the quasilocal energy of the complete spacetime region enclosed inside the box . on the other hand , the lw hamiltonian evolves spatial slices that are restricted to lie outside the event horizon . with our choice of the boost parameter @xmath23 , this corresponds to the helmholtz free energy of the system , which is less than the quasilocal energy : this is expected since the lw slices span a smaller region of the spacetime compared to the by slices . also , the fact that the lw slices are limited to lie outside the event horizon implies that the energy on these slices can be harnessed by an observer located outside the box . this is consistent with the fact that it corresponds to the helmholtz free energy of the system which is the amount of energy in a system that is available for doing work by the system on its surroundings . using the thermodynamical roles played by the lw hamiltonian and the by hamiltonian ( see section [ sec : thermoconsi ] ) , we now derive , at the classical level , many of the thermodynamical quantities associated with the schwarzschild black hole enclosed inside a box . we begin by finding the temperature on the box . from eq . ( [ lwh ] ) , the helmholtz free energy is @xmath171 the above equation , along with eqs . ( [ f ] ) and ( [ e ] ) , implies that @xmath172 or , @xmath173 where @xmath174 . equation ( [ s1 ] ) gives an expression for the entropy in terms of the geometrical quantity @xmath23 . on the other hand one can find @xmath48 also from the thermodynamic identity @xmath175 where @xmath0 is the partition function defined by eq . ( [ partitionh ] ) and eq . ( [ lwhf1 ] ) . ( [ s2 ] ) gives the entropy to be @xmath176 the above equation gives another expression for the entropy , now in terms of the derivative of @xmath23 . comparing eqs . ( [ s1 ] ) and ( [ s3 ] ) we find @xmath177 where @xmath178 is some undetermined quantity that is independent of @xmath5 . the exact form of @xmath47 as a function of @xmath15 is found by noting that the free energy @xmath49 should be a minimum at equilibrium . since in a canonical ensemble the box - radius @xmath70 and the temperature @xmath47 ( which is proportional to @xmath23 ) are fixed , the only quantity in @xmath49 that can vary is @xmath15 . thus the question we ask is the following : for a fixed value of the curvature radius @xmath20 and the boost parameter @xmath23 , what is the value of @xmath15 that minimizes @xmath49 ? > from the expression for @xmath49 in eq . ( [ lwhf1 ] ) one finds this value of @xmath15 , to be a function of @xmath23 . inverting this relation gives @xmath179 > from ( [ n0beta ] ) and ( [ n0 m ] ) we find that the equilibrium temperature on a box of radius @xmath16 enclosing a black hole of mass @xmath15 obeys @xmath180 in agreement with known results . significantly , eq . ( [ n0beta ] ) shows that the equilibrium temperature geometrically corresponds to a particular value of the boost parameter . > from eqs . ( [ s1 ] ) and ( [ n0beta ] ) we find that the entropy of a schwarzschild black hole is quadratic in its mass . unfortunately , in this formalism one can not determine the correct constants of proportionality in @xmath48 and @xmath47 . however , notice that our derivation is purely classical . although simple mathematically , this derivation is incomplete due to the lack of the constant of proportionality @xmath178 in eq . ( [ n0beta ] ) . the correct value for this constant , @xmath181 , can be obtained only from a quantum treatment . finally , we note that for the spatial slices that obey @xmath182 , the free energy can be obtained from ( [ lwh ] ) to be @xmath183 the above equation shows that if the radius of the box @xmath16 is kept fixed , then the free energy of the system is minimum for the configuration with a black hole of mass @xmath184 . in this work our goal was to seek a geometrical basis for the thermodynamical aspects of a black hole . we find that the value of the brown - york hamiltonian can be interpreted as the internal energy of a black hole inside a box . whereas the value of the louko - whiting hamiltonian gives the helmholtz free energy of the system . after finding these thermodynamical roles played by the by and lw hamiltonians , we ask what the geometrical significance of these hamiltonians is . in this regard the geometrical role of the lw hamiltonian was already known . it was recently shown by lw that their hamiltonian evolves spatial hypersurfaces in a special foliation of the kruskal diagram . the characteristic feature of this foliation is that it is limited to only the right exterior region of this spacetime ( see fig . 2 ) and the spatial hypersurfaces are required to converge onto the bifurcation 2-sphere , which acts as their inner boundary ( the box itself being the outer boundary ) . on the other hand , the geometrical significance of the by hamiltonian as applied to the black hole case was not fully known , although it had been argued that its value is the energy of the schwarzschild spacetime region that is enclosed inside a spherical box . in this work we establish the geometric role of the by hamiltonian by showing that it is the generator of time - evolution of spatial hypersurfaces in certain foliations of the schwarzschild spacetime . establishing the thermodynamic connection of the by and lw hamiltonians allowed us to obtain a geometrical interpretation for the equilibrium temperature of a black hole enclosed inside a box , i.e. , as measured by a stationary observer on the box . geometrically , the temperature turns out to be the rate at which the lw spatial hypersurfaces are boosted at the bifurcation 2-sphere . one could however ask what happens if the lw hypersurfaces are evolved at a different rate , i.e. , if the label time @xmath24 is chosen to be boosted with respect to the proper time of a stationary observer on the box . in that case , it can be shown that the by hamiltonian and the rate at which the lw hypersurfaces are evolved at the bifurcation 2-sphere get `` blue - shifted '' by the appropriate boost - factor . on the other hand , the entropy of the system can still be interpreted as the change in free energy per unit change in the temperature of the system . we thank abhay ashtekar , viqar husain , eric martinez , jorg pullin , lee smolin , and jim york for helpful discussions . we would especially like to thank jorma louko for critically reading the manuscript and making valuable comments . financial support from iucaa is gratefully acknowledged by one of us ( sb ) . this work was supported in part by nsf grant no . phy-95 - 07740 . the approach we describe above in studying the thermodynamics of 4d spherically symmetric einstein gravity can also be extended to the case of the 2d vacuum dilatonic black hole @xcite in an analogous fashion . in the case of a 2d black hole , the event horizon is located at a curvature radius @xmath185 , where @xmath186 is a positive constant that sets the length - scale in the 2d models . the quasilocal energy of a system comprising of such a black hole in the presence of a timelike boundary situated at a curvature radius @xmath16 can be shown to be @xmath187 which strongly resembles the 4d counterpart in ( [ quasih ] ) . @xmath188 evolves constant @xmath24 spatial hypersurfaces that extend from an inner timelike boundary lying on a constant killing - time surface in the dynamical region up to a timelike boundary ( the box ) placed in the right exterior region ( see fig . 3 ) . the hamiltonian that evolves the two - dimensional counterpart of the louko and whiting spatial slices that extend from the bifurcation point up to the box ( see fig . 2 ) is @xmath189 where in general @xmath22 and @xmath190 are functions of time . the above hamiltonian @xmath191 was found in ref . there it was found that @xmath192 is the rate at which the spatial hypersurface are boosted at the bifurcation point . on the other hand , @xmath193 , @xmath26 being the time - time component of the spacetime metric on the box . if one restricts the spatial hypersurfaces to approach the box along constant proper - time hypersurfaces , then , as in 4d , @xmath31 . using fall - off conditions on the adm variables at the bifurcation point analogous to the lw fall - off conditions ( [ ls - r ] ) , it can be shown that on solutions @xcite @xmath194 where @xmath33 is the killing time , and @xmath195 is the surface gravity of a witten black hole . the time - evolution of these restricted spatial hypersurfaces is given by the hamiltonian @xmath196 like the 4d case , here too it can be shown that @xmath188 is analogous to the internal energy , whereas @xmath197 denotes the helmholtz free energy of the 2d system . a similar analysis also shows that @xmath198 and @xmath199 which is inverse of the blue - shifted temperature on the box . the temperature of a 2d black hole at infinity on the other hand is @xmath200 , which is independent of the black hole mass . in section [ sec : thermoconsi ] , we found a choice of spatial hypersurfaces that were evolved by the by hamiltonian under a specific set of boundary conditions . in this appendix we find a different choice of spatial hypersurfaces , i.e. , with a different inner boundary , that is evolved by the by hamiltonian under a different set of boundary conditions . we begin by stating the boundary conditions and specifying the spacetime foliation they define . at the inner boundary @xmath87 , we fix @xmath70 and @xmath201 to be prescribed positive - valued functions of @xmath24 . this means fixing the metric on the three - surface @xmath87 , and in particular fixing this metric to be spacelike there . on the other hand , at @xmath101 , we fix @xmath70 and @xmath102 to be prescribed positive - valued functions of @xmath24 . this means fixing the metric on the three - surface @xmath101 to be timelike . in the classical solutions , the surface @xmath101 is located in the right exterior region of the kruskal diagram . we now wish to give an action principle appropriate for these boundary conditions . note that the surface action @xmath103 $ ] given by eq . ( [ s - ham ] ) is well defined under the above conditions . consider the total action @xmath104 = s_\sigma [ \lambda , r , p_\lambda , p_r ; n , n^r ] + s_{\partial\sigma } [ \lambda , r , p_\lambda , p_r ; n , n^r ] \ \ , \label{app : s - total}\ ] ] where the boundary action is given by @xmath105 \nonumber \\ & & = \int dt { \biggl [ n r r ' \lambda^{-1 } - n^r \lambda p_\lambda - \case{1}{2 } r { \dot r } \ln \left| { n + \lambda n^r \over n - \lambda n^r } \right| \biggr]}_{r=0}^{r=1 } \ \ , \label{app : s - boundary}\end{aligned}\ ] ] where @xmath106_a^b$ ] implies the difference in the values of the _ term _ evaluated at @xmath202 and at @xmath107 . the variation of the total action ( [ app : s - total ] ) can be written as a sum of a volume term proportional to the equations of motion , boundary terms from the initial and final spatial surfaces , and boundary terms from @xmath87 and @xmath101 . the boundary terms from the initial and final spatial surfaces take the usual form @xmath203 with the upper ( lower ) sign corresponding to the final ( initial ) surface . these terms vanish provided we fix the initial and final three - metrics . the boundary term from @xmath87 and @xmath101 read @xmath109}_{r=0}^{r=1 } \ \ , \label{app : bt-1}\end{aligned}\ ] ] where @xmath106^b_a$ ] implies the difference between the values of the _ term _ evaluated at @xmath202 and at @xmath107 . as @xmath70 and @xmath110 are fixed at @xmath87 and @xmath101 , the first three terms in ( [ app : bt-1 ] ) vanish . the integrand in the last term in ( [ app : bt-1 ] ) is proportional to the equation of motion ( [ plambda ] ) , which is classically enforced for @xmath111 by the volume term in the variation of the action . therefore , for classical solutions , also the last term in ( [ app : bt-1 ] ) will vanish by continuity . we thus conclude that the action ( [ app : s - total ] ) is appropriate for a variational principle which fixes the initial and final three - metrics , the three - metric on the spacelike boundary at @xmath87 and the three - metric on the timelike boundary at @xmath101 . each classical solution belongs to that region of a kruskal diagram that lies to the future of the null line at killing time @xmath204 . the constant @xmath24 slices are spacelike everywhere between the spacelike boundary at @xmath87 , and the timelike boundary at @xmath101 . the canonical transformation given in kucha @xcite from the variables @xmath112 to the new variables @xmath113 is readily adapted to our boundary conditions . as mentioned earlier , we shall assume that @xmath80 . recall that the new variables @xmath113 have been defined in subsection [ subsec : transformation ] by equations ( [ trans ] ) and ( [ f - def ] ) . the new lagrange multipliers are defined in eq . ( [ n - def ] ) . it can be shown that the transformation ( [ trans ] ) is a canonical transformation also under the new boundary conditions being considered in this appendix . we wish to write an action in terms of the new variables . using eqs . ( [ n - def ] ) , one finds that the constraint terms @xmath118 in the old surface action ( [ s - ham ] ) take the form @xmath119 and the new surface action is the same as that given in ( [ s2-ham ] ) . therefore , the equations of motion remain unchanged and are given by ( [ eom2 ] ) . we now turn to the boundary conditions and boundary terms . as before , we define @xmath205 in general , @xmath127 need not be positive for all values of @xmath71 , even for classical solutions . as in the preceeding section , we shall introduce boundary conditions that fix the intrinsic metric of the three - surfaces @xmath87 and @xmath101 to be spacelike and timelike , respectively , and under such boundary conditions @xmath127 is negative at @xmath87 but positive at @xmath101 . from ( [ app : q2 ] ) it is then seen that @xmath124 is nonzero at @xmath87 and @xmath101 . recalling that we are assuming @xmath206 , eq . ( [ sfn - def ] ) shows that @xmath124 is positive at @xmath101 for classical solutions with the schwarzschild slicing , since in this slicing one has @xmath207 . continuity then implies that @xmath124 must be positive at @xmath101 for all classical solutions compatible with our boundary conditions . on the other hand , at @xmath87 we now put the additional condition that @xmath208 ( or , equivalently , @xmath209 ) . on classical solutions , this extra condition restricts the surface @xmath87 to lie either in the past or the future dynamical region of the schwarzschild spacetime . although the final expression for the quasilocal energy is independent of this choice , for definiteness we will choose the spatial boundary at @xmath87 to lie in the future dynamical region ( see fig . 4 ) . then at @xmath87 , eq . ( [ sfn - def ] ) shows that @xmath124 has to be negative because @xmath210 there . we can therefore , without loss of generality , choose to work in a neighborhood of the classical solutions such that @xmath124 is positive at @xmath101 whereas @xmath124 is negative at @xmath87 . consider now the total action @xmath130 = s_\sigma [ m , { \sf r } , p_m , p_{\sf r } ; { \sf n } , n^{\sf r } ] + s_{\partial\sigma } [ m , { \sf r } , p_m , p_{\sf r } ; { \sf n } , n^{\sf r } ] \ \ , \label{app : s2-total}\ ] ] where the boundary action is given by @xmath131 \nonumber \\ & = & \int dt { \left [ { \sf r } \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \case{1}{2 } { \sf r } \dot{\sf r } \ln \left ( { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right ) \vphantom { { \left| { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right|}^q_q } \right]}_{r=1 } \nonumber \\ & -&\int dt { \left [ { \sf r } \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \case{1}{2 } { \sf r } \dot{\sf r } \ln \left ( { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right ) \vphantom { { \left| { \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } - \dot{\sf r } \over \sqrt{{\sf f}q^2 + { \dot{\sf r}}^2 } + \dot{\sf r } } \right|}^q_q } \right]}_{r=0 } \label{app : s2-boundary}\end{aligned}\ ] ] with @xmath132 . note that the argument of the logarithm in ( [ app : s2-boundary ] ) is always positive . the variation of ( [ app : s2-total ] ) contains a volume term proportional to the equations of motion , as well as several boundary terms . these boundary terms vanish if on the initial and final three - surfaces we fix the new canonical coordinates @xmath79 and @xmath121 , and at @xmath87 and @xmath101 we fix @xmath121 and the intrinsic metric on these three - surfaces . we now reduce the action @xmath134 $ ] ( [ s2-total ] ) to the true dynamical degrees of freedom by solving the constraints ( [ eom2-mm ] ) and ( [ eom2-psfr ] ) as before . this gives the true hamiltonian action to be @xmath138 = \int dt \left ( { \bf p } { \dot { \bf m } } - { \bf h } \right ) \ \ , \label{app : s - red}\ ] ] where @xmath148 and @xmath15 are defined as in section [ subsec : reduction ] . the reduced hamiltonian @xmath167 in ( [ app : s - red ] ) takes the form @xmath211 with here @xmath142 ( @xmath213 ) and @xmath143 ( @xmath214 ) are the values of @xmath121 and @xmath127 at the timelike ( spacelike ) boundary @xmath101 ( @xmath87 ) , and @xmath144 . @xmath142 , @xmath143 , @xmath213 , and @xmath214 are considered to be prescribed functions of time , satisfying @xmath215 and @xmath216 . note that @xmath140 is , in general , explicitly time - dependent . the interpretation of ( [ app : red - eom1 ] ) remains unchanged . to interpret equation ( [ app : red - eom2 ] ) , note that @xmath148 equals by ( [ bfp ] ) the difference of the killing times at the left and right ends of the constant @xmath24 surface . as the constant @xmath24 surface evolves in the schwarzschild spacetime , the first term in ( [ app : red - eom2 ] ) gives the evolution rate of the killing time at the left end of the hypersurface , where the hypersurface terminates at a spacelike surface located completely in the future dynamical region ( see fig . 4 ) . the second term in ( [ app : red - eom2 ] ) gives the negative of the evolution rate of the killing time at the right end of the surface , where the surface terminates at the timelike boundary . the two terms are generated respectively by @xmath218 ( [ app : bfhs ] ) and @xmath219 ( [ app : bfhb ] ) . the case of interest is when the ` inner ' spacelike boundary lies on the schwarzschild singularity , i.e. , @xmath220 , and when the radius of the ` outer ' boundary two - sphere does not change in time , @xmath150 . in that case @xmath218 ( [ app : bfhs ] ) and the second term in @xmath151 ( [ app : bfhb ] ) vanish . one can also make the first term in @xmath221 ( [ app : red - eom2 ] ) vanish provided one restricts the slices to approach the surface at @xmath87 in such a way that @xmath222 vanishes faster than @xmath223 there . the second term in ( [ app : red - eom2 ] ) is readily understood in terms of the killing time of a static schwarzschild observer , expressed as a function of the proper time @xmath152 and the blueshift factor @xmath153 . the reduced hamiltonian is given by @xmath154 where @xmath16 is the time - independent value of @xmath142 . following the same arguments as given in sec . [ subsec : reduction ] , we find that the appropriate hamiltonian under the new boundary conditions of this appendix is @xmath224 which is the by hamiltonian . similarly , from the time - reversal symmetry of the kruskal extension of the schwarzschild spacetime , the by hamiltonian could also be interpreted to evolve spatial slices that extend from the box upto an inner boundary that is the past white hole spacelike singularity . m. srednicki , phys . lett . * 71 * , 666 ( 1993 ) ; + g. t hooft , nucl . phys . * b256 * , 727 ( 1985 ) ; + l. susskind and j. uglum , phys . d * 50 * , 2700 ( 1994 ) ; + m. maggiore , nucl . * b429 * , 205 ( 1994 ) ; + c. callan and f. wilczek , phys . * b333 * , 55 ( 1994 ) ; + s. carlip and c. teitelboim , phys . d * 51 * , 622 ( 1995 ) ; + s. carlip , phys . d * 51 * , 632 ( 1995 ) ; + y. peleg , `` quantum dust black holes '' , brandeis university report no.brx-th-350 , hep - th/93077057 ( 1993 ) . figure 1 : a bounded spacetime region with boundary consisting of initial and final spatial hypersurfaces @xmath29 and @xmath30 and a timelike three - surface @xmath19 . @xmath19 itself is the time - evolution of the two - surface @xmath18 that is the boundary of an arbitrary spatial slice @xmath17 . figure 2 : the louko - whiting choice of a foliation of the schwarzschild spacetime . the spatial slices of this foliation extend from the bifurcation two - sphere to the box . the initial and final spatial hypersurfaces have label time @xmath225 and @xmath226 , respectively . figure 3 : a choice of foliating the schwarzschild spacetime that is different from the louko - whiting choice . here the spatial slices extend from the box to a timelike inner boundary that is located completely inside the hole .

in this work , we extend the analysis of brown and york to find the quasilocal energy in a spherical box in the schwarzschild spacetime . quasilocal energy is the value of the hamiltonian that generates unit magnitude proper - time translations on the box orthogonal to the spatial hypersurfaces foliating the schwarzschild spacetime . we call this hamiltonian the brown - york hamiltonian . we find different classes of foliations that correspond to time - evolution by the brown - york hamiltonian . we show that although the brown - york expression for the quasilocal energy is correct , one needs to supplement their derivation with an extra set of boundary conditions on the interior end of the spatial hypersurfaces inside the hole in order to obtain it from an action principle . replacing this set of boundary conditions with another set yields the louko - whiting hamiltonian , which corresponds to time - evolution of spatial hypersurfaces in a different foliation of the schwarzschild spacetime . we argue that in the thermodynamical picture , the brown - york hamiltonian corresponds to the _ internal energy _ whereas the louko - whiting hamiltonian corresponds to the _ helmholtz free energy _ of the system . unlike what has been the usual route to black hole thermodynamics in the past , this observation immediately allows us to obtain the partition function of such a system without resorting to any kind of euclideanization of either the hamiltonian or the action . in the process , we obtain some interesting insights into the geometrical nature of black hole thermodynamics . = 10000 epsf

You are an expert at summarizing long articles. Proceed to summarize the following text: it is inherent to the lattice approach to quantum field theory , that one has to extrapolate from finite lattices , finite statistics and non - critical coupling parameters to infinite lattices , infinite statistics and critical points . since the result supposedly is a non - trivial , non - perturbatively defined quantum field theory , this process is plagued by uncertainties . a typical example of such a situation , where all these aspects combine , is the study of the thermal transition in qcd for small quark masses . one is interested in the continuum limit ( gauge coupling @xmath1 ) , small or vanishing fermion masses ( @xmath2 ) , close to critical temperature ( @xmath3 ) in the thermodynamic limit ( @xmath4 ) a formidable problem . the extrapolations are always based on assumptions on the asymptotic behavior . well known examples are scaling functions based on renormalization group and chiral perturbation theory an expansion around a ground state with goldstone bosons . here we will examine another such approach , which should allow the extrapolation to infinite volume and vanishing fermion mass : chiral random matrix theory ( chrmt ) . rmt attempts to identify universal features of ensembles of ( random ) matrices with common symmetry properties . its chiral version , if successful , allows to separate two aspects of a theory like qcd : the general universal properties shared with other theories from the model - specific `` dynamical '' content of the theory . microscopic eigenvalue distribution shapes are an example for the first aspect , expectation values of the fermion condensate for the second . the limitations for validity of the chrmtconsiderations ( for a given @xmath5 in the phase of broken chiral symmetry ) are set by @xcite @xmath6 where @xmath7 is the linear size of the system and @xmath8 is the mass of the lightest ( pseudo-)goldstone boson . the second restriction imposes that the pion does not fit into the space - time volume and it therefore appears to be unphysical . however , various correlators in the dirac operator spectrum can be computed precisely in this limit . chrmthas been proven to give exact analytical predictions for the spectrum of the dirac operator in the microscopic limit @xcite . the microscopic scaling region is simply a blowup of the origin in the spectrum . to be specific , one considers eigenvalues @xmath9 on the scale @xmath10 where @xmath11 is the chiral condensate , related to the spectral density per unit volume @xmath12 via the banks - casher @xcite relation , @xmath13 . this regime is , by definition , only well defined in the spontaneously broken phase where @xmath14 . in the phase with restored symmetry the scale of interest is set by the density of states in the vicinity of the onset of @xmath12 . here we present a study of the microscopic correlators in the spectrum of the staggered dirac operator in su(3 ) gauge theory with dynamical fermions at finite temperature . specifically we examine the low lying eigenvalue statistics at temperatures below , near , and above the critical temperature of the chiral phase transition . our analysis is based on the evaluation of the milc collaboration s gauge configurations @xcite . we therefore concentrate on the new aspects connected to rmt ideas for the spectral correlators of the dirac operator . in particular for @xmath15 we study the singularity at the inner endpoint of the spectral density . strictly speaking , in the continuum limit @xmath16 , non - zero temperature is realized for lattices @xmath17 with @xmath18 and @xmath19 . in that limit , for vanishing quark mass @xmath20 , one expects a phase transition at @xmath0 . in @xcite the critical temperature was estimated to lie between 143 and 154 mev . for @xmath21 chiral symmetry is broken spontaneously , with massless pseudoscalars and @xmath22 ; above @xmath0 we expect restoration of this symmetry . whereas for pure yang - mills theory the deconfinement transition is associated with a breaking of the center - symmetry with the polyakov loop as order parameter , this symmetry is explicitly broken by the fermion action . nevertheless , remnants of the original breaking feature of the polyakov loop persist even for small fermion masses . the nature of the chiral phase transition depends on the number of flavors @xmath23 . an argument based on a 3d @xmath24-models analysis @xcite predicts a first order phase transition for @xmath25 . for @xmath26 one expects a second order phase transition with @xmath27 scaling behavior . however , even first order behavior may be arguable @xcite . for staggered fermions at non - vanishing lattice spacing the correct counting of flavors is unclear since flavor symmetry is restored only in the continuum limit . staggered fermions ( as simulated by milc with the hybrid r - algorithm @xcite ) correspond to the case @xmath26 in the continuum limit . on coarse lattices the so defined fermions should show at least @xmath28 scaling behavior . for a discussion of the various scenarios cf . @xcite . it is unclear whether the phase transition at @xmath0 extends towards @xmath29 or whether , when moving from lower to higher temperature , one just observes crossover - like behavior . some evidence points towards this second scenario @xcite . in the following , we denote the crossover ( phase transition ) position by @xmath30 or simply @xmath0 . according to the nature of the elements in the random matrix , chrmtappears in three universality classes . in this paper we consider the su(3 ) gauge theory with quarks in the fundamental representation , which belongs to the universality class of the chiral unitary ensemble ( chue ) . the partition function under study is @xcite @xmath31 where @xmath32 is a @xmath33 block hermitian matrix ( the elements of @xmath34 being random complex numbers ) , and @xmath5 is a deterministic , i.e. not - random , off - diagonal block matrix @xmath35 here d@xmath32 denotes the haar measure , @xmath36 is an analytic function . the predictions from chrmtconcern the correlations between the eigenvalues @xmath9 of @xmath37 on the scale of individual eigenvalues in the thermodynamic limit @xmath38 . the matrix @xmath39 is the analogue of the massless dirac operator in qcd . the chiral phase transition within chrmtis identified through the value of the spectral density of the eigenvalues of @xmath39 near zero , i.e. using the banks - casher relation . modeling the chiral phase transition in chrmtamounts to driving a depletion of eigenvalues of @xmath39 near the origin by means of some temperature parameter . two separate approaches have been examined in the literature . first , the unitary invariant chrmt@xcite , corresponding to ( [ z ] ) with @xmath40 , in which the chiral phase transition is driven by tuning @xmath41 . second , the non - unitary invariant chrmt@xcite , corresponding to ( [ z ] ) with @xmath42 , where the deterministic block matrix @xmath5 mimics the effect of the temperature . in this paper we do not need to distinguish between the two approaches as they are consistent for the quantities measured here . below @xmath0 , i.e. when @xmath14 , chrmtpredicts @xcite the probability distribution for the smallest eigenvalue ( for the trivial topological sector ) @xmath43 with @xmath44 where @xmath45 is the spectral density at the origin for the _ massless _ situation ( i.e. when @xmath46 in ( [ z ] ) ) . @xmath47 denotes the @xmath48th modified bessel function . this result is universal in the chrmtcontext , that is , the analytic form of @xmath49 does not change under deformations of @xmath41 provided that @xmath14 . after the identification @xmath50 ( @xmath51 is the physical volume in lattice units ) , ( [ pmin ] ) allows to extract @xmath52 ( the fermion condensate in the chiral limit ) from finite - volume dirac spectra of course , in this the mild condition ( [ qcdrange ] ) must be satisfied . above @xmath0 , when there is a finite gap in the spectral density , the repulsion between the eigenvalue pair @xmath53 becomes negligible ; chrmthence predicts a _ soft _ inner edge , known as the airy - solution @xcite . at @xmath0 signaled in chrmtby a power - like behavior of the spectral density at small @xmath9 @xmath54 , the prediction from chrmtis not unique . it turns out that the spectral correlators depend on the value of @xmath55 @xcite . the distribution of the smallest eigenvalue is a spectral one - point correlation function and is quite sensitive to statistical fluctuations ( see below ) . as an additional measure we also study a two - point correlator : the level spacing distribution @xmath56 . note that the level spacings , @xmath57 , are determined in the unfolded spectrum @xmath58 . unfolding separates the fluctuation properties of the spectrum from the supposedly smooth background behavior . the unfolded variable is defined in terms of the eigenvalue spectrum and the local average spectral density by @xmath59 the rmt prediction for the level spacing distribution is well approximated by the unitary wigner surmise @xmath60 the level spacing distribution is not expected to be affected by temperature and masses in chrmt , see e.g. @xcite . chrmtmakes predictions for average spectral correlators in sectors with definite topological charge @xmath61 , i.e. derived assuming exact zero modes ( these are not included in the predicted distributions ; cf . @xcite for the result of the weighted summation of all topological sectors ) . for ginsparg - wilson fermions @xcite , which realize chiral symmetry on the lattice , one may identify exact zero modes as resulting from topological excitations according to the atiyah - singer index theorem ( for wilson fermions one can hypothesize that zero modes are replaced by real modes ) . this is not the case for staggered fermions , where exact zero modes are absent away from the continuum limit @xcite and even gauge configurations with non - vanishing topological charge do not give zero eigenvalues . exact zero modes are here replaced by `` almost '' zero modes which accumulate to the origin in the continuum limit . in the strong coupling region the microscopic staggered dirac spectra summed over all topological sectors show @xcite good agreement with the analytical prediction for the topologically neutral , @xmath62 , sector from chrmt . however , approaching weaker coupling observations contradicting this scenario have been found in a two - dimensional context @xcite . before turning to the numerical studies , let us comment on the validity of the chrmtpredictions . the condition for application of chrmtin lattice analyses is well established when @xmath63 : the range in the unfolded spectrum over which the chrmtcorrelations dominate is @xmath64 @xcite , where @xmath65 is the pion decay constant . an equivalent statement for @xmath66 is not known and no stringent tests of the low - lying eigenvalue statistics have been carried out so far . let us emphasize that even though the larger part of the studies of chrmthave been focused on the situation where @xmath14 , there is nothing wrong from first principles in using chrmtwhen @xmath67 . by courtesy of the milc collaboration @xcite there are sets of gauge configurations @xcite available to the lattice community . these were generated with two species of dynamical staggered fermions , at various lattice sizes , temperatures , values of the gauge coupling and small values of the bare fermion mass . in table [ tabconf ] the samples used in the present study are listed . for further details on the method of determination of the gauge configurations and the physical parameters we refer to @xcite . . summary of the milc configurations used in our analysis ( for ft01 we only considered a subset of the total of 149 configurations available ) . the suggestions in the rightmost column are based on milc s results . the transition is near @xmath68=5.26 for @xmath69 and @xmath68=5.725 for @xmath70 . [ cols= " < , > , < , > , < , < , < " , ] [ tab : fit ] in order to further investigate this feature , we studied ( for the set with @xmath71 ) the influence of the configurations where the eigenvectors @xmath72 of the lowest eigenvalues exhibit a large contribution to the total chirality , i.e. @xmath73 . according to the index theorem , these configurations with large chirality , which make up roughly one half of the ensemble , tend to carry non - vanishing topological charge and therefore zero modes . indeed we find that a substantial part ( 75% ) of the first peak may be explained from these contributions . these findings suggest that indeed topological modes are responsible for a low - lying peak in the distributions . below @xmath0 all topological sectors are present and the low - lying eigenvalues have comparable magnitudes ( their average position being roughly proportional to @xmath61 ) . when the temperature approaches @xmath0 the theoretical expectation is that the topological fluctuations begin to be suppressed , although still present in the ensemble , quasi superimposed on the background distribution , which starts to broaden significantly with increasing temperature . sufficiently far above @xmath0 only the topologically trivial sector survives and there may be no small eigenvalues at all . this is indeed what we actually observe for the lattice @xmath74 . in a recent study of quenched configurations @xcite there have been indications for a dilute gas of instanton anti - instanton pairs producing a poissonian distribution of small eigenvalues above @xmath0 . these may be suppressed or absent when considering dynamical fermions . in our context this seems to be the case for the finest lattices ( @xmath75 with @xmath76 ) at our disposal . [ secondevs ] gives the histograms for the 2nd smallest eigenvalues . again we notice a dramatic change of the distribution shape around @xmath30 . we interpret the sudden flatness of the distribution of the smallest eigenvalues as being ( i ) due to the vanishing spectral density and ( ii ) due to increasing statistical fluctuations near the chiral phase transition . the latter effect is not reproduced in chrmtsince this is a zero dimensional and non - dynamical theory . furthermore , the mutual overlap of @xmath77 and @xmath78 increases for @xmath79 . this is also inconsistent with chrmt . in order to study this effect further we now turn to the level spacing distributions . another observable with definite predictions from rmt ( cf . [ secrmt ] ) is the distribution of level spacings . the advantage here is , that the level spacing should not be influenced by possible distortions of the smallest eigenvalues due to the unknown topological charge of the configurations ( if the smallest eigenvalues are removed from the data ) . the studies of the level spacing statistics in lattice data so far have shown a uniform picture consistent with the rmt prediction ( [ wue ] ) . the agreement extends on both sides of the confinement - deconfinement phase transition @xcite . however , to the authors knowledge there is no analytical prediction from chrmtfor the level spacing distribution when focusing on the soft edge or at @xmath0 . so one might worry that the standard prediction , eq . ( [ wue ] ) , is not appropriate when @xmath67 . within the t - model of @xcite for @xmath80 and @xmath81 , we have performed a numerical high statistics simulation to eliminate such doubt , and we there confirmed the distribution ( [ wue ] ) . ( the t - model is defined by ( [ z ] ) with @xmath42 and @xmath82 in ( [ tmodel ] ) chosen proportional to the unit matrix . ) usually it is possible to get high statistics on the level spacing distributions since each configuration provides a large number of eigenvalue spacings . however , already the lowest 10 ( or 8) eigenvalues allow for a crude estimate of the distribution shape . recall that the level spacing distribution is measured in the unfolded spectrum , see ( [ defunf ] ) . here we use the average spacings @xmath83 between contiguous eigenvalues to define the unfolded level spacings @xmath84 in fig . [ levelspc ] we compare the data with the parameter - free theoretical expectation . whereas below and above @xmath0 we find reasonable agreement with the theoretical expectation , there are clear discrepancies near @xmath0 . we observe unexpected high histogram entries . since the average value by definition is 1 this then leads to a shift of the central peak to the left . in order to further check our unfolding procedure , we also considered other approaches , e.g. using a average density as in ( [ defunf ] ) by smoothing our distribution in various ways . we furthermore tried to discard the higher lying eigenvalues , e.g. using only the lowest 5 level spacings or introducing a cutoff near the peak of the distributions in fig . [ firstevs ] . in all those checks we found essentially the same behavior with discrepancies near @xmath0 . in conclusion of this section , we observe at @xmath85 a breakdown of the otherwise universal microscopic spectral correlations . the dynamics of qcd plays an essential rle in the phase transition . a rmt model where such dynamics is not there fails to account for the increased fluctuations in the eigenvalue level spacings . we now turn to the results for @xmath15 . in our results for @xmath75 at @xmath76 ( fig . [ all - dist](c ) ) a gap in the spectral distributions is obvious . however even at @xmath86 we may speculate , that a clear signal of a gap is only prevented by the ( topological ) quasi - zero modes responsible for the small bump at small eigenvalues . recall , that chrmt@xcite predicts the presence of a gap in the spectral density @xmath12 of the dirac operator centered around @xmath87 . furthermore , the inner edge of this gap is predicted to show a singularity , at a point @xmath88 , in the macroscopic spectral density @xcite @xmath89 where @xmath90 is a known constant . the constant @xmath91 takes the value @xmath92 in the generic chrmt , i. e. without fine - tuning the matrix potential in ( [ z ] ) ( which would be necessary in order to obtain higher values of @xmath91 ) . this corresponds to a square - root - like eigenvalue density near @xmath88 . one concern here is to measure @xmath91 . with the limited amount of data available it is not possible to do this based on the spectral density only . instead we propose to study the average distance between the smallest and the sequel eigenvalues @xmath93 ; the extraction of @xmath91 is carried out by noting the following scaling relation in the index @xmath94 , ordering by size the eigenvalues which follow the smallest , @xmath95 this proportionality follows by integration in ( [ aisingularity ] ) . in fig . [ and ] we display the seven average distances , @xmath96 , from the ensemble of 146 configurations on a @xmath97 lattice for @xmath98 . we also exhibit the best fit to ( [ qi ] ) with respect to @xmath91 , giving @xmath99 . also shown are the corresponding curves for @xmath92 and @xmath100 . the value @xmath92 is clearly favored . since the two - point correlations behave as expected from rmt , we now turn to the one - point distribution . the microscopic behavior of the spectral density in the vicinity of this singularity is universal in the chrmtsense , but depends on the value of @xmath91 @xcite . for @xmath92 the exact analytical prediction for the microscopic spectral density in the vicinity of the inner edge is @xcite @xmath101 here the origin has been moved to the inner spectral endpoint @xmath88 by means of the rescaled eigenvalue @xmath102 , which is defined through @xmath103 \ .\ ] ] the consistency with the prediction ( [ qi ] ) for @xmath92 and the approximate validity of chrmtcorrelations in the level spacing statistics above @xmath0 suggest that the airy - density ( [ aidens ] ) corresponding to the value @xmath92 should fit the spectral density . if it does , then we can extract the inner - endpoint @xmath88 of the spectrum in the thermodynamic limit , by fitting @xmath104 with respect to @xmath88 to the lowest part of the spectral density , see fig . [ airyfig ] . this fit does not convincingly confirm airy density . however , the statistical fluctuations at this @xmath68-value affect the _ one - point distribution _ substantially and prevent a decisive comparison . we have examined the manifestations of the chiral phase transition in the microscopic spectral correlators for the dirac operator . for the level spacing distribution , we find agreement with rmtbelow and above @xmath0 . below @xmath0 the chrmtdistributions allow us to determine condensate values with implicit consideration of lattice volume and quark mass dependence . this could in principle serve to improve the scaling analysis of the condensate near the chiral transition . near @xmath0 , however , the microscopic spectral statistics differs from the chrmtprediction . by measuring the monte carlo time evolution of the chiral condensate , aoki et al . @xcite have shown , that there are mixed phase signals , which , however , vanish towards larger volumes . the existence of a mixed phase would offer an explanation for the observed deviations from chrmtnear @xmath0 . in that case the level spacing distribution near @xmath0 would be a mixture of those from the two phases . such a mixture would lead to large spacings : the spacings are unfolded according to the average spacings of the total ensemble and not according to that of the separate phases . the observed discrepancy from the rmt level spacing statistics may also be interpreted as an inclination towards poissonian statistics ; distribution shapes interpolating between wigner and poissonian statistics have been suggested by brody @xcite . as may be seen from the @xmath75 ensembles at @xmath86 and @xmath105 , a gap develops the spectral density for @xmath15 . this is consistent with the observed suppression of topological fluctuations in the latter ensemble @xcite . for the @xmath76 ensemble we have measured the critical exponent characterizing the steepness of the density at the inner edge . the value is found to be compatible with 1/2 . this is exactly as predicted by chrmtwhere the chiral phase transition is manifested by the crossover from the bessel hard edge to the airy soft edge . the indications of the bessel to airy scenario are suggestive but simulations with extended statistics are needed in order to quantify the observation . however , even with low statistics the @xmath68-dependence of the distance between e.g. 8th and 1st eigenvalue provides an excellent means to identify the change of the phase . at low @xmath68 , on coarse lattices , staggered fermions appear to be blind with regard to the topological charge of the gauge configurations , and the smallest eigenvalue distribution agrees with the chrmtdistribution for the @xmath62 sector . as the lattice becomes finer , topology becomes more relevant . although this is maybe `` good '' for the continuum limit of staggered fermions , it affects unfavorably the agreement with chrmtsince the want - to - be - zero modes and the non - zero modes have similar eigenvalues , and begin to separate only when the non - zero modes are pushed to larger values when increasing the temperature . we want to thank the milc collaboration for making available the gauge configurations , that we used in our analysis . special thanks go to jim hetrick and doug toussaint for their help in accessing those and for the support in making additional ensembles public . . de f. thanks jim hetrick and jean - franois laga for their contribution at a preliminary stage of this project . k. s. would like to thank andrew jackson for discussions .

we re - analyze data from available finite - temperature qcd simulations near the chiral transition , with the help of chiral random matrix theory ( chrmt ) . statistical properties of the lowest - lying eigenvalues of the staggered dirac operator for su(3 ) lattice gauge theory with dynamical fermions are examined . we consider temperatures below , near , and above the critical temperature @xmath0 for the chiral phase transition . below and above @xmath0 the statistics are in agreement with the exact analytical predictions in the microscopic scaling regime . above @xmath0 we observe a gap in the spectral density and a distribution compatible with the airy distribution . near @xmath0 the eigenvalue correlations appear inconsistent with chrmt . 2

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Proceed to summarize the following text: bose - einstein condensation of quasiparticles in solid - state systems has been observed in excitons in quantum hall bilayers , @xcite exciton - polaritons in semiconductor microcavities , @xcite gases of magnons , @xcite cavity photons @xcite and indirect excitons . @xcite exciton - polaritons , mixed light - matter quasiparticles behaving as bosons , form condensates which exhibit not only the fundamental properties of quantum gases , but also new fascinating phenomena related to their out - of - equilibrium character . @xcite the photonic component of polaritons is responsible for their light mass , which makes condensation possible up to room temperature @xcite , and for their easy creation , manipulation and detection by using simple optical - microscopy setups . on the other hand , their excitonic component yields strong coulomb repulsive interactions that make them promising candidates for future non - linear optical technologies . the peculiar quantum fluid properties of polariton condensates are under intense research nowadays . recent findings include : robust propagation of coherent polariton bullets @xcite and elucidation of the validity of the landau criterion for frictionless flow in the presence of weak structural defects , @xcite persistent quantized superfluid rotation , @xcite and solitary waves resulting from compensation between dispersion and particle interaction . @xcite moreover , the intrinsic out - of - equilibrium character of polariton condensates has motivated recent theoretical studies on how to describe properly the energy flow from an optically - injected hot exciton reservoir to the coherent polariton modes , @xcite which we carefully address in this work . the functionalities of microcavities in the strong coupling regime , as integrated optical elements , promote polaritons as an undreamt platform to create new logical devices . @xcite thanks to their interactions with non - condensed excitons , polaritons can be easily accelerated , propagating over macroscopic distances in high finesse microcavities . @xcite in this case , new interferometric devices can be built by properly shaping the excitation profile @xcite as well as the microcavity etching . @xcite extra confinement can be achieved by lateral bounding the optical modes through patterning the microcavity , @xcite by sculpting the pumping profile creating blueshift - induced traps , @xcite or by a combination of both methods . @xcite this paves the way for studies of atom - like scenarios in which the energy spectrum becomes discrete . in a recent work using quasi 1d - microwire ridges , a polariton condensate transistor switch has been realized through optical excitation with two beams . @xcite one of the beams creates a polariton condensate which serves as a source ( _ s _ ) of polaritons ; their propagation is gated using a second weaker gate beam ( _ g _ ) that controls the polariton flow by creating a local blueshifted barrier ( a list of symbols used in the manuscript are given in the appendix a ) . the on state of the transistor ( absence of _ g _ ) corresponds to forming a trapped condensate at the edge of the ridge ( collector , _ c _ ) labelled as @xmath0 . the presence of _ g _ hinders the propagation of polaritons towards _ c _ , remaining blocked between _ s _ and _ g _ ( off state ) . an insight of the energy relaxation and dynamics of the condensed polariton propagation in this system has been obtained lately by a time - resolved study of the on / off states . @xcite in the present work , we make a systematic study of the influence of the density of polaritons created in _ s _ and _ _ g__on the propagation and the gating of polariton bullets , of their energy and density relaxation and of the optimal conditions for realizing an all - optical polariton condensate transistor switch . our experiments are compared with simulations of the polariton condensate dynamics based on a generalized gross - pitaevskii equation , modified to account for incoherent pumping , decay and energy relaxation within the condensate . we investigate a high - quality @xmath1 algaas - based microcavity with 12 embedded quantum wells , with a rabi splitting of @xmath2 mev . ridges have been sculpted through reactive ion etching with dimensions @xmath3m@xmath4 ( further information about this sample is given in refs . and ) . figure [ fig : fig0 ] ( a ) shows a scanning electron microscopy image of such a ridge , including the excitation scheme ; a temporal scheme of the excitation and emission processes is given in panel ( b ) . in our sample lateral confinement is insignificant as compared to much thinner , 1d polariton wires . @xcite the chosen ridge is in a region of the sample corresponding to resonance ( detuning between the bare exciton and bare cavity mode is @xmath5 0 ) . the sample , mounted in a cold - finger cryostat and kept at 10 k , is excited with 2 ps - long light pulses from a ti : al@xmath6o@xmath7 laser , tuned to the first high - energy bragg mode of the microcavity ( 1.612 ev ) . we split the laser beam into two independent beams , whose intensities , spatial positions and relative time delay ( zero for these experiments ) can be independently adjusted . we focus both beams on the sample through a microscope objective to form 5 @xmath8m-@xmath9 spots spatially separated by @xmath1040 @xmath8 m along the ridge . the same objective is used to collect ( angular range @xmath11 ) and direct the emission towards a spectrometer coupled to a streak camera obtaining energy- , time- and spatial - resolved images , with resolutions of 0.4 mev , 15 ps and 1 @xmath8 m , respectively . in our experiments polaritons propagate along the @xmath12 axis of the ridge . there is also some diffusion of polaritons in the @xmath13 direction , but it is not relevant for the operation of our device . all the images in the manuscript show the emission collected along the @xmath12 axis from a 10-@xmath8 m wide , central region of the ridge . the power threshold for condensation of polaritons is @xmath14 mw . figure [ fig : fig1 ] shows , as an example , under cw conditions , the intensity distribution of the polariton emission as a function of energy and of the position in the ridge : when we only use the _ s _ beam , fig . [ fig : fig1 ] ( a ) , we place it @xmath1075 @xmath8 m away from the right ridge border ; in the _ _ s__+_g _ beam excitation , _ g _ is placed @xmath1035 @xmath8 m away from the border , fig . [ fig : fig1 ] ( b ) . in fig . [ fig : fig1 ] ( a ) , exciting with a power @xmath15 , the blue - shifted ( @xmath16 mev ) emission at the source , together with a weak condensate emission @xmath0 , are clearly observed . @xmath0 emits from an energy lower than that of the propagating polaritons as a result of an unintentional modification of the microcavity structure created by the etching process at the edge of the ridge . polaritons propagate at a constant energy towards the left and right sides of the ridge . exciting with both laser beams , under different excitation conditions for a typical off state , @xmath17 and @xmath18 , the gating state of the switch is readily seen with the stopped , condensed polaritons just before the _ g _ position . in this work , we time - resolve the different excitation configurations presented in ref . : fig . 2 ( where the @xmath19 is varied whilst @xmath20 ) and fig . 3 ( @xmath21 and @xmath22 is varied ) . our study obtains intensity- and energy - dynamics of exciton and polariton emission in the ridge . in the former case varying @xmath19 , we fully characterize the on state response of a polariton transistor switch ; in the latter one , we modulate the polariton condensate trapping potential . @xcite in this section we present three time - resolved cases for different @xmath19 pump powers , figs . [ fig : fig2]-[fig : fig4 ] . figure [ fig : fig2 ] shows the dynamics of the emission when @xmath15 . for each panel the time is displayed at the right upper corner , being the temporal origin set at the instant when the _ s _ intensity is maximum . panel ( a ) shows that the emission from _ s _ at -24 ps occurs at 1.545 ev ; at 31 ps , ( b ) , the emission redshifts and a small spatial expansion around 0 @xmath8 m is observed ; propagating polaritons , expanding more rapidly towards the border , at an energy of1.542 ev , are detected at 194 ps ( c ) , eventually reflecting backwards , interfering coherently and creating the @xmath0 condensate , weakly emitting at 1.539 ev at later times , 483 ps ( d ) . since the pump power is at threshold , the emission intensity of polaritons is weak and slightly higher than the noise level in all panels of fig . [ fig : fig2 ] . let us note that at early times , the emission observed in fig . [ fig : fig2 ] appears blueshifted from the lower polariton minimum by an amount comparable to one - half of the rabi splitting . this suggests that the emission at the source comes from polaritons with a strong excitonic character . for this reason we will refer to the emission from the source as arising from excitons , although the decrease in the blueshift over time corresponds to a continuous transition from excitonic polariton states to those with roughly equal excitonic and photonic fractions . the emission from the propagating states and collector region , at lower energy , is clearly a polaritonic emission . it is also important to note that the duration over which the condensate is present greatly exceeds the polariton lifetime . this is because the condensate is continuously fed by high energy excitons ( not visible in the spectrum and distinct from the excitonic states emitting at the source ) excited by the non - resonant pulse . the emission at _ s _ is determined by repulsive interactions with hot excitons which contribute a blueshift to the potential energy . as hot excitons decay from the system , either through recombination or condensation , this potential energy decreases over time . once the polariton condensate has formed , due to the low density of polaritons , there are minimal energy relaxation processes , such that the propagating polaritons tend to conserve their energy as they spread out from the source ( see fig . [ fig : fig2 ] ( d ) ) . the dynamics of the emission increasing @xmath19 to @xmath23 is shown in fig . [ fig : fig3 ] : the initial excitonic emission at the source takes place at 1.546 ev ( a ) , slightly higher than before , due to increased blueshift due to a larger hot - exciton repulsion . at @xmath24 ps ( b ) , an essential difference with respect to the case of fig . [ fig : fig2 ] ( b ) is revealed : polaritons emit from a lower energy than that of the source , which is @xmath102 mev blueshifted ; this situation holds during the first @xmath10200 ps of the decay process . panel ( c ) shows the arrival of polaritons at the ridge border at 140 ps , and the eventual condensation of @xmath0 ( d ) . this final relaxation phase in the dynamics takes place into a state defined in a minimum of the wire structural potential located at the wire edge . a clear indication of the polariton coherence is evidenced by the interference at 1.540 ev between counter - propagating wave - packets . the source population at _ s _ , still 1 mev blueshifted with respect to propagating polaritons , expands around @xmath25 as it decays in energy , and continuously feeds the propagating polariton condensate , increasing its effective lifetime ( e ) . finally , as shown in ( f ) at 617 ps , polaritons at _ s _ merge with those propagating along the ridge . the emission is still observed for times as large as @xmath26 ns ( not shown ) . the case of the highest source power used in our experiments is shown in fig . [ fig : fig4 ] : at @xmath27 ps the excitonic source population emits at 1.547 ev ( a ) . the progressive spatial expansion of the excitonic population and the fast relaxation of the polariton condensate , as it propagates towards the right side , at 1.541 ev , reaching the ridge edge at 35 ps , is shown in panels ( b ) and ( c).@xmath0 is now slightly blueshifted , with respect to its energy at lower @xmath19 conditions , to 1.540 ev , due to the higher density condensate population at this place of the ridge ( d ) . at later times , as those shown in ( e , f ) for 300 and 508 ps , the population at _ s _ decreases and expands in space whilst @xmath0 redshifts its energy emission due to its reduced occupancy . movies corresponding to figs . [ fig : fig2]-[fig : fig4 ] are provided as supplementary material . @xcite the introduction of a new secondary pulse , dubbed before as gate ( _ g _ ) , between _ s _ and _ c _ , adds new interaction phenomena . the existence of two condensates becomes very clear in this case : one of them located initially between _ s_-_g _ , @xmath28 , which eventually becomes propagating , and a second one , already labelled as @xmath0 . the polariton propagation towards _ c _ along the ridge can be hindered with a below - threshold intensity gate beam , see fig . [ fig : fig1 ] ( b ) and fig . [ fig : fig5 ] , rendering the @xmath0 switch - off and creating the trapped condensate @xmath28 . as the _ g_-repulsive potential gradually decreases in time , a tiny fraction of @xmath28 is able to tunnel through the barrier and it spreads between 40 and 80 @xmath8 m , see fig . [ fig : fig5 ] ( d ) . this configuration has been already discussed in detail in ref . . figure [ fig : fig6 ] shows the dynamics for @xmath17 and @xmath29 . the visibility of the emission at _ g _ is delayed by @xmath1020 ps with respect to that at _ s _ , despite of the fact that both beams reach the sample simultaneously , panels ( a , b ) . this delay is due to the power dependence of the emission rise time , which increases with decreasing power . the @xmath28 condensate lies at a constant energy , 1.541 ev , remaining trapped , ( c ) . on their own account , the emission energy of _ s _ and _ g _ decay , at a rate determined by the carrier density and the carrier - carrier interactions , until they reach 1.542 ev for both . the population between _ g _ and _ c _ , mainly created by the _ g _ pulse , propagates towards the border ( c ) , being reflected ( d ) and forming the @xmath0 condensate ( e ) . when the _ g _ barrier further decays the @xmath28 condensate becomes propagating and coherent interference patterns are generated from counter - propagation , see panel ( f ) . for completeness , fig . [ fig : fig7 ] depicts the case corresponding to large values of @xmath22 . panel ( a ) depicts the excitonic emission at 1.547 ev when the laser beams arrive at _ s _ and _ g_. @xmath28 is trapped around @xmath12=20 @xmath8 m , blueshifted up to 1.542 ev , due to repulsive interactions ( b ) , whilst polaritons between _ g _ and _ c _ propagate towards the border . at 76 ps , a new condensate , @xmath0 , becomes trapped at 1.540 ev , and the emission energy of _ s _ and _ g _ reaches that of @xmath28 ( c ) . due to the barrier reduction at _ g _ , @xmath28 propagates along the ridge from 0 to 60 @xmath8 m ( d ) . @xmath0 remains confined for later times at a constant energy , whereas @xmath28 decays and interferes with itself , panels ( e , f ) . movies corresponding to figs . [ fig : fig5]-[fig : fig7 ] are provided as supplementary material . @xcite in our sample , the emission above 1.544 ev is coming from excitonic states . the polariton emission lies at lower energies , down to 1.538 ev at the collector region . in this section , we analyze the dynamics of the energy and population relaxation along the full region of propagation of the condensates between _ s _ and _ c _ both in the presence or absence of _ g _ , obtaining quantitative values for the energy time - decays and the optimal working conditions for the on - state . figures [ fig : fig8 ] and [ fig : fig9 ] show spatial - temporal maps of the energy ( a - c)/intensity ( d - f ) evolution of the emission for the same power values as those used in figs . [ fig : fig2]-[fig : fig4 ] and [ fig : fig5]-[fig : fig7 ] , respectively . panels ( a - c ) have been obtained identifying the time at which the maximum emission intensity takes place , at a given _ x _ position on the ridge , for every energy : this gives a point in the map whose energy is coded with the false - color scale shown on the right - hand side of the upper row . note that all the information concerning the strength of the emission , and therefore the polariton population , is lost in this representation . @xcite the complementary information is encoded in the second row in figs . [ fig : fig8 ] and [ fig : fig9 ] , giving in this case the polariton population from integrating all emission energies . these plots provide a straight and precise insight on the energy / intensity decay of the population at every position along the ridge . let us start by considering the one - beam excitation compiled in fig . [ fig : fig8 ] . in panels ( a , b ) the energy trap created at @xmath30 m due to the potential discontinuity close to the border of the ridge is clearly observed : @xmath0 , emitting at 1.539 ev , is separated by a small energy gap from the polaritons propagating above at 1.540 ev . the horizon , @xmath31 , on the right upper corners , between white and colored points is given by the arrival of polaritons at different positions along the ridge . another discontinuity is observed between the decay of carriers at _ s _ and the propagating polaritons , @xmath32 . the power dependence of both discontinuities is evident in these panels and gives information about the speed of propagation of different emitting species . at @xmath33 , the border , @xmath32 , between carriers at _ s _ and polaritons , whose propagation is seen for @xmath34 m , is absent ( a ) , because the energy of excitons and polaritons decay at the same rate , but it becomes very clear in panels ( b , c ) . the speed of propagation of the carriers can be obtained from the slope of @xmath32 and @xmath31 lines . for the carriers at _ s _ in panel ( b ) , @xmath32 is almost straight , therefore a mean speed value , @xmath35 , can be obtained amounting to @xmath36m / ps . at the highest power ( c ) , @xmath37 initially has increased by a factor of @xmath103 as compared to @xmath35 , but the strong non - linearities associated with the high carrier densities lead to the appearance of deceleration rendering a gradual decrease of @xmath38 . the spatial extension of the carriers around _ s _ also widens with increasing power , almost doubling its value from @xmath1030 to @xmath1060@xmath39 m at 400 ps as seen in ( b ) and ( c ) panels . it is also noticeable that the energy decay of the carriers is spatially flat in the region enclosed by @xmath32 . the acceleration / deceleration of the propagating polaritons is distinct in the slope changes of @xmath31 , panels ( a - c ) . for @xmath15 , a rough estimation of the speed obtains @xmath40@xmath41m / ps ; @xmath42 increases to @xmath43 and @xmath44m / ps for @xmath45 and @xmath46 , respectively . in the later case @xmath42 amounts to @xmath47 of the speed of light in vacuum . the formation of @xmath0 at threshold , ( a ) , is seen by the purple ( 1.539 ev ) oval shape at ( @xmath30 m , 400 - 600 ps ) . the enhancement of @xmath42 together with that of stimulated scattering processes with power give rise to an earlier appearance of @xmath0 at @xmath10280 ps lasting for 400 ps , almost doubling its spatial extent , at @xmath48 ( b ) . the values for @xmath42 are in agreement with others reported in the literature ( see , for example , ref . ) . the much smaller values for @xmath38 are due to the larger exciton mass compared to that of polaritons . the energy gap between @xmath0 and the propagating polaritons dissolves at @xmath49 due to the very large number of polaritons and the very fast formation of this condensate . finally , let us remark that the ballistic propagation of polaritons is evidenced in panel ( b ) by the constant energy ( same color ) , for a given time , seen in the region enclosed by the @xmath32-border and _ c_. however , in case ( c ) a gradual change in energy ( color ) is observed , indicating the energy loss during the polariton propagation towards _ we briefly discuss now the density maps for different power excitation shown in the lower row of fig . [ fig : fig8 ] , in a normalized , logarithmic false - color scale shown on their right - hand side . panels ( d - f ) show that the main emission intensity arises from the population at _ s _ , with a gradual expansion towards _ c _ with a much lower polariton population . the emission - intensity decay becomes faster with increasing @xmath19 power . for @xmath48 , ( b ) , an enhanced emission following the @xmath32 is apparent ; interferences of polaritons in the region between @xmath1020 and @xmath1060 @xmath8 m are visible ; the formation of @xmath0 appears at 280 ps . panel ( f ) shows several reflections of condensed polaritons between the ridge edge and the left - bouncing positions marked with white bars , which are determined by the potential delimited by @xmath32 and the energy of the bouncing condensates : the longer the time , the larger the energy loss of the polaritons , which become less able to climb the barrier side , as borne out by the progressively increasing distance between the bars and the @xmath32 line , obtained from panel ( c ) and depicted with a white dotted line . at @xmath1060 @xmath8 m and 100 ps a considerable amount of population forms the @xmath0 condensate . we turn now to the two - beam excitation compiled in fig . [ fig : fig9 ] . panel ( a ) displays the energy decay of the polaritons in the off - state for @xmath17 and @xmath18 . the @xmath28 condensate , extending 20 @xmath8 m , reveals an almost constant energy emission in time . the contrast of the off - state is high as assessed by the negligible amount of polaritons that goes through the _ g _ potential ( d ) ; only a hint of the polaritons that were able to tunnel through is seen at ( 80 @xmath8 m , 400 ps ) in panel ( a ) , that codifies the energy but not the intensity of the signal . the ratio @xmath50is much larger than @xmath51 , obtained in the one beam case since @xmath28 is trapped closer to _ s _ and its feeding process is more efficient . increasing @xmath22 to @xmath52 both _ s _ and _ g _ beams contribute to the formation and trapping of polariton condensates ( b ) , @xmath28 and @xmath0 . figure [ fig : fig9 ] ( c ) , for @xmath53 , shows that , for the first @xmath54 ps , the energy decays at _ s _ and _ g _ are much faster than those of the polariton condensates . for longer times , @xmath55 ps , the energy decay of the populations at _ s _ , @xmath28 and _ g _ is almost identical ; however , @xmath0 is always at a lower energy due to the trapping at _ c_. a further inspection of the energy - integrated intensity maps shows that in panel ( e ) , at 300 ps , when the _ g_-barrier has considerably decayed , so that its energy coincides with that of @xmath28 , the @xmath28 condensate starts expanding along the ridge ; concomitantly a slanted interference pattern is obtained , revealing the dynamics of merging counter - propagating polaritons . the @xmath28 formation time , @xmath1070 ps , is much shorter than of @xmath0 , @xmath10350 ps , due to the fact that @xmath19 is much larger than @xmath22 and that both beams contribute to feed @xmath28 while only the population at _ g _ refills the @xmath0 condensate , which reaches its maximum intensity emission at 400 ps . in panel ( f ) , the high _ s_- and _ g_-pump powers make the @xmath28 condensate very intense at 40 ps . the confluence of the _ s_- and _ g_-population with @xmath28 takes place at 100 ps and 1.542 ev . then @xmath28 doubles its spatial width , as observed by the spreading cone of polaritons extending 20 @xmath8 m at 40 ps to 40 @xmath8 m at 250 ps . a clear back and forth bouncing of the @xmath0 condensate between the _ g_-barrier and the ridge edge is observed during the first 100 ps . after losing its kinetic energy at @xmath56 ps , @xmath0stops and emits for more than 600 ps , as its population is continuously fed by propagating polaritons at @xmath57 ev . the energy maps shown in figs . [ fig : fig8]-[fig : fig9 ] ( a - c ) allow to quantitatively analyze the energy decay at every @xmath12-position ; in particular we present in fig . [ fig : fig10 ] this decay at the _ s _ position for @xmath58 . the solid white line in fig . [ fig : fig10 ] ( a ) corresponds to its best fit to the sum of two exponentially decaying functions , shown separately by the dashed and dot - dashed lines . the double fashion decay is attributed to two different physical processes : a fast decay due to relaxation driven by exciton - exciton interactions and a slow one , attributed to the decreasing blueshift caused by the diminishing exciton and polariton populations . the rate of condensation can be expected to be faster at early times due to larger densities of carriers resulting in stronger stimulated scattering processes . a fast condensation rate results in an initial fast drop in the exciton density since excitons condense rapidly into polaritons that quickly decay . this drop in the exciton population gives a corresponding drop in the polariton population and so both blueshifts , due to polariton - exciton and polariton - polariton repulsion , drop sharply at early times . at longer times polariton condensation proceeds slower , due to weaker stimulated scattering and the exciton populations decay with a slow exponential dependence due to exciton recombination . figure [ fig : fig10 ] ( b ) compiles the power dependence of the decay times : both decrease with increasing power , more markedly for @xmath59 ( circles ) , which decreases by @xmath60 for a 20 fold increase of power , whilst @xmath61 ( squares ) only diminishes by @xmath62 , revealing the larger influence of density in exciton - exciton scattering processes than in exciton - polariton ones . the spatial integration of the data shown in figs . [ fig : fig2]-[fig : fig7 ] reveals the total energy and intensity decay dynamics for the different configurations under study as shown in fig . [ fig : fig11 ] : panels ( a - c)/(i - iii ) , correspond to one / two beam excitation under different @xmath19/@xmath22 powers . the addition of contributions from different population species gives rise to a very rich dynamics . figures [ fig : fig11 ] ( a - c ) exhibit a critical difference in the power dependence of the total decay : panel ( a ) shows a collective energy decay for * all * spatial positions along the ridge . panels ( b , c ) show a low energy streak corresponding to a polariton condensate drop that propagates along the ridge with an almost constant energy , unveiling the ballistic propagation of the condensate . the two streaks presented in panel ( i ) correspond to the decay of population at _ s _ ( high energy one ) and the emission of @xmath28 for a typical switch off state ( low energy one ) : the dynamics of both streaks is similar to those shown in panel ( c ) , with the difference , not appreciated in the figure , that polaritons now are stopped just before the _ g _ barrier . the three traces appearing in panel ( ii ) , ordered by decreasing energy , compile the emission from : the population at the source and the gate ( _ s+g _ ) , the @xmath28 condensate and the @xmath0 condensate , respectively . it is worthwhile noting the identical decay dynamics of the _ s _ and _ g _ populations , observed by the existence of only one streak for both populations . the @xmath0 condensate shows an emission at @xmath101.539 ev , with a dynamics similar to that shown in panel ( b ) . as the _ s _ power is kept constant in this subset of experiments , the _ g _ power permits manipulating on demand the amount of condensed polaritons at @xmath28 : if it would have been formed only by the _ s _ pulse , its energy should decay slightly ; however , the extra population injected by the _ g _ pulse contributes with an additional blueshift giving rise to an increase of the @xmath28 emission energy , hinted at @xmath10 300 ps in panel ( ii ) , which becomes clearly visible in panel ( iii ) . in this latter panel , s+g _ decays are also superimposed and @xmath0 emits at a constant energy of 1.540 ev for @xmath63 ps . it is important to note that in panels ( ii , iii ) , the additional polaritons provided by the _ g _ pulse make the @xmath28 condensate the highest populated state in the device with an emission intensity even larger than that of @xmath64 together . let us consider the optimal power conditions for the on state for a _ s - c _ spatial separation of @xmath65 @xmath8 m . we present in fig . [ fig : fig12 ] the main effects of the @xmath19 power on the transistor switch on state , which was illustrated before in figs . [ fig : fig2]-[fig : fig4 ] . figure [ fig : fig12 ] ( a ) plots the normalized intensity dynamics of the source , at @xmath25 , ( shadowed traces ) and that of @xmath0 , at @xmath66 @xmath8 m , ( full lines ) , for different @xmath67 values . we define the switch on time , @xmath68 , as the temporal delay between the _ s _ maximum intensity and that of the @xmath0 . it is clearly observed that @xmath68 decreases , and the shape of the @xmath0 time - evolutions becomes more asymmetric with increasing @xmath19 . the asymmetry of the @xmath0 temporal evolution ( see fig . [ fig : fig12 ] ( a ) ) , which strongly depend on @xmath19 , is characterized in fig . [ fig : fig12 ] ( b ) , where we define a raise time , @xmath69 ( up triangles ) , given by the time spent to raise from an intensity of 0.5 up to the maximum value of 1 . similarly , @xmath70 ( down triangles ) is given by the time interval in which the intensity falls from a value of 1 to 0.5 . a non monotonic dependence of @xmath70 on power is observed with a sharp raise at low @xmath19 values and a gradual fall for high ones : if the aim is to create a long lived on state , the optimal power corresponds to @xmath71 , where @xmath72 ps . on its own hand , the raise time , @xmath69 , decreases monotonically with increasing @xmath19 , reaching a minimum value of @xmath73 ps : a marked dependence at small powers , followed by an almost negligible decay at high ones , results in an optimum power to create a fast response transistor at similar powers than those required for a long lived on state . figure [ fig : fig12 ] ( c ) shows the power dependence of @xmath68 ( full circles ) together with the initial energy shift of the emission at _ s _ ( @xmath74 , open diamonds ) . the monotonous decrease of @xmath68 with power ( increase of switching rate ) is linked to the increase of @xmath74 , due to the enhanced polariton acceleration from _ s _ to _ c _ produced by the augmented photo - generated repulsive excitonic potential , but other contributions as , for example , increase of stimulated scattering processes in the creation of @xmath0 are also responsible for the quickening of @xmath68 . the minimum value of @xmath68 , @xmath75 ps , corresponds to a polariton propagation speed of @xmath76 @xmath8m / ps in agreement with the values of @xmath42 obtained from the horizon established by @xmath31 in the energy maps of fig [ fig : fig8 ] ( a - c ) . finally , we should mention that our results indicate that the optimal conditions for gating are obtained for @xmath77 , in agreement with the previous results of ref . . at this power , the maximum attenuation of the @xmath0 condensate is obtained yielding the highest contrast for the off state ; at lower values of @xmath22 the traveling polaritons are not gated efficiently and at higher values the gate starts feeding @xmath0 . to model our experimental results theoretically , we make use of a phenomenological treatment of polariton energy - relaxation processes taking place in the system . such processes are not only responsible for the relaxation of hot excitons ( injected by the pump ) into polaritons in the form of a condensate , but also for the further relaxation in energy of the polariton condensate as it propagates . this latter energy relaxation process can be strongly influenced by a spatially dependent potential coming from repulsion from the hot excitons . let us first introduce the description of the polariton condensate , which we will later couple to a description of higher energy excitons . a fundamental feature of bose - einstein condensates is their spatial coherence that allows them to be well described with a mean - field approach . @xcite the gross - pitaevskii equation has been developed to describe the non - equilibrium dynamics of condensed polaritons , where losses due to the short polariton lifetime @xcite and gain due to non - resonant pumping @xcite were included phenomenologically . in such form , a variety of recent experiments can be modeled , including , for example , experiments on polariton transport @xcite , spatial pattern formation @xcite and spin textures . @xcite the gross - pitaevskii equation for the polariton wave - function , @xmath78 , is : @xmath79\psi(x , t)\notag\\ & \hspace{5mm}+i\hbar\mathfrak{r}\left[\psi(x , t)\right]\label{eq : gp}\end{aligned}\ ] ] here @xmath80 represents the kinetic energy dispersion of polaritons , which at low wavevectors can be approximated as @xmath81 , with @xmath82 the polariton effective mass . @xmath83 represents the strength of polariton - polariton interactions . being repulsive ( @xmath84 ) , these interactions allow both a spatially dependent blueshift of the polariton condensate energy and energy - conserving scattering processes . our analysis shows , however , that neither of these effects play a dominant role in our experiments . the effective potential acting on polaritons caused by repulsive interactions between polaritons and higher energy excitons is more significant , and is responsible for the blocking of polariton propagation in the presence of a gate pump . the effective potential @xmath85 can be divided into a contribution from three different types of hot exciton states , which will be described shortly , as well as a static contribution due to the wire structural potential , @xmath86 : @xmath87\notag\\ & \hspace{5mm}+v_0(x)\end{aligned}\ ] ] @xmath88 , @xmath89 and @xmath90 correspond to density distributions of active " , inactive " and dark excitons , respectively , as described below . experimental characterization has revealed that the static potential , @xmath86 , is non - uniform along the wire and exhibits a slight dip in the potential near the wire edge . @xmath91 , @xmath92 and @xmath93 define the strengths of interaction with the various hot exciton states . @xmath88 represents the density distribution of an `` active '' hot exciton reservoir . @xcite these excitons have the correct energy and momentum for direct stimulated scattering into the condensate and so appear as an incoherent pumping term in eq . ( [ eq : gp ] ) with @xmath94 the condensation rate . to describe the dynamics of the system , it is important to note that not all excitons in the system are in this active form . in fact , the non - resonant pumping creates excitons with very high energy and they must first relax in energy before becoming active . we can thus identify an `` inactive '' reservoir of hot excitons that is excited by the non - resonant pump but not directly coupled to the condensate . the dynamics of the exciton densities are described by rate equations : @xmath95 @xmath96 @xmath97 when solving the equations we start from the initial condition @xmath98 , @xmath99 and introduce a density proportional to the pump intensity profile in the inactive reservoir , @xmath100 . this represents an instantaneous injection by the non - resonant ultra - short pulse used in the experiment . the inactive reservoir is coupled by both linear and non - linear terms to the active reservoir , described by @xmath101 and @xmath102 , respectively . we also account for a linear coupling to a dark exciton reservoir , @xmath103 , described by coupling rate @xmath104 . dark excitons are long - lived states that are optically inactive yet can nevertheless be populated as high energy excitations from the non - resonant pump relax in energy . the dark excitons introduce a long - lived repulsive contribution to the effective polariton potential , @xmath85 , and are thus efficient at gating propagating polaritons at long - times . @xmath105 , @xmath106 and @xmath107 describe the decay rates of each of the reservoirs . the feeding of dark excitons from the inactive reservoir represents processes where higher energy electron - hole pairs relax in energy forming dark exciton states . we have neglected any further conversion between bright and dark excitons . nonlinear conversion has been shown to generate oscillations between bright and dark excitons . @xcite however , these processes require coherent excitation of exciton - polaritons near the dark exciton resonance . in our case , we do not expect accumulation of exciton - polaritons at such an energy . furthermore , the fact that no oscillations in the polariton condensate density were observed , suggests that any coupling between bright and dark states is slower than the condensation rate . it is worthwhile mentioning that we have also considered hot - exciton diffusion along the wire when solving eqs . ( [ eq : na]-[eq : nd ] ) , using typical exciton diffusion rates ; however no noticeable effect on polariton dynamics was observed . returning to eq . ( [ eq : gp ] ) , the decay of polaritons is accounted for by the decay rate @xmath108 . the final term in eq . ( [ eq : gp ] ) accounts for energy relaxation processes of condensed polaritons . polaritons are expected to condense at the source into the lowest energy state , where they have zero kinetic energy and potential energy given by @xmath85 ( and an additional blueshift due to polariton - polariton interactions ) . while this is the lowest energy state available at the source , one notes that the potential energy can be reduced if polaritons propagate away from the source ( @xmath85 decreases away from the source , where the reservoir densities are weaker ) . if polaritons were to conserve their energy , then they would convert this potential energy into kinetic energy as they move away from the source , accelerating down the potential gradient . however , the polariton kinetic energy can be lost as polaritons scatter with acoustic phonons @xcite or hot excitons . @xcite surface scattering could be also responsible for this loss ; however since we have considered a phenomenological energy relaxation , the actual mechanism that causes that relaxation does not play a direct role in our calculations . previous methods to introduce energy relaxation into a description of polariton condensates have been based on the introduction of an additional decay of particles depending on their energy @xcite ( occasionally known as the landau - khalatnikov approach ) . the polariton number can be conserved in such a process via the introduction of an effective chemical potential . @xcite the energy relaxation term is : @xmath109=-\left(\nu+\nu^\prime|\psi(x , t)|^2\right)\left(\hat{e}_\mathrm{lp}-\mu(x , t)\right)\psi(x , t),\label{eq : relax}\ ] ] where @xmath110 and @xmath111 are phenomenological parameters determining the strength of energy relaxation . @xcite we do not attempt here a microscopic derivation of the energy relaxation terms , we only note that we can expect some energy relaxation at low polariton densities ( described by the parameter @xmath110 ) as well as a stimulated component of the relaxation proportional to the polariton density , @xmath112 ( described by the parameter @xmath111 ) . note that the energy relaxation rate is assumed proportional to the kinetic energy of polaritons ; polaritons will relax in energy until they decay from the system or until their kinetic energy is zero ( such that they have zero in - plane wave - vector ) . the local effective chemical potential , @xmath113 , can be obtained from the condition : @xmath114 where , @xmath115 for @xmath116 real , and : @xmath117\notag\\ & = \left.\frac{\partial\sqrt{n(x , t)}}{\partial t}\right|_\mathfrak{r}e^{i\theta(x , t)}\notag\\ & \hspace{10mm}+\left.i\sqrt{n(x , t)}e^{i\theta(x , t)}\frac{\partial\theta(x , t)}{\partial t}\right|_\mathfrak{r}.\end{aligned}\ ] ] @xmath118 denotes the components of the derivatives due to the term @xmath119 $ ] in eq . ( [ eq : gp ] ) . note that other terms in eq . ( [ eq : gp ] ) , namely the pumping and loss terms , do not conserve the number of condensed polaritons . although it would be desirable to define a mean free path between scattering events , this is not trivial because the energy - relaxation rate is both energy and density dependent . equations ( [ eq : gp]-[eq : nd ] ) were solved numerically for different initial density profiles @xmath100 , corresponding to the different source and gate configurations studied experimentally . we used the following parameters in the theory : @xmath120 ( obtained from fits to the dispersions measured in ref . ; @xmath121 is the free electron mass ) , @xmath122mev@xmath123 ( ref . ) , @xmath124 ps@xmath125 , @xmath126 ps@xmath125 , @xmath127 ps@xmath125 , @xmath128 ps@xmath125 , @xmath129 ps@xmath125 , @xmath130 ps@xmath131m@xmath4 , @xmath132 ps@xmath131m@xmath4 , @xmath133 ps@xmath131m@xmath4 , @xmath134 , @xmath135m@xmath4 . figure [ fig : time_1 ] shows the evolution of the spectrum in real space for @xmath15 . as in the experimental case ( fig . [ fig : fig2 ] ) , condensation initially takes place into a state blueshifted due to the repulsive interactions from the hot exciton reservoirs contributing to the effective potential @xmath85 . over time , this blueshift decays resulting in progressively lower energy of the source state . in addition , a propagating polariton state can be observed , which forms interference fringes due to reflection from the end of the ridge . energy relaxation is slow , appearing only at very long times due to the lack of stimulation by the low polariton density . for higher power ( @xmath45 ) , fig . [ fig : time_2 ] shows the onset of stimulated energy relaxation processes . as in the experimental case ( fig . [ fig : fig3 ] ) the relaxation takes place in two subsequent stages : first there is relaxation from the source state into the extended state with energy set by the ridge potential , followed by relaxation into the @xmath0 condensate at low energy . at @xmath46 , fig . [ fig : time_3 ] shows that the energy relaxation occurs rapidly . as in the experimental case ( fig . [ fig : fig4 ] ) the collector state is rapidly populated . it is interesting to note that , as shown for the experiments in fig . [ fig : fig12 ] ( c ) , the blueshift of the condensate at the source position does not increase linearly with the pump power . this is because even though the injected hot exciton population can be expected to increase linearly , the increased carrier density results in a faster condensation rate due to the stimulation of scattering processes ( hot exciton relaxation processes as well as processes that cause excitons to relax into condensed polaritons ) . polaritons decay much faster than uncondensed hot excitons , such that a high intensity pumping of hot excitons is quickly depleted giving rise to a limited blueshift of polaritons at the source . in the presence of the gating pulse , the propagation of the @xmath28 condensate is blocked , as shown in fig . [ fig : time_4 ] . this is due to the injected hot exciton density at the gate position that adds to the polariton effective potential profile , @xmath85 . at long times , the theory predicts a small transmission across the gate pulse , due to the decay of the potential barrier . spatial - temporal maps of the peak emission energy with one - beam excitation are shown in fig . [ fig : peak ] . in panel ( a ) there is a fast propagation of a high energy mode from the source followed by a decrease in energy of the emission over the whole space . at longer times , one identifies relaxation into @xmath0 , near the wire edge . this behaviour is in qualitative agreement with the experimental result , however , it can be noted that the speed of propagation appears overestimated in the theory . this is because the theory neglects changes in the shape of the polariton dispersion caused by the hot - exciton induced blueshift , which can be particularly important at early times when the particles are strongly excitonic with a larger effective mass and slower group velocity . figures [ fig : peak ] ( b ) and ( c ) show the peak emission energy maps for increasing source intensity , where the relaxation into an extended state with lower energy than the source can be identified . the relaxation is stronger at the highest pump power , due to increased stimulated energy relaxation processes . this is also evidenced by the shorter time required for @xmath0 to appear with increasing pump power . we discuss now the simulated energy- and intensity maps under two - beam excitation conditions compiled in fig . [ fig : peak2 ] . panel ( a ) shows the case when a gate pulse , @xmath18 , is present . at short times the energy of the collector state is low , although it should be noted that it is populated with a negligible density , panel ( d ) . the weak tunneling of particles across the gate is better evidenced by the increase of the collector state energy , since the tunneling particles have higher energy than the collector ground state . at these low gate powers , the theory appears to predict a high number of polaritons passing the gate . these polaritons have relatively high momentum and are expected to be less visible experimentally due to reduced photonic fractions . an increase of the gate power above threshold , panels ( b , c , e , f ) , leads to an enhanced collector signal , as in the experimental case , see figs . [ fig : fig9 ] ( b , c , e , f ) , and the device leaves the off state . this is expected as additional excited polaritons move directly from the gate to the collector . figure [ fig : it_all ] shows the time evolution of the spatially integrated spectra . in agreement with the experimental results ( fig . [ fig : fig11 ] ) there is a two timescale decay of the emission energy . at early times , the fast drop is due to the fast condensation rate in the presence of strong stimulated scattering . as mentioned earlier , this fast condensation rapidly depletes the hot exciton reservoir and the total particle density quickly drops as polaritons quickly decay . at longer times , reduced relaxation between the active and inactive reservoirs limits the effective condensation rate . the condensate is continuously fed while the reservoir intensities slowly decay . the short lifetime of the condensate for pumping at threshold ( fig . [ fig : it_all ] ( a ) ) is expected from the theoretical definition of threshold where the incoming rate @xmath136 is slightly larger than the polariton decay rate @xmath108 in eq . ( [ eq : gp ] ) . as soon as condensation starts , the reservoir density @xmath88 drops below threshold such that continued condensation can not take place . for higher pump powers , [ fig : it_all ] ( b , c ) show that both emission into a high energy mode , corresponding to the source , and a lower energy emission coexist . this fact is in close agreement with the experimental data shown in figs . [ fig : fig11 ] ( b , c ) and ( i ) , with the relaxation into the lower energy state occurring earlier for increased pumping power . in summary , we have time - resolved the energy and intensity relaxation processes of excitons and polaritons in a microcavity ridge . two different excitation configurations have been studied with one and two non - resonant , pulsed laser beams , permitting polariton condensate trapping on demand . a detailed analysis of the decay processes has been accomplished by mapping the energy and intensity emission along the ridge . decay times of the source emission are reported under one - beam excitation , where we show the acceleration of the decaying processes as a function of increasing @xmath19 . the time response of the polariton transistor switch is characterized and optimized.we used a generalized gross - pitaevskii model to describe the spatial dynamics of our propagating polariton condensates , which includes a phenomenological treatment of energy - relaxation processes that cause condensates to further thermalize as they travel in a non - uniform effective potential . the nonlinearity of energy - relaxation processes throughout the system , those causing relaxation between polariton states as well as relaxation between higher energy exciton states , is necessary to explain features of the experimental results . approximating the system as a 1d system , we are able to describe the main qualitative features of the experiment . while the system is essentially 1d , lateral expansion could result in a lower propagation speed than that predicted theoretically . we intend to investigate lateral propagation effects in future work . the optimization of individual condensate transistor elements , as we have reported here , is an essential step towards developing information processing devices with the present scheme . in the future , an important goal is the achievement of cascadability and fan - out of multiple elements for the construction of extended circuits . such a feat was very recently achieved in polariton based systems with coherent near - resonant excitation . @xcite achieving the same with the gating of incoherently generated polariton condensates , as we study here , would be particularly promising as it would open up routes toward electrically injected devices and consequently hybrid electro - optical processing systems . c.a . and g.t . acknowledge financial support from spanish fpu and fpi scholarships , respectively . p.s . acknowledges greek gsrt program aristeia " ( 1978 ) for financial support . the work was partially supported by the spanish mec mat2011 - 22997 , cam ( s-2009/esp-1503 ) and fp7 itn s clermont4 " ( 235114 ) , spin - optronics " ( 237252 ) and index ( 289968 ) projects . in this appendix we define the symbols used as abbreviations along the manuscript . 1@|c|>p0.8| * symbol & * meaning @xmath74 & energy shift of the emission at the source position@xmath59 & energy fast decay time at the source@xmath61 & energy slow decay time at the source__c _ _ & collector @xmath137 & polariton condensate trapped at the collector@xmath28 & polariton condensate trapped between source and gate positions@xmath138 & horizon of propagating polaritons along the ridge@xmath32 & horizon given by the interface between carries at the source and propagating polaritons__g _ _ & gate @xmath139 & emission intensity of the condensate trapped at the collector@xmath140 & emission intensity of the condensate trapped between source and gate@xmath141 & emission intensity at the source position@xmath22 & gate beam power@xmath19 & source beam power@xmath33 & pump power threshold for polariton condensation__s _ _ & source @xmath70 & time time for the collector intensity to drop from 1 to 0.5@xmath68 & time delay between maxima of the source and collector intensities@xmath69 & time time for the collector intensity to rise from 0.5 to 1@xmath42 & mean speed of propagating polaritons @xmath38 & mean speed of carriers at the source position * * 1@|c|>p0.8| * symbol & * meaning@xmath83 & polariton - polariton interaction strength@xmath108 & polariton decay rate@xmath105 & decay rate of the active exciton reservoir@xmath107 & decay rate of the dark exciton reservoir@xmath106 & decay rate of the inactive exciton reservoir@xmath110/@xmath111 & phenomenological parameters for the strength of energy relaxation@xmath78 & polariton wavefunction@xmath142 & lower polariton branch energy dispersion@xmath93 & polariton - exciton interaction strength for dark excitons @xmath92 & polariton - exciton interaction strength for indirect excitons @xmath91 & polariton - exciton interaction strength for reservoir excitons @xmath82 & polariton effective mass@xmath121 & free electron mass@xmath88 & density distribution of active excitons@xmath89 & density distribution of inactive excitons@xmath90 & density distribution of dark excitons * * @xmath94 & polariton condensation rate@xmath104 & linear coupling to the dark exciton reservoir@xmath101 & linear coupling to the inactive exciton reservoir@xmath102 & non - 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we present a time - resolved study of energy relaxation and trapping dynamics of polariton condensates in a semiconductor microcavity ridge . the combination of two non - resonant , pulsed laser sources in a gaas ridge - shaped microcavity gives rise to profuse quantum phenomena where the repulsive potentials created by the lasers allow the modulation and control of the polariton flow . we analyze in detail the dependence of the dynamics on the power of both lasers and determine the optimum conditions for realizing an all - optical polariton condensate transistor switch . the experimental results are interpreted in the light of simulations based on a generalized gross - pitaevskii equation , including incoherent pumping , decay and energy relaxation within the condensate .

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Proceed to summarize the following text: the effect of inhomogeneity on the critical behaviour of magnetic systems has been considered in various contexts ( e.g. disorder , coupling randomness , quasiperiodic structures ) ; in particular , discrete - spin models defined on fractal topologies possess critical properties significantly different and richer than those found for translationally invariant systems @xcite . the interest in fractal structures is not purely theoretical : many condensed - matter systems display strong nonuniformity on all length scales and can therefore be characterized as fractal objects ; examples include the backbone of percolation clusters , aggregates obtained from diffusion - limited growth processes , and absorbent surfaces . one of the most known fractals is the sierpinski gasket ( sg ) , which , due to its exact decimability , allows analytical approaches ; in particular , by means of renormalization group techniques , it was proved that the ising model on the sg exhibits phase transition only at zero temperature , while at any finite temperature the system breaks into domains and loses long - range order @xcite . while this result was found in the thermodynamic limit , at the mesoscopic sizes peculiar and interesting thermodynamic properties arise @xcite . more precisely , the ising model defined on a finite sg exhibits critical - like features at nonzero , low temperatures and the solution found in the thermodynamic limit turns out to be a poor approximation . this anomalous behavior has been investigated from a thermodynamic point of view and derives from long - range , slowly decaying correlations at low temperatures @xcite . the way such a behavior is reflected by the evolution of spin configurations is an item so far overlooked ( even if , for rectangular lattices , the idea of studying cluster dynamics may be traced back to peierls and griffith @xcite ) . given the importance of the dynamics of ising - like clusters in many research areas , from condensed matter to biological systems @xcite , the definition and the development of proper tools for this kind of analysis would be very useful . moreover , it would be particularly intriguing for inhomogeneous substrates , due to the emerging non - trivial thermodynamic behavior ; on the other hand , it is just on such structures that the definition of a proper metrization or evolutionary dynamics can be more awkward . the interest in cluster mobility actually extends to an extremely wide class of models : indeed the ising model on the sg may be seen as a particular realization of cellular automata on graphs , i.e. discrete time dynamical systems assigning to each node of a graph @xmath0 a value chosen in a alphabet @xmath1 , along a rule depending only on a finite neighborhood at previous time @xcite . in this work we aim to introduce and develop proper algorithms for the quantitative characterization of cluster dynamics on graphs , and we use this approach for the ising model on a finite sg , meant as a prototype of cellular automata on graphs . the procedure requires a projection of evolving configurations into an appropriate _ partition space _ , where an information - based metrics ( rohlin distance ) and a method measuring the effective emergence of configurational novelty ( reduction process and amplification parameter ) may be naturally defined and worked out in order to focus the changing and the stable components of configurations . the algorithmic implementations of rohlin distance and related quantities are deeply affected by the topological features of the substrate @xcite . for instance , in previous implementations designed specifically for automata on regular lattices , the very passage from one to two dimensions yields a much higher order of complexity , and could not be exported on different substrates @xcite . on the contrary , the algorithm developed here can be directly applied to generic cellular automata for which the metric characterization of cluster dynamics gets feasible ; a brief description is given in the appendix @xmath2 . our investigations on the ising sg highlight the existence of two `` critical '' temperatures , @xmath3 and @xmath4 demarcating three main regimes which recover , both qualitatively and quantitatively , the results of the previous thermodynamic analysis ; in addition , within the above mentioned regimes , we obtain a more detailed dynamical and geometrical characterization . in particular , at very low temperatures ( @xmath5 ) , a long - range order is established , and the few small - sized clusters display poor overlap from one time step to the next one : this makes the distance close to zero , and the reduction ineffective . as the spot sizes start to increase , the slowness of the evolution is still such that both non - similarity and overlap between successive configurations rapidly grow . at greater temperatures ( @xmath6 ) , large - scale correlations start to decay , clusters of all sizes appear , and overlap gets easier ; then , for @xmath7 , any trace of order has vanished and , even if overlaps are very important , the complexity of magnetic pattern is irreducible . the progression of such different kinds of non - similarity and dynamical overlap is well described by our parameters ( distance , amplification and intersection , see below ) . we will also evidence the existence of different scales controlling the disappearance of local and correlated order . thus , the phenomenology provided by our method , confirms with a deeper geometric insight the peculiar critical - like behavior exhibited by the system we finally notice that , in view of future extensions to non - equilibrium situations , we will adopt a microcanonical dynamics @xcite , which allows implementations even in the presence of temperature gradients . in the next two sections , after recalling basic notations on graphs , we review some facts concerning the thermodynamic properties of the ising model on the sierpinski gasket ( sec . [ sec : sg ] ) , and the microcanonical dynamics working as evolutionary dynamics ( sec . [ sec : micro ] ) . then , we introduce general procedures for cluster identification and reduction ( sec . [ sec : metrica ] ) and we show our results on the sg ( sec . [ sec : numerics ] ) . finally , we present our conclusions and perspectives ( sec . [ sec : con ] ) . technical remarks about partitions on graphs can be found in the appendices . a generic graph @xmath8 is mathematically specified by the pair @xmath9 consisting of a non - empty , countable set of points @xmath10 joined pairwise by a set of links @xmath11 . the cardinality @xmath12 of @xmath10 represents the number of sites making up the graph , i.e. its volume . from an algebraic point of view , a graph is completely described by its adjacency matrix @xmath13 . every entry of this off - diagonal , symmetric matrix corresponds to a pair of sites , and it equals @xmath14 if and only if this couple is joined by a link , otherwise it is @xmath15 . here we consider the sierpinski gasket which can be built recursively with the following procedure : the initial state ( @xmath16 ) is a triangle and the @xmath17th stage @xmath18 is obtained joining two of the three external corners of three @xmath19 to form a bigger triangle ( see fig . [ fig : sierp ] ) . in this way , the volume of @xmath20 is @xmath21 . the gasket is obtained as the limit for @xmath22 of this procedure . ( color on line ) sierpinski gasket of generation @xmath23 ( a ) , @xmath24 ( b ) and @xmath25 ( c ) , with volume @xmath26 , @xmath27 and @xmath28 , respectively . ] the ising model on a generic graph @xmath29 is defined associating the spin variable @xmath30 to every site @xmath31 of the graph , and considering a nearest - neighbours interaction between points @xmath31 and @xmath32 , such that @xmath33 . the hamiltonian is therefore @xmath34 where @xmath35 denotes the magnetic configuration of the system and the coupling @xmath36 is assumed to be the same for any couple ; in the following we will set @xmath37 . as it is well known , a magnetic model defined on a finite lattice can not exhibit critical behaviour at nonzero temperatures ; critical features can only emerge when the underlying lattice becomes infinitely large , i.e. in the thermodynamic limit . another necessary condition in order to have a nonzero critical temperature concerns the topology of the ( infinite ) substrate : for euclidean structures it was rigorously shown that the dimension must be larger than 1 . analogously , it has been shown that the discrete symmetry ising model on finitely ramified fractals can not have a nonzero critical temperature @xcite . however , the critical behaviour of one - dimensional systems and finitely ramified fractals can be markedly different , since for the latter it is further governed by additional geometric aspects such as ramification , lacunarity , and connectivity @xcite . the thermodynamic properties of the ising model on the sg were studied in details in @xcite , where it was shown that its scale - invariant , fractal structure leads to highly cooperative correlations and , at sufficiently low temperatures , the correlation length @xmath38 becomes extremely large and slowly decaying ( compared with the one - dimensional case ) , so that any system with size smaller than @xmath38 displays long - range order . indeed , one can define an apparent magnetic transition temperature @xmath39 as the point where @xmath40 ; for a system of generation @xmath17 ( @xmath41 ) , @xmath42 , for example , for generation @xmath43 and @xmath44 one finds @xmath45 and @xmath46 , respectively . hence , as the system size is enlarged , @xmath39 diminishes slowly ; more precisely , being @xmath47 the number of nodes for a gasket of generation @xmath17 , one has @xmath48 $ ] . on the other hand , the specific heat @xmath49 does not display any anomaly associated with long - range order , yet it exhibits a peak at a temperature @xmath50 @xcite . conversely , the `` reduced '' specific heat @xmath51 , ( basically the derivative of the magnetic energy per link with respect to the inverse temperature @xcite ) , evidences a qualitative difference between the sg and a one - dimensional system , since for the former it exhibits a peak , while for latter it grows monotonically as the temperature is increased . indeed , for the sg , as @xmath52 is increased from small values , the energy decreases rather slowly as a result of a relatively large cost due to the large ( bulk ) coordination number ; at larger temperatures it becomes progressively easier to reduce the energy and this reflects the fact that the large fluctuations begin to develop ; finally , since the energy must ultimately vanish , the rate of change gets smaller and smaller @xcite . in conclusion , the sg displays a non - trivial thermodynamic behavior which can be summarized as follows . at @xmath53 a long - range order is established ; for @xmath54 a short - range order is still present with large fluctuations on all length scales . at larger temperatures @xmath55 any trace of order has disappeared and a paramagnetic state is approached . when studying transport properties , microcanonical dynamics are usually chosen as they allow to describe an isolated system , or its isolated bulk , without any assumption on the equilibrium state between the system and the surrounding . here we adopt a recently introduced microcanonical dynamics @xcite , which features a high degree of flexibility , being ergodic in any temperature range and implementable on a generic structure , even in the presence of disorder . on regular lattices ( e.g. cylinder , torus ) such a dynamics has already been shown to be able to lead the system to thermalized states compatible with those expected from a canonical dynamics and to allow the study of out - of - equilibrium properties @xcite . although we will focus only on equilibrium regimes , the reason for choosing this dynamics is twofold : first we test its reliability on an inhomogeneous structure ; second , we pave the ground for the study of transport properties on such a substrate . in the following , we briefly resume how it works having in mind as substrate a generic graph @xmath8 described by the adjacency matrix @xmath13 . for each pair of connected sites , namely each link @xmath56 , such that @xmath33 , besides the magnetic energy @xmath57 , we introduce a local _ kinetic energy _ @xmath58 which is , in principle , unbounded . now , the dynamical rule proceeds as follows : 1 . start from a ( discrete ) distribution of energies @xmath59 ; 2 . choose randomly a link @xmath60 ; 3 . extract one over the possible four spin - configurations for the couple of sites @xmath61 , and evaluate the magnetic energy variation @xmath62 induced by the move ; 4 . if @xmath63 , accept the move and increase the link energy @xmath64 of @xmath65 . when @xmath66 , the move is accepted only if @xmath67 and the link energy is consequently decreased of @xmath65 ; otherwise the move is not accepted and the link energy is not updated . it is worth remarking that @xmath65 allows for energy variations occurred on the link @xmath60 as well as on those pertaining to links adjacent to sites @xmath31 or @xmath32 : @xmath68 where @xmath69 if the @xmath31th spin has undergone a spin flip , otherwise it is zero . it is therefore clear that , due to the discreteness of the system and to the fact that @xmath36 is constant over all links , both @xmath70 and @xmath64 are discrete variables : the former can only assume two different values corresponding to the aligned and non aligned configurations of the adjacent spins @xmath31 and @xmath32 ; the latter can only assume integer values deriving by proper combinations of the pertaining @xmath65 . we also notice that @xmath64 works as an additional degree of freedom and the above dynamics conserves the total energy given by the following hamiltonian function @xmath71 as shown in @xcite , the magnetic and kinetic energies result to be non - correlated : this allows a natural definition of temperature at equilibrium , which , in a very natural way , depends only on the average kinetic energy . in fact , the link energy satisfies the boltzmann distribution @xmath72 , and the fitted constant @xmath73 just corresponds to the expected inverse temperature of the system . finally , we stress that the possible coupling with thermostats set at a temperature @xmath74 can be realized straightforwardly by selecting a subset of links @xmath75 ( or , analogously a subset of nodes ) and by extracting the pertaining kinetic energies according to the boltzmann distribution , being @xmath76 @xcite . as foresaid , an interesting characterization of the ising model on the sg may be accomplished by a configurational analysis defined in the wider context of cellular automata on graphs . such an analysis can be realized by a particular metrization referring non directly to the configuration space , whose points are the states @xmath77 of the system , but to a peculiar _ partition space _ containing - among other elements - the cluster distributions of the system . the mathematical framework , which is summarized in the appendix @xmath78 , requires that the graph is endowed with the structure of a probability space . precisely , we consider the triple @xmath79 , where the measure @xmath80 on the subset algebra @xmath81 of the gasket @xmath0 is simply given by the normalized number of nodes in every subset . a _ configuration _ ( or _ state _ ) on @xmath0 is a function assigning to each node a value in an alphabet @xmath1 . the set of all possible configurations will be denoted as @xmath82 , the configuration space . since we consider the ising model on @xmath83 , the alphabet is binary , but all we are going to say is independent of the number @xmath84 . the adjacency matrix , combined with the list of values on nodes , allows an easy definition of clusters on an arbitrary structure ( the procedure is identical to the recognization of connected subset in graph colouring ) : two homogeneous nodes belong to the same cluster if they are connected through a path of nodes sharing the same value . thus , every state @xmath85 determines in a natural way a _ partition _ of @xmath83 , i.e. an exhaustive collection @xmath86 of disjoint subsets @xmath87 , each connected and homogeneous , commonly called _ atoms _ of the partition . the set of all partitions of @xmath0 constitutes the _ partition space _ @xmath88 . the application @xmath89 is many - to - one , since , for instance , permutations in @xmath1 produce different states but the same partition . obviously , @xmath90 contains much more partitions than those derived from clusterization , e.g. since , in general , atoms do not require to be connected sets , as clusters are . in the present case , @xmath91 is discrete and finite because @xmath0 is such , but the formalism applies in abstract probability spaces ( see @xcite ) . clearly , when a dynamics is defined on the graph , this determines configuration orbits @xmath92 starting from any initial state @xmath93 , and the corresponding partition orbits @xmath94 , where @xmath95 basic operations between two arbitrary partitions @xmath96 and @xmath97 are the minimal common multiple @xmath98 , and the maximal common factor ( m.c.f . ) or _ intersection _ @xmath99 ( see fig . [ fig : example ] and appendix @xmath78 for details ) . the entropy @xmath100 of the intersection @xmath101 , when calculated between partitions at next steps along an orbit , is an index of the relevance of the ( instantaneous ) non evolving part . the metrization of the partition spaces is based on the rohlin distance , which describes the non - similarity of two arbitrary partitions @xmath96 and @xmath97 . this distance , requiring the shannon conditional entropy @xmath102 of measurable partitions ( see eq . [ condiz ] ) , is given by the functional of eq . [ rohlin ] , we anticipate here : @xmath103 in order to amplify non similarity , in the appendix @xmath104 we present also a method , referred to as `` reduction process '' and denoted @xmath105 , which acts on couples of partitions and uses both operations @xmath106 and @xmath107 . more precisely , given two partitions , say @xmath96 and @xmath97 , their reduction is obtained by first defining their intersection @xmath99 and by keeping from both partitions only those subfactors @xmath108 and @xmath109 , prime with @xmath101 , namely such that @xmath110 . then , the reduced partitions are given by , @xmath111 and @xmath112 respectively ( see fig . [ fig : all ] , lower panel ) . the process @xmath105 gives evidence of the essentially different sub - partitions of any couple in @xmath113 , and therefore amplifies their distance : @xmath114 . hence , by comparing the distance between reduced and non - reduced couples , it is possible to introduce an _ amplification ratio _ , that is @xmath115)/d_r ( \alpha , \beta)$ ] , which provides further information about the cluster distribution and mobility . however , all this analysis has to be performed in correlation with other observables . as explained in appendix @xmath116 , the reduction process is effective , namely gives rise to a large amplification ratio , whenever one of the two partitions , say @xmath96 , displays at least one cluster which in @xmath97 is exactly decomposed into smaller ones . now , in the case under study @xmath96 and @xmath97 are partitions defined by cluster configurations at two successive steps , and the existence of a common cluster is furthered by the special topology of the sg : clusters corresponding to subgraphs which are ( combinations of ) gaskets of generation @xmath117 are rather stable ( the border is made by two vertices only ) , nonetheless internal fluctuations may occur and hence decompose the cluster itself . it is worth underlining that while the @xmath98 operation is rather trivial , the @xmath99 operation and the reduction process @xmath105 are quite tricky . reduction in particular is the main algorithmical obstacle in handling large graphs ( @xmath118 ) . of course , the rohlin distance is deeply different from the well known hamming distance @xmath119 in the configuration space , i.e. @xmath120 where @xmath121 and @xmath122 are the values of the @xmath123th node , @xmath124 is a distance functional in the alphabet . the simplest distance in @xmath125 is @xmath126 , leading to hamming distance 0.5 for purely random configurations . looking for instance to fig . [ fig : all ] ( upper panel ) , we would get the maximal hamming distance for the configurations @xmath127 and @xmath128 , and a minor distance for @xmath127 and @xmath129 , while the rohlin metrics on the corresponding partitions gives null distance in the former case and a high distance in the latter . in other terms , @xmath130 and @xmath119 are deeply different in principle not only because they refer to different objects ( defined in @xmath131 and @xmath82 , respectively ) , but also because the former has do to with mutual distribution of clusters , which could involve geometrical features and long range correlations , while the latter is the sum of strictly local differences . therefore , even if the correspondence @xmath132 is many - to - one , the loss of information should be compensated by the fact that by @xmath130 we get a global estimate on cluster distributions , instead of a bare sum of uncorrelated differences . finally , more details concerning the implementation of the algorithm for measure of the above mentioned metric observables can be found in appendix @xmath133 . we now focus on the ising model on the sg , with the microcanonical dynamics described in sec . [ sec : micro ] . the equilibrium regime is ensured by coupling the system to thermostats set at the same temperature ; in this particular case the simplest way is to consider , for a given generation @xmath17 , the external triangle , i.e the @xmath134 links defining the perimeter of the gasket ; if then we exclude the six angular links , we get @xmath24 separated thermostats . this could be useful in the future , as it consistutes a quite simple way of imposing temperature gradients to the system . however , it should be noticed that , due to the fact that the contact between the thermostats and the system gets vanishingly small with respect to the bulk as @xmath17 is increased , the time for thermalization is expected to grow with the size of the system . anyway , as mentioned , we are now interested only in the equilibrium behavior . before proceeding , we underline that our checks strongly confirm that the thermalized states reached by the microcanonical dynamics are consistent with those expected from a canonical dynamics , e.g. based on the metropolis algorithm . in particular , we verified that , for a given temperature , macroscopic observables like the magnetization and the energy measured with the two kinds of dynamics are indistinguishable . for any given temperature @xmath52 , the geometric observables have been calculated as time series starting after a thermalization time , and lasting an observation time @xmath135 , where time is measured in units of monte carlo ( mc ) steps ; i.e. @xmath136 elementary moves . in particular , we consider finite segments of trajectories @xmath137 in @xmath138 or @xmath139 in @xmath140 , for @xmath141 , as well as the related intersections @xmath142 and the couples of reduced partitions at successive time steps @xmath143 . from such equilibrium trajectories we obtain segments of time series for the following quantities : 1 . the entropy @xmath144 , applied either to the orbits @xmath145 or @xmath146 ; 2 . the rohlin distance @xmath147 , ( see eq . [ rohlin ] ) ; 3 . the amplified rohlin distance @xmath148)$ ] ; 4 . the amplification ratio @xmath149 ; 5 . the hamming distance @xmath150 , ( see eqs . [ eq : dh ] and [ hamming ] ) . here we focus to the case of unitary time steps ( @xmath151 equals one mc step ) , leaving the study of the role of the time gap to future investigations . after setting the experimental parameters ( size , thermalization time , confidence length of trajectories @xmath135 ) , we calculate time averages , variances etc . for each of the series above . results found for different choices of size ( we especially focused on gaskets of generation @xmath152 , @xmath43 and @xmath44 , corresponding to @xmath153 , @xmath154 and @xmath155 sites respectively ) and of @xmath135 are qualitatively in very good agreement . moreover , to approach the thermalized state , the initial configuration is taken ferromagnetic ; this minimizes the likelihood of pinning effects during the evolution @xcite . in the following we report only the essential information , dropping redundant numerical outputs . in general , the observables analyzed highlight the existence of different regimes , demarcated by remarkable temperatures @xmath156 and @xmath157 . more precisely , @xmath3 corresponds to the flex in the rohlin distance and to the peak in the amplification ratio , while @xmath158 corresponds to the peak in the variance of rohlin distance and to the crossover in the intersection entropy ( see figs . @xmath159 ) . interestingly , @xmath3 and @xmath158 recover the `` critical '' temperatures @xmath39 and @xmath160 evidenced by thermodynamic analysis ( see sec . @xmath23 and @xcite ) . indeed , consistently with thermodynamic results , the highlighted regimes correspond to a long - range order region and to a disordered region with a critical - like transition region in between . more precisely : [ fig : d_r ] as a function of the temperature ; three different sizes are compared.,title="fig:",scaledwidth=100.0% ] [ fig : d_h ] as a function of the temperature ; three different sizes are compared.,title="fig:",scaledwidth=100.0% ] * for @xmath161 , @xmath162 is close to @xmath14 ( see @xcite ) , while the distances @xmath119 and @xmath130 are approximately @xmath15 ( see figs . @xmath23 and @xmath24 ) ; in fact , at such small temperatures a ferromagnetic order is established over large length - scales : clusters are constituted by few single spots in the large `` sea '' of equally oriented spins and spin - flips are rather unlikely to happen . so , configurations - and partitions - at consecutive steps differ for such small spots that distances are extremely small . moreover , during a mc step , the rare spin flips occurring are yet able to change the atoms in such a way that overlaps between consecutive partitions are quite improbable . this inhibits the reduction , and the amplification is close to 1 . for @xmath52 approaching @xmath3 , such spots get larger , but , due to the slowness of the evolution at low temperature , their borders can remain sufficiently unchanged for several steps . hence , the emergence of spots within such clusters allows the reduction , processing couples of consecutive partitions , to get more effective , and the time average of the amplification is manifestly increased . the growth of the spots can be retrieved by the intersection entropy increase , as shown in the inset of fig . @xmath25 ( left panel ) . * for @xmath163 , @xmath162 is close to @xmath164 and both @xmath119 and @xmath130 start to be significantly larger than @xmath15 . this is a consequence of the fact that spin flips are getting more frequent . more interestingly , the amplification ratio reaches a maximum . we have seen that , at the middle of the previous regime , two conditions cooperated to start the growth of the amplification ratio : non empty intersection @xmath101 , and the fact that at consecutive steps there exist uniform large clusters which are decomposed internally , yielding an effective reduction ( as explained in appendix @xmath116 ) . here , the peak in the amplification ratio means that cluster sizes and the speed of the dynamics optimally fit such conditions ; moreover , fixed clusters are more likely . the exact determination of the peak temperature is tricky , due to the growing complexity of the configuration dynamics , but we argue that @xmath3 ( or @xmath39 ) is a good approximation . this behavior is consistent with the apparent transition occurring at @xmath39 as a long - range order breakdown . * for @xmath6 , @xmath162 is approaching zero . clusters get more intricate due to the growing temperature , and @xmath100 , which measures the relevance of overlapping between successive configurations , exhibits a rapid growth ( see fig . @xmath25 , left panel ) . the coexistence of fragmentation of large clusters , which is a signature of decay for long range correlations , and overlapping may be related to `` critical slowing down '' effects @xcite . clearly , fragmented clusters have a higher probability to overlap , but are unlikely to include small spots ; as a consequence the amplification ratio is still larger than @xmath14 , though rapidly decreasing . note that @xmath100 is rather far from saturation ( namely , uncorrelated chaos ) and this suggests the persistence of a local short - length order . * for @xmath165 , the amplification ratio is practically 1 : such a breakdown of the reduction , as forementioned , has a completely different meaning than in the case of very low temperatures , where overlapping was insignificant for the smallness of sparse clusters in the `` big sea '' of dominant magnetic orientation . here , due to the smallness of the @xmath101 atoms , it is unlikely that a sufficiently large cluster is internally decomposed at the next step ( see also the appendix @xmath116 ) . + at this temperature the variance of @xmath130 has a maximum ( whose value scales like @xmath166 ) , and this constitutes a signature that , along the trajectory , the fluctuation of the distance is particularly important . indeed , the complexity of clusters shape and their mobility can give rise to wide fluctuations of distances in time . also , @xmath100 exhibits a crossover : from @xmath158 , the growth is due only to fragmentation , while the `` critical slowing down '' is over . both the persistent growth of @xmath100 and the value of @xmath167 ( neatly below @xmath164 ) indicate that the complete chaos is far from being established at this temperature and for these sizes . moreover , the large fluctuations make the system more susceptible to spin - flips , i.e. energy changes , and this is consistent with the peak in the specific heat . * for @xmath168 , the rohlin variance decreases roughly as @xmath169 toward an asymptotic value corresponding to the uncorrelated chaos , whose onset may be recognized by @xmath170 . we argue that the fragmentation and the mobility of clusters stabilize the behavior of the time series for global quantities like distances . + [ fig : ampl ] ( left panel ) and amplification ratio @xmath171 ( right panel ) as a function of the temperature . for the latter , accurate numerical data are presented only for generation @xmath43 , due to time consuming calculations ; nonetheless we have checked that for other values of @xmath17 the qualitative behavior is robust . the inset of the left panel shows a magnification of the departure from zero of the entropy ; notice the semilogarithmic scale.,title="fig:",scaledwidth=100.0% ] + we note however that the maximum for the @xmath119-variance occurs at @xmath172 , well above the corresponding maximum of the @xmath130-variance at @xmath173 . we deduce that in this temperature interval big clusters of all sizes , typical of the previous fractal configurations , have started their fragmentation , so that the global shape diversity between subsequent configurations is stabilizing and this explains why the @xmath130-variance decreases . however , the fragmentation process is still slow enough that the local matching between different or equal spins at next steps is highly unstable in time , allowing for the growth of the @xmath119-variance . indeed , spins belonging to the inner part of a cluster ( even if of small size ) result from the dynamics to be more stable than those belonging to its periphery . for @xmath174 the fragmentation is such that both local overlaps and medium length correlations tend to stabilize the time behavior . there is therefore a temperature interval exhibiting a subtle interplay between correlation length and time stability . in other terms , in the way to the chaos we recognize two time scales : one in terms of global similitude ( shape ) of clusters along the orbits , which starts to stabilize just at the end of the `` critical slowing down '' ( @xmath173 ) ; the other in terms of local overlaps , whose stabilization requires a higher temperature ( @xmath172 ) . the direct visual inspection could be awkward for such properties , undetectable also from the mean magnetization . in this work we have performed a quantitative characterization of cluster dynamics for the ising model defined on a finite sierspinski gasket , whose thermodynamic behavior is known to be non - trivial . the analysis is based on a set of geometric observables , such as hamming and rohlin distances , and on reduction operations among partitions which allow to detect the evolving and the stable components of clusters . the phenomenology evidenced by previous thermodynamic analysis is qualitatively and quantitatively confirmed by the present metric approach , which provides in addition a geometric characterization of the anomalous behavior of the system . indeed , we highlight first the existence of two `` critical '' temperatures : @xmath3 corresponding to a peak in the amplification ratio , meaning that there an abrupt change in the clusters behavior occurs , and @xmath158 , corresponding to a peak in the rohlin variance and other crossovers , evidencing the loss of short range order . more precisely , the amplification ratio gives detailed information about the development of clusters , starting from little independent spots ( amplification ratio equal to 1 ) to first overlaps ( amplification ratio larger than 1 and growing to the maximum ) while the next phase of decreasing ratio indicates the complex effect of decreasing efficiency of the reduction due to the fragmentation , up to @xmath158 when the fragmentation is such that , notewithstanding the large intersection , the reduction process is inhibited by the extreme improbability of clusters with a decomposable inner part ( see [ ampli ] ) . another new information we obtain in the high temperature regime is the existence of different scales in the destruction of local and correlated order , as evidenced by the hamming and rohlin variances , with an anticipated peak for the latter quantity . thus , the analysis above enabled us to distinguish different behaviors and phases demarcated by @xmath3 and @xmath158 as main milestones ; we can not speak of course of `` phase transitions '' , since the onset of these distinct behaviors seems to be a rather smooth process . however , from such a detailed description of the collective motion we get an insight on the interplay between dynamic and statistical features , especially the decay of spatial and temporal correlations . this work opens a lot of possible extensions and further insights . the next step is the metric characterization of clusters dynamics in condition of non - equilibrium ; a kind of analysis allowed by the chosen dynamics even in the presence of a disordered coupling pattern @xcite . in particular , it could be interesting to deepen the effect of the double scale in the approach to uncorrelated chaos on the conductivity . in the same context , another interesting item to explore is the relevance of topology for the cluster diffusion . indeed , the whole set of operations performed on the sg , from cluster identification to reduction , immediately applies to any automaton on arbitrary , connected graphs : the process depends only on the adjacency matrix , on a generic alphabet and on a dynamics , working as a proper external engine generating a succession of configurations . therefore , by the bare substitution of the adjacency matrix , the algorithm is ready to fit a great variety of statistical models ( e.g. pott s model on lattices of arbitrary dimension or graphs ) , or other network problems where a dynamics is defined . if necessary , nodes could also be weighted , defining alternative probability measures . true algorithmic problems and non trivial extensions could only arise from alternative definitions of partitions ( assuming e.g. atoms of a different kind with respect to the clusters ) or from a different factorization , modifying the reduction process . the formalism and general results for partition spaces and rohlin metrics may be recovered e.g. in @xcite . let @xmath175 be a probability space , that is an arbitrary set @xmath176 , a @xmath177-algebra @xmath178 of subsets of @xmath176 , and a normalized measure @xmath179 on @xmath178 . in our case the set @xmath176 is just given by @xmath180 . a partition of @xmath176 is a finite collection @xmath181 of measurable disjoint subsets covering @xmath176 , i.e. @xmath182 and @xmath183 . the @xmath184 s are called the `` atoms '' of @xmath185 . the set of all finite measurable partitions is denoted @xmath186 . the unit partition @xmath187 consists of the single atom @xmath176 . a partial order in @xmath188 , i.e. a relation @xmath189 , means that @xmath97 is a refinement of @xmath96 ; equivalently , every @xmath190 is exactly composed with some @xmath191 included in @xmath97 . in such case , @xmath96 is said to be a `` factor '' of @xmath97 . clearly , @xmath192 for every @xmath96 . such terms as `` unit '' and `` factor '' depend on a commutative and associative pseudo - product , or composition , @xmath193 , denoting the less refined of all partitions greater or equal to both @xmath96 and @xmath97 , whose atoms are the non empty intersections of the @xmath96 and @xmath97 atoms . if not ambiguous , we can also write @xmath194 . obviously , @xmath195 whenever @xmath196 , and in particular @xmath197 for every @xmath96 . such properties make the result of this operation a kind of `` minimal common multiple '' . conversely , @xmath198 is the greatest partition such that @xmath199 and @xmath200 . in this case , @xmath201 for every @xmath202 , and @xmath203 implies that @xmath202 and @xmath204 are `` relatively prime '' ( i.e. they have no common factor ) . therefore , the result is a sort of `` maximal common factor '' . see fig . [ fig : example ] for an example of product and intersection among partitions . product ( upper panel ) and intersection ( lower panel ) between two examples of partition realized on a square lattice.,title="fig : " ] product ( upper panel ) and intersection ( lower panel ) between two examples of partition realized on a square lattice.,title="fig : " ] a partition may represent a probabilistic experiment with non overlapping outcomes @xmath205 , where the `` atomic '' event @xmath190 has probability @xmath206 . a factor is therefore a sub - experiment of the finer experiment , grouping several outcomes as equivalent : for instance , `` odd or even '' is a two - atoms sub - experiment of the @xmath207 dice experiment . the shannon s entropy @xmath208 defined on every partition as @xmath209 is a measure of the mean information obtained from the experiment . if @xmath210 is another partition , the conditional entropy of @xmath96 with respect to @xmath97 is @xmath211 where , as usual , one takes @xmath212 for @xmath213 . this conditional entropy is the mean residual information obtained from @xmath185 when the result of @xmath214 is known . note that the shannon entropy depends only on the distribution of the atom measures , not on their nature or `` shape '' ( this term coud have no meaning in abstract spaces ) . on the contrary , the mutual relations among atoms ( and possibly their shapes ) directly influence the conditional entropy ( see figure a.2 , upper row ) . now , the rohlin distance @xmath130 is defined in @xmath215 by @xmath216 and it may be considered as a measure of the overall non - similarity between @xmath217 and @xmath218 . if @xmath219 is finite , a _ configuration _ or _ state _ @xmath220 on @xmath219 is a function assigning to each point @xmath221 a value @xmath222 in an alphabet @xmath223 . all possible configurations form a space @xmath224 . in @xmath138 the hamming distance @xmath119 is defined by @xmath225 where @xmath226 is a distance in @xmath223 and @xmath227 a possible normalization coefficient . to each configuration corresponds an exhaustive collection of clusters , i.e. connected subsets of @xmath228 with homogeneous value in @xmath229 , defining a particular partition in @xmath230 . this establishes a many - to - one correspondence @xmath231 , making possible the comparison between @xmath232 in @xmath138 and @xmath233 in @xmath140 , where @xmath234 and @xmath235 . upper row : the three configurations depicted evidence the difference between hamming and rohlin distances : @xmath119 is maximal for configurations @xmath127 and @xmath236 , and minor for @xmath127 and @xmath237 ; @xmath130 is null in the former case and large in the latter . lower row : for the partitions @xmath96 and @xmath97 the common factor is given by @xmath101 , while @xmath238 and @xmath239 are the reduced partitions respectively ] 30.0 pt upper row : @xmath96 and @xmath97 are two partitions of the square , each of four atoms ; @xmath101 is their intersection or m.c.f . of their atoms . lower row : list of relevant elementary dichotomic factors , where the black specifies the atom and the white the complementary set . ] the partitions @xmath240 and @xmath241 are the reduced partitions of those appearing in figure [ fig : redaa ] : atoms are now three , individuated by black , white and grey . the partition @xmath242 is the reduced intersection . notice that the grey atom is non connected . ] 10.0 pt the essential non - similarity between two partitions could be confused and weakened by the presence of a tight common factor , i.e. a common sub - partition ( see fig . [ fig : all ] , lower panel ) . therefore , we would eliminate common factors as far as possible , a `` reduction '' which is expected to amplify the distance . however , this operation ( analogous to the reduction to minimal terms for fractions @xcite ) is not uniquely definite because partitions , differently from integers , do not admit a unique factorization into primes . the role of primes ( i.e. indecomposable ) factors can be played by dichotomic sub - partitions , which are still extremely redundant ( @xmath243 indeed for a partition with @xmath244 atoms ) . for @xmath181 we shall define therefore a restricted family @xmath245 of `` elementary '' dichotomic factors @xmath246 such that 1 . @xmath245 must be well defined for every @xmath247 ; 2 . @xmath245 does not contain more than @xmath244 ( the number of atoms in @xmath217 ) elementary factors ; 3 . @xmath248 . a universal choice consists in taking as dichotomic factors @xmath249 , the partitions formed by single atoms and their complements to @xmath250 . elementary factors of this form , used throughout in the present paper , will be called `` simple '' . for a couple @xmath217 and @xmath214 , once their elementary factors @xmath245 and @xmath251 have been defined , the reduction process consists in the following steps : 1 . define the maximal common factor @xmath198 ; 2 . drop from @xmath245 and @xmath251 those factors which are not relatively prime with @xmath177 , and note the surviving factors @xmath252 and @xmath253 respectively ( this means @xmath254 ) ; 3 . define @xmath255 and @xmath256 . the reason of step @xmath257 , which seems to be cumbersome with respect to the simple dropping of common factors in @xmath245 and @xmath251 , is that in general two families could have no common factors and , nevertheless , @xmath258 . this happens , for instance , when @xmath259 with no common elementary factors . then @xmath260 and @xmath261 with the reduction above , while @xmath262 with the dropping of common factors . it results that @xmath263 , as requested @xcite . note however that while two relatively prime partitions are already reduced , not necessarily two reduced partitions are relatively prime : see for instance the case of figure a.3 , which could represent small portions of wider partitions deduced from almost chaotic configurations . in the next subsection we give more precise details on this . the correspondence @xmath264 is many - to - one and idempotent , i.e. @xmath265 . it is a sort of projection from @xmath266 on the subset of irreducible pairs . the reduction process , therefore , essentially depends on the choice of the family @xmath245 of elementary factors . besides simple factors , other families exist , which in particular cases could prove more convenient for algorithmic reasons or for the observer s attitude in the probabilistic experiment . while simple dichotomic factors correspond to looking at the `` occurrence or not '' of single atomic outcomes , other attitudes could isolate dichotomic outcomes enjoying supplementary properties which are not universal but depend , typically , on some additional geometrical structure of @xmath250 ( order , connection , orientation etc . ) . for instance , in previous works on rectangular lattices @xcite elementary factors were identified by external contours of clusters , a choice intended to optimize the simple connection of the factor atoms . this is not convenient on general graphs , where @xmath267 , because the determination of external surfaces could be cumbersome or impossible . therefore , in the present work , elementary factors will always be the simple factors @xmath268 . 10.0 pt let @xmath96 and @xmath97 such that @xmath269 , and @xmath270 . every @xmath271 is the union of some subsets @xmath272 and @xmath273 of atoms in @xmath96 and @xmath97 . possibly , such subsets may be of one single atom or more : we indicate @xmath274 and @xmath275 the single or multiple atoms cases for @xmath96 , and analogously @xmath276 and @xmath277 for @xmath97 . clearly , at fixed @xmath278 , @xmath279 and @xmath280 , but it is useful to keep a distinct notation in order to remember that such atoms are not only in the partitions but also in their m.c.f . therefore , @xmath281 may be composed in four forms : @xmath282 , @xmath283 , @xmath284 and @xmath285 . for instance , in figure [ fig : all ] , the m.c.f . @xmath101 has one @xmath286 , one @xmath287 and three @xmath288 atoms , while , in figure [ fig : redaa ] , @xmath101 has one @xmath289 and two @xmath288 atoms . * proposition 1 : * the atoms of @xmath238 are all those of @xmath96 contained in the @xmath290 groups , plus one atom constituted by the intersection of their complementary sets , or equivalently by @xmath291 ( and similarly for @xmath239 ) . the proof immediately follows from the fact that the @xmath292 elementary factors are dropped in the reduction process if and only if @xmath293 . this is true in any abstract partition space , whenever simple factors are used ; notice however that , speaking of partitions generated by _ connected _ configurations , as those considered in the present paper , this last atom could be non - connected ( an example in the figure [ fig : redbb ] ) . * proposition 2 : * the atoms of @xmath294 are : _ i ) _ all the @xmath271 of the @xmath295 form ; _ ii ) _ one more atom ( the complementary part to the union of the previous ones ) if at least one term @xmath282 exists ; or _ iii ) _ two more atoms if , besides the @xmath295 , only mixed terms @xmath283 and @xmath284 exist . the point _ i ) _ follows from proposition 1 , since the @xmath296 and @xmath297 groups reconstitute common subpartitions in @xmath238 and @xmath239 ; point _ iii ) _ depends on the fact that , in absence of @xmath282 terms , there is an exact correspondence between @xmath298 and @xmath299 ( and equivalently @xmath300 and @xmath301 ) , where the @xmath278 index runs over the mixed - type @xmath281 s only ; the two supplemetary atoms are therefore @xmath302 and @xmath300 ; as to point _ ii ) _ , we observe that the complementary set of the @xmath303 , i.e. @xmath304 , can not be exactly decomposed by the atoms of @xmath305 , and vice versa , just because the presence of the supplementary simple terms . several easy corollaries follow . for instance , when simple factors are used , propositions 1 and 2 constitute a direct constructive proof that the reduction @xmath105 is a projection . or that there is no reduction at all when in @xmath101 there are only @xmath285 terms , or when @xmath101 has only two atoms . or else that in order to have @xmath306 , no terms @xmath285 are allowed and at least one term @xmath282 is necessary moreover , if for one partition , say @xmath96 , no @xmath307 group exists , then @xmath308 and @xmath309 . an important item is the relation between reduction ( meant as pattern simplification , as stated in proposition 1 ) and the amplification ratio @xmath310 , measuring the metric effectiveness of the reduction . * proposition 3 : * the necessary and sufficient conditions in order to have @xmath311 , and therefore amplification ratio @xmath312 , are the existence in @xmath101 of at least one atom of mixed form , @xmath283 or @xmath284 , and the existence of at least two distinct group @xmath313 from the same partition ( @xmath314 or @xmath97 ) . this may be proved using the important equality @xmath315 ( see @xcite ) . in the computation of all entropies appearing in eq . [ dr ] , one can split the shannon s sums ( see eq . [ shannon ] ) along the atoms @xmath316 of @xmath101 , because the intersections @xmath317 appearing in @xmath318 are surely empty for different @xmath281 s . therefore , using propositions 1 , for atoms of the @xmath285 form , the partial contributes to entropies in eq . [ dr ] are the same in @xmath319 and @xmath320 . in absence of mixed terms , the remaining part of both @xmath96 and @xmath97 is constituted by the same atoms @xmath321 , and the contribute to the distance is 0 , exactly as for the supplementary single atom @xmath322 common to @xmath323 and @xmath324 . assume now that also mixed terms @xmath283 and @xmath284 are present ( and possibly also terms @xmath282 ) in such a way that at least two @xmath325 or two @xmath326 appear . using again eq . [ dr ] for these components , @xmath327 and @xmath328 are equal , because the intersection of the multiple atoms in one partition give the same result with the single atoms in the other partition ( before reduction ) or with their union ( after reduction ) . as to the subtracted quantities in eq . [ dr ] , the contributes from @xmath208 and @xmath329 are strictly greater than @xmath330 and @xmath331 because there are contributes from separate atoms in the former case , and from their union in the latter ( this is why at least two @xmath325 or two @xmath326 are requested ) . clearly , elementary convexity properties of the @xmath332 function used here , see e.g. @xcite . therefore @xmath333 is strictly greater than @xmath334 , and after the subtraction in eq . [ dr ] the distance is increased , i.e. the amplification ratio is greater than one . proposition 3 clarifies that the metric effectiveness of the reduction , besides the pattern simplification , is due to the difference between @xmath335 and @xmath336 , where @xmath337 is @xmath96 or @xmath97 . in particular , if @xmath96 and @xmath97 are partitions defined by cluster configurations at two successive steps as those considered in the present paper , then the amplification requires that there exist big clusters at one time exactly decomposed at the next ( or previous ) time into smaller ones . 10.0 pt the @xmath338 adjacency matrix @xmath13 of the graph @xmath0 and its state @xmath339 , i.e. the @xmath340-valued list of sites labelled @xmath341 , constitute the essential information necessary to work out the partition algebra and the metrization algorithm , which is independent of the way the states are generated ( dynamics ) . by standard `` colouring '' techniques , connected sites are iteratively recognized testing their value in @xmath340 , and the partition is easily produced as a list of grouped labels , one group for every atom . for a ten sites graph , for instance , a particular partition could be @xmath342 , with measures @xmath343 for @xmath344 respectively . a useful representation for the atoms is a binary string of length @xmath136 : e.g. , in the example above , @xmath345 , etc . note that from now on the topological nature of the graph does not influence the operations , which regards only the label lists . the `` simple factors '' of the family @xmath346 are immediately defined . in the example , @xmath347 , etc . ( the corresponding binary strings are obviously complementary ) . clearly , the extreme simplicity of this procedure could be replicated with other choices of dichotomic factors , e.g. by taking the `` internal - close '' and the `` external - open '' parts of jordan contours ( provided that such contours are well defined ) . only in the very special case of one - dimensional chains the latter choice ( internal - external , i.e. left - right ) proves not only more effective but easier than the former , because contours are left extremities of semi - open intervals . the @xmath348 operation may be easily implemented by boolean intersections on the atom lists . this is enough for the rohlin distance computation by formula [ shannon ] , thanks to the equality of eq . [ dr ] above . more attention is required for the @xmath107 operation , which is the key step to define the maximal common factor @xmath99 , but the task may equally accomplished by boolean operators : every atom of @xmath101 is built site by site , testing the simultaneous belonging to some atoms of @xmath96 and @xmath97 , up to exhaustion . once the maximal common factor is defined , most of the computation time is spent in the storage and management of the surviving simple factors @xmath349 and @xmath350 , along the criteria of the subsection [ ampli ] . however , all this does not imply conceptually new operations . reduced partitions , and their atoms , which in principle are defined by intersections on the atoms of such surviving factors , can take advantage of proposition 1 in subsection [ ampli ] , completing the @xmath105 process . for the amplified distance @xmath351 , the functional in eq . [ dr ] is applied to the couple @xmath352 , and the observations of the previous subsection reveal useful again . this work is partly supported by the firb grant : @xmath353 y. gefen , a. aharony , b.b . mandelbrot and s. kirkpatrick , 1981 _ phys . lett . _ * 47 * 1771 . y. gefen , a. aharony and b.b . mandelbrot , 1983 _ j. phys . a _ * 16 * 1267 . y. gefen , a. aharony , y. shapir and b.b . mandelbrot , 1984 j. phys . a. * 17 * 435 . m. knezevic , j. joksimovic and d. knezevic , 2005 physica a. * 367 * 207 . r. campari and d. cassi , 2010 phys . rev . e * 81 * 021108 . a. vezzani , 2003 j. phys . a * 36 * 1593 . r. burioni , d. cassi and l. donetti , j. phys . a * 32 * , 5017 ( 1999 ) . liu , phys . b * 32 * 5804 ( 1985 ) j.h . luscombe and r. c. desai , phys . b * 32 * , 1614 ( 1985 ) . r. peierls , 1936 _ proc . cambridge phys . soc _ * 32 * , 477 . v.i.arnold and a. avez , _ problmes ergodiques de la mcanique classique _ , gauthier - villars , paris , 1967 . n. f. g. martin and j , w. england , _ mathematical theory of entropy _ , addison - wesley , reading ma 1981 .

we develop and implement an algorithm for the quantitative characterization of cluster dynamics occurring on cellular automata defined on an arbitrary structure . as a prototype for such systems we focus on the ising model on a finite sierpsinski gasket , which is known to possess a complex thermodynamic behavior . our algorithm requires the projection of evolving configurations into an appropriate partition space , where an information - based metrics ( rohlin distance ) can be naturally defined and worked out in order to detect the changing and the stable components of clusters . the analysis highlights the existence of different temperature regimes according to the size and the rate of change of clusters . such regimes are , in turn , related to the correlation length and the emerging `` critical '' fluctuations , in agreement with previous thermodynamic analysis , hence providing a non - trivial geometric description of the peculiar critical - like behavior exhibited by the system . moreover , at high temperatures , we highlight the existence of different time scales controlling the evolution towards chaos . + _ keywords _ : dynamical processes ( theory ) , classical phase transitions ( theory ) , classical monte carlo simulations

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Proceed to summarize the following text: bose - einstein condensates ( bec ) in optical lattices ( ol ) have recently attracted a great deal of attention due to the possibility of investigating , both at the theoretical and at the experimental level , interesting physical phenomena such as bloch oscillations , landau zener tunneling , mott transitions , etc . @xcite . the interplay between the nonlinearity ( intrinsic in the interatomic interaction ) and the periodic structure ( induced by the ol ) leads to the formation of localized states trough the mechanism of the modulational instability of the bloch states at the edges of the brillouin zone of the underlying linear periodic system @xcite . these states , also known as gap - solitons , can exist in presence of both attractive and repulsive interactions @xcite , this last fact being possible only due to the presence of the ol . the existence of gap solitons in repulsive bec was experimentally demonstrated in @xcite . the phenomena of bloch oscillations , generation of coherent atomic pulses ( atom laser)@xcite , superfluid - mott transition@xcite , were also experimentally observed . the ols considered in these experiments act as external potentials ( and therefore linearly ) on the condensate , this introducing an intrinsic ( state independent ) periodicity in the system . in the following we shall refer to this type of lattices as _ linear _ ols ( lols ) . in higher dimensions lols were shown to be very effective in stabilizing localized states against collapse or decay , leading to the formation of stable multidimensional solitons @xcite . besides lols , it is also possible to consider _ nonlinear _ ols ( nols ) with symmetry properties which depend on the wavefunction characterizing the state of the system . a nol can be obtained by inducing a periodic spatial variation of the two body interatomic interaction strength ( atomic scattering length ) , leading to a periodic space modulation of the nonlinear coefficient in the gross - pitaevskii equation ( gpe ) governing the mean field dynamics of the ground state . this periodic modulation can be experimentally achieved either by means of the standard feshbach resonance method @xcite , taking an external magnetic field near the resonance which is spatially periodic @xcite , or by the optically induced feshbach resonance technique . in the last case the nonlinear periodic potential can be produced by two counter propagating laser beams with parameters near the optically induced feshbach resonance @xcite . a periodic variation of the laser field intensity in space and a proper choice of the resonance detuning lead to a spatial dependence of the scattering length @xcite and hence to a spatial dependent nonlinear coefficient in the gpe . different interesting phenomena occurring in bec in presence of a nol have already been studied , such as the transmission of wave packets through nonlinear barriers , generation of atomic solitons , and existence of localized states refs . mathematical properties of the ground state and the existence of localized states of quasi-1d bec in nol have also been studied in @xcite . all these studies @xcite concern mainly with scalar ( single component ) 1d bec in nol . the possibility of stabilizing multi - dimensional scalar solitons by means of nols is presently under investigation ( preliminary studies show that nols are unable to stabilize 2d solitons if the average nonlinearity is negative ) , while multi - component becs in nol have not been considered yet neither theoretically nor experimentally . this last problem arises when two or more bec atomic species interact in presence of periodic spatial modulations of the scattering lengths , which can occur between the species ( inter - species ) and/or within the species ( intra - species ) . the interaction between the two bec components leads to an inter - species nol which can play a stabilizing role for localized states . spatial modulations of the intra - species scattering length ( giving rise to intra - species nols ) can also lead to the existence of new types of soliton states . the aim of the present paper is to study the properties of the localized states of two - component bec mixtures in nols . the case of a sinusoidal variation in space of the intra- and inter - species scattering lengths will be considered . in particular we show the existence of new types of solitons and study their stability by means of analytical and numerical methods . the symmetry properties of the localized with respect to the nol are also investigated . we show that the nol allows the existence of bright soliton modes with equal symmetry in both components , bright localized modes of mixed symmetry type , as well as bright - dark bound states and bright modes on periodic backgrounds . we also show that , in spite of the quasi 1d nature of the problem , the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the non linear optical lattice is varied . this transition is associated with the existence of an unstable solution which exhibit a shrinking ( decaying ) behavior for slightly overcritical ( undercritical ) variations in the number of atoms . the phenomenon of the delocalizing transition was also investigated in @xcite for the case of multidimensional single component bec solitons in lol and in @xcite for the case of one - dimensional bec s with combined linear and nonlinear ol s . delocalizing transitions in binary bec mixtures have not been previously investigated . for the analysis of strongly localized modes ( i.e. localized in one or few cells of the nol ) we will apply the variational approach which was shown to be effective for such type of problems , while for delocalizing transitions and broad solitons we use a vectorial gross - pitaevskii equation averaged over rapid variations in space of the nonlinear potential . results are then compared with those obtained by direct numerical simulations of the coupled gpe system . as numerical tools to investigate the above problems we use both self - consistent exact diagonalizations @xcite and generalized relaxing methods @xcite . the paper is organized as follows . in section ii we describe the physical model for the two component bec under action of a nol , based on the optical manipulation of the scattering length by optically induced feshbach resonances . the model equations are introduced in the mean field approximation in terms of two coupled 1d gross - pitaevskii equations with intra- and inter - species interaction terms . the problem of existence of soliton solutions ( when the inter- and intra - species atomic scattering lengths are periodically modulated in space ) , the symmetry properties of localized modes and their stability are discussed in sections iii . the delocalizing transitions of fundamental modes and the existence of unstable solutions associated with them are studied in section iv . the analytical predictions are confirmed by direct numerical simulations of the full gp equation ( sections ii - iv ) . finally , in section v , the main results of the paper are summarized . two component condensate represent the mixture of atoms in the different hyperfine states@xcite . we consider here the dynamics of two - component bec in presence of a nonlinear optical lattice produced either by spatially varying magnetic fields near a feshbach resonance ( fr ) value or by optically induced feshbach resonances @xcite . according to this last approach , the scattering length @xmath0 can be optically manipulated if the incident light is close to the resonance with one of the bound @xmath1 levels of electronically excited molecules . virtual radiative transitions of a pair of interacting atoms to this level , can change value and/or reverse the sign of the scattering length . the periodic variation of the laser field intensity in the standing wave , @xmath2 , produces periodic variation of the atomic scattering length , such that @xmath3,\ ] ] where @xmath4 is the scattering length in the absence of light , @xmath5 is the frequency detuning of the light from the fr and @xmath6 is a constant factor . for weak intensities , when @xmath7 , we have that @xmath8 . periodic variation of the scattering length by a spatially varying external magnetic field @xmath9 near a feshbach resonance ( fr ) , can be described by @xmath10 where @xmath11 is the resonant value and @xmath12 the corresponding width . examples are : a multicomponent bec of @xmath13na atoms@xcite or the mixture of @xmath14k - @xmath15rb atoms on the surface of a chip . the periodic variation of @xmath16 can be controlled by the current in a magnetic wire on the chip surface@xcite . for the mixture @xmath14k - @xmath15rb it was shown recently that the inter - species scattering length @xmath17 can be tuned using the feshbach resonances by varying the external magnetic field in the interval @xmath18 g @xcite . the mean field equations for the ground state wavefunction of a quasi-1d two - component bec under the action of a nol are given by the following coupled gp equations @xcite : @xmath19 where @xmath20 refer to the component index , and the full - dimensional space and time variables are given by @xmath21 and @xmath22 . in the above , @xmath23 is the parameter giving the strength of the inter - species nol and @xmath24 is directly related to the atomic scattering length of the species @xmath25 ( @xmath26 ) . in the following we fix the spatial dependence of @xmath27 and @xmath28 as @xmath29 introducing the dimensionless variables @xmath30 and @xmath31 where @xmath32 , @xmath33 , @xmath34 , @xmath35 . below we will consider the particular case @xmath36 , @xmath37 . we can rewrite the above pair of equations as @xmath38 where the normalization of the total wave - function @xmath39 is related to the components @xmath40 and to the number of atoms @xmath41 by the equation @xmath42 we remark that in experiments the magnitude and sign of both the inter- and the intra - species scattering lengths can be controlled by external magnetic fields @xcite or by counter propagating laser fields @xcite . in the case of immiscibility , when @xmath43 , the repulsive cross - interaction between the components affects strongly the self - interaction between components . in this section we perform an analytical study in the framework of the variational approach ( va ) for the case of localized ( soliton ) solutions of the form : @xmath44 , where @xmath45 are the chemical potentials . from eq . ( [ gp ] ) we have @xmath46 the total energy can be obtained from eqs . ( [ gp ] ) and ( [ norm ] ) : @xmath47 where @xmath48 @xmath49 + \right . \nonumber\\ & + & \left . \int_{-\infty}^{\infty}dx\ ; \sigma_{12}(x)\;|u_1|^2|u_2|^2 \right\}\frac{1 } { n_1 + n_2}.\end{aligned}\ ] ] the corresponding lagrangian is given by @xmath50 -\sigma_{12}(x)|u_1|^2 |u_2|^2 \nonumber \\ & = & \sum_{i=1}^2 \left [ \mu_i |u_i|^2 - \left|\frac{\partial u_i}{\partial x}\right|^2 -\frac { ( \gamma_{i0}+\gamma_i\cos(2x ) ) \nonumber\\ & - & ( g_0+g_1\cos(2x))|u_1|^2 |u_2|^2 .\end{aligned}\ ] ] in our variational approach we consider @xmath40 given by @xmath51 ^ 2}{2a_i^2}\right)\;\;(i=1,2),\ ] ] where the normalization @xmath41 is related to the number of atoms of the species @xmath25 , @xmath52 is the corresponding width , and @xmath53 is a parameter given the relative initial position of the two components . by substituting this ansatz in eq . ( [ energy ] ) and in the averaged lagrangian @xmath54 , we obtain : @xmath55 + \frac{n_1n_2}{\sqrt{\pi}}g\right\}\frac{1}{n_1+n_2 } , \end{aligned}\ ] ] @xmath56 - \frac{n_1 n_2}{\sqrt{\pi}}g,\end{aligned}\ ] ] where @xmath57.\nonumber\end{aligned}\ ] ] from the euler - lagrange equations @xmath58 , @xmath59 and @xmath60 we obtain the equations for the chemical potentials @xmath45 and number of atoms @xmath41 : @xmath61,\end{aligned}\ ] ] with @xmath62,\end{aligned}\ ] ] @xmath63 where @xmath64 from ( [ mui ] ) and ( [ energy1 ] ) , it also follows that @xmath65 for the particular choice of parameters for the symmetric case , when @xmath66 , @xmath67 , we have @xmath68 , @xmath69 , @xmath70 , and @xmath71 , with @xmath72 the equations for the chemical potential @xmath73 , energy @xmath74 , and number of atoms @xmath75 become ( @xmath76 ) @xmath77 + \nonumber \\&+ & \frac{n}{\sqrt{2\pi}\;a } \left[(g_0 + g_1e^{-a^2/2})e^{-x_0 ^ 2/2a^2}\right],\label{mus } \\ e & = & \frac{\mu}{2}+\frac{1}{4a^2 } , \\ \frac{n}{\sqrt{2\pi } a}&=&\frac{1}{\sqrt{2}a(p+q)},\;\;\;\;\ ; ( q\equiv q_i , p\equiv p_i ) \label{ns}\end{aligned}\ ] ] @xmath78 { e^{-\frac{x_0 ^ 2}{2a^2 } } } \right\}^{-1 } \nonumber\end{aligned}\ ] ] @xmath79 . \label{gx0}\end{aligned}\ ] ] for @xmath80 , we have @xmath81 , \label{mu1}\end{aligned}\ ] ] @xmath82 } \label{n1 } .\end{aligned}\ ] ] by using ( [ n1 ] ) in ( [ mu1 ] ) for the symmetric case with @xmath80 we obtain @xmath83 \label{mu2}.\end{aligned}\ ] ] , @xmath84 , @xmath85 , @xmath86 , with different values of @xmath53 , we show the va results for the number of particles @xmath75 , chemical potential @xmath73 and energy @xmath74 , versus the width @xmath87 . @xmath75 is given in the upper panel , with @xmath88 ( scale in the lhs ) and @xmath89 ( scale in the rhs ) given in the lower panel . , width=283,height=207 ] , @xmath84 , @xmath85 , @xmath86 , with different values of @xmath53 , we show the va results for the number of particles @xmath75 , chemical potential @xmath73 and energy @xmath74 , versus the width @xmath87 . @xmath75 is given in the upper panel , with @xmath88 ( scale in the lhs ) and @xmath89 ( scale in the rhs ) given in the lower panel . , width=302,height=207 ] , in the symmetric case with the same parameters as in fig . [ fig01 ] . exact results are also shown for the case of @xmath80 ( solid line with empty circles , in both panels ) . here we observe that the small unstable region , @xmath90 , presented by the va is not confirmed by the full numerical results.,width=283,height=207 ] , in the symmetric case with the same parameters as in fig . [ fig01 ] . exact results are also shown for the case of @xmath80 ( solid line with empty circles , in both panels ) . here we observe that the small unstable region , @xmath90 , presented by the va is not confirmed by the full numerical results.,width=283,height=207 ] , @xmath84 , @xmath91 , and @xmath92 , with different values of @xmath53 , we show the va results for the number of particles @xmath75 , chemical potential @xmath73 and energy @xmath74 , versus the width @xmath87 . @xmath75 is given in the upper panel , with @xmath88 ( scale in the lhs ) and @xmath89 ( scale in the rhs ) given in the lower panel . , width=275,height=207 ] , @xmath84 , @xmath91 , and @xmath92 , with different values of @xmath53 , we show the va results for the number of particles @xmath75 , chemical potential @xmath73 and energy @xmath74 , versus the width @xmath87 . @xmath75 is given in the upper panel , with @xmath88 ( scale in the lhs ) and @xmath89 ( scale in the rhs ) given in the lower panel . , width=283,height=207 ] , in the symmetric case with the same parameters as in [ fig03 ] . as we can see , the stable region ( @xmath93 ) is more pronounced for @xmath94 than for @xmath80.,width=283,height=207 ] , in the symmetric case with the same parameters as in [ fig03 ] . as we can see , the stable region ( @xmath93 ) is more pronounced for @xmath94 than for @xmath80.,width=283,height=200 ] the stability of the soliton solution can be investigated by using the vakhitov - kolokolov criterion @xcite ( in the present case , implying that for a stable system we should have @xmath95 ) , and also by studying the total energy @xmath74 and chemical potentials @xmath45 as functions of the width @xmath87 . for the symmetric cases ( when @xmath96 and @xmath97 ) , the results of such study is presented in figs . [ fig01 ] to [ fig04 ] , considering an attractive inter - species scattering length ( @xmath98 ) in figs . [ fig01 ] and [ fig02 ] ; and repulsive inter - species scattering length ( @xmath99 ) in figs . [ fig03 ] and [ fig04 ] . from figs . [ fig01 ] and [ fig03 ] we obtain the behavior of @xmath75 , chemical potential @xmath73 and energy @xmath74 as functions of @xmath87 . the behavior of @xmath73 versus @xmath75 , in order to check the vk criterion , is shown in figs . [ fig02 ] and [ fig04 ] . this stability study was done mainly by using the variational approach ( va ) , considering different values of the parameter @xmath53 , which gives the position of the soliton solution in respect to the optical lattice . in case of @xmath53 the va solutions are also compared with full numerical results in figs . [ fig02 ] and [ fig04 ] ( solid lines with empty circles ) . as observed , the va gives a good qualitative picture of the exact results , with improved quantitative results for large values of @xmath100 . the dominant @xmath101 behavior of the chemical potential @xmath73 and energy @xmath74 , as functions of the width @xmath87 , are removed in the bottom panels of figs . [ fig01 ] and [ fig03 ] ( by a multiplicative factor proportional to @xmath102 ) , in order to enhance their @xmath53 dependence . as we can verify , in both the cases , the most stable configuration is obtained when @xmath94 . as we can see in fig . [ fig02 ] , the single soliton is stable for @xmath103 . the va predicts the existence of small instability region , that is not confirmed by the numerical simulations of the system of gp equations . this instability region corresponds to the broad soliton case with @xmath104 , where the va approach is not applicable . ) with nol parameter values @xmath105 , @xmath84 , @xmath85 , @xmath86 . the top panel refers to the case of an equal number of atoms @xmath106 in the two components with equal chemical potentials @xmath107 . the bottom panel refers to the case of different number of atoms @xmath108 in the two components and different chemical potentials @xmath109 . dashed lines refer to second components.,width=283,height=207 ] ) with nol parameter values @xmath105 , @xmath84 , @xmath85 , @xmath86 . the top panel refers to the case of an equal number of atoms @xmath106 in the two components with equal chemical potentials @xmath107 . the bottom panel refers to the case of different number of atoms @xmath108 in the two components and different chemical potentials @xmath109 . dashed lines refer to second components.,width=283,height=207 ] in this section we investigate the symmetry properties of localized modes with similar number of particles in each component . these modes can be of equal symmetry or of mixed symmetry type . in order to find these solutions we use both the self - consistent exact diagonalization method and the generalized relaxing method described in the appendix ( these methods provide identical results for all the cases studied below , with the only exception of the state in fig . [ fig08 ] , for which the relaxation method was not effective ) . in fig . [ fig05 ] we show the fundamental modes obtained in the attractive case ( @xmath110 ) with equal and different number of atoms in the two components . ) of symmetry type is - is ( top panel ) and is - os ( bottom panel ) . the number of atoms and chemical potentials of the is - is mode are @xmath108 , @xmath111 , while for the is - os mode are @xmath112 , @xmath113 . the parameters of the nol for the is - is mode are fixed as @xmath105 , @xmath114 , @xmath85 , @xmath115 , while for the is - os mode are fixed as : @xmath105 , @xmath116 , @xmath117 , @xmath115 . dashed lines refer to second components.,width=283 ] ) of symmetry type is - is ( top panel ) and is - os ( bottom panel ) . the number of atoms and chemical potentials of the is - is mode are @xmath108 , @xmath111 , while for the is - os mode are @xmath112 , @xmath113 . the parameters of the nol for the is - is mode are fixed as @xmath105 , @xmath114 , @xmath85 , @xmath115 , while for the is - os mode are fixed as : @xmath105 , @xmath116 , @xmath117 , @xmath115 . dashed lines refer to second components.,width=283 ] . left panels refer to first components.,width=283,height=207 ] . left panels refer to first components.,width=283,height=207 ] in both cases we have that the maximum of the atomic densities are symmetric around the minimum of the corresponding effective potentials ( see appendix a ) . adopting the same terminology introduced in @xcite for the case of a lol , we shall refer to these modes as os - os ( onsite symmetric in both components ) . note that os - os - modes with equal number of atoms have the same chemical potentials , while for different number of atoms the component with a lower number of atoms has also a lower chemical potential . for sufficiently strong nol ( see below ) these modes are very stable under gpe time evolution and represent the fundamental ground states of the system in the case of all attractive interactions . in particular , the gpe time evolution of the density of the os - os mode in fig . [ fig05 ] with a different number of atoms does not show any deviation from the starting density for a time @xmath118 going from @xmath119 to @xmath120 besides onsite symmetry modes , it is also possible to have modes that are intersite symmetric ( is ) in one or in both components , i.e. symmetric around a maximum of the effective nonlinear potential instead than a minimum . such modes can be of type is - is ( intersite symmetric in both components ) such as the one shown in the top panel of fig . [ fig06 ] or of mixed type ( os - is or is - os ) such as the one shown in the bottom panel of fig . [ fig06 ] . in contrast with the os - os mode , the intersite symmetric localized modes are found to be unstable under gpe time evolution as one can see from figs . [ fig07 ] and [ fig08 ] for is - is and os - is modes , respectively . notice that in both cases the states decay into an os - os mode which is the true ground state of the system , and that in the is - os case the decay of the is component give rise to internal oscillations ( relative motion between the two final os components ) which can last for a long time . internal oscillations of the os - os modes can also be excited trough scattering with other modes . besides modes that are localized in both components it is also possible to couple a localized mode in one component with an extended mode in the other component such that the extended state acts as a periodic potential for the localized mode and forming a bound state . such an example is presented in figs . [ fig08 ] and [ fig09 ] for the case of a binary mixture with an average repulsive interaction for the first component ( @xmath121 , @xmath122 ) and an average attractive interaction for the second component ( @xmath123 , @xmath124 ) . this combination of signs for the interactions makes the ground state of the system to be extended for the first component and localized in the second one , leading to the formation of the dark - bright bound state depicted in the upper panel of fig . [ fig08 ] . another possible solution for the same combination of parameters is also verified , as shown in the lower panel of fig . [ fig08 ] , with the formation of a bright - bright state , having one bright solution on top of the background . for the considered parameters , both the solutions presented in fig . [ fig08 ] are quite stable under the gpe time evolution . in fig . [ fig09 ] we show the time evolution of the dark - bright state shown in the upper panel of fig . [ fig08 ] . ) , obtained with the nol parameters @xmath125 , @xmath84 , @xmath85 , and @xmath86 . in both the cases , we have the same chemical potentials @xmath1260.9476 and @xmath1272.746 . the resulting number of atoms in the two components are , respectively , @xmath1282 and @xmath1294.5 for the dark - bright solution ( upper panel ) ; and @xmath12864.246 and @xmath1290.444 for the bright - bright solution ( lower panel).,title="fig : " ] ) , obtained with the nol parameters @xmath125 , @xmath84 , @xmath85 , and @xmath86 . in both the cases , we have the same chemical potentials @xmath1260.9476 and @xmath1272.746 . the resulting number of atoms in the two components are , respectively , @xmath1282 and @xmath1294.5 for the dark - bright solution ( upper panel ) ; and @xmath12864.246 and @xmath1290.444 for the bright - bright solution ( lower panel).,title="fig : " ] . , width=328,height=366 ] in this section we investigate the existence of a delocalizing transition for the fundamental os - os symmetry mode . to this regard we recall that for a single component 1d bec with combined lol and nol there exists a threshold in the number of atoms below which the state becomes delocalized . in the limit of rapidly varying nols one can show , using the averaging method , that the system can be reduced to a nonlinear schrodinger equation with cubic and quintic nonlinearities for which the existence of a delocalizing transition is known . for binary bec mixtures the same method leads to a coupled system of cubic - quintic nls equations for which delocalizing transitions are also expected to exist . at the transition point the localized state becomes spatially more extended and displays properties similar to townes solitons of the 1d quintic nls system or of the 2d nls equation with cubic nonlinearity . for broad soliton states , i.e. when the soliton width @xmath130 becomes much larger than the periodicity scale @xmath131 , we can consider the expansion @xmath132 , with @xmath133 . at the leading order @xmath134 we obtain @xmath135 substituting into eq . ( [ gp ] ) and averaging over rapid oscillations we get for the slowly varying functions @xmath136 the following coupled system with cubic - quintic interactions @xmath137 ) at the critical strength of the inter - species nol @xmath138 for a delocalizing transition to occur . the mode has equal critical number of atoms @xmath139 with the same chemical potential @xmath140 and same profiles in both components . the dashed line represents the corresponding effective potential @xmath141 in eqs . ( [ self1 ] ) and ( [ self2 ] ) . other parameters are fixed as @xmath142 , @xmath84 , @xmath91 . bottom panel . time evolution of the townes soliton mode in the top panel as obtained from eq . ( [ gp]).,title="fig:",width=287 ] ) at the critical strength of the inter - species nol @xmath138 for a delocalizing transition to occur . the mode has equal critical number of atoms @xmath139 with the same chemical potential @xmath140 and same profiles in both components . the dashed line represents the corresponding effective potential @xmath141 in eqs . ( [ self1 ] ) and ( [ self2 ] ) . other parameters are fixed as @xmath142 , @xmath84 , @xmath91 . bottom panel . time evolution of the townes soliton mode in the top panel as obtained from eq . ( [ gp]).,title="fig : " ] when @xmath143 , we obtain a system of coupled quintic nls equations . for the symmetric case @xmath144 , the system reduces to the quintic nls equation @xmath145 with @xmath146 the townes soliton solution of eq . ( [ qnls ] ) is @xmath147 with norm given by @xmath148 this solution behaves as a separatrix between collapsing and decaying solutions of the quintic nlse . here @xmath149 and the vk criterion gives marginal stability . the total hamiltonian is equal zero on this solution @xmath150 . for example , for parameters values @xmath151 , we obtain the critical number @xmath152 . a comparison with the numerical results in fig . [ fig11ms ] shows that the averaged nls quintic equation overestimates the critical number @xmath153 by about @xmath154 percent(notice that for the same parameters values we have @xmath155 in fig . [ fig11ms ] ) . from this we conclude that the quintic nls can be used only as a qualitative model for the delocalizing transitions of two component bec in nol . in the following we shall investigate townes solitons and delocalizing transitions by recurring to numerical methods . delocalizing transitions in binary bec mixtures with nol and in coupled nls equations with cubic - quintic nonlinearities have not been previously investigated . to show the existence of this phenomenon in a binary bec mixture in a nol we vary in time the parameter @xmath156 characterizing the intra - species interaction while keeping fixed the inter - species nols to which the two component are subjected . starting from a given value of @xmath157 , for which a stable os - os mode exists , we adiabatically decrease @xmath156 to a value @xmath158 and then increase it back to the original value . in absence of delocalizing transitions the state will restore to its original form for any decrement @xmath159 , while in presence of a delocalizing transition a threshold value for @xmath159 will appear above which the state becomes irreversibly delocalized ( it can not be restored to its original form ) . in fig . [ fig10 ] we show the time evolution of an os - os symmetric state with an equal number of atoms in the two components , during a variation of the inter - species interaction in time according of the form @xmath160\}. \label{g1}\ ] ] . the top panel shows the time evolution of the mode when the inter - species parameter @xmath156 is varied according to eq . ( [ g1 ] ) with @xmath161 , @xmath162 , @xmath163 . the bottom panel shows the same evolution but for the case @xmath164 . other parameters are fixed as @xmath142 , @xmath84 , @xmath91 , @xmath165 . the chemical potentials of the initial states at @xmath166 are @xmath167.,title="fig:",width=321 ] . the top panel shows the time evolution of the mode when the inter - species parameter @xmath156 is varied according to eq . ( [ g1 ] ) with @xmath161 , @xmath162 , @xmath163 . the bottom panel shows the same evolution but for the case @xmath164 . other parameters are fixed as @xmath142 , @xmath84 , @xmath91 , @xmath165 . the chemical potentials of the initial states at @xmath166 are @xmath167.,title="fig:",width=321 ] from the top panel of this figure it is clear that for a small decrement @xmath159 the state is able to restore the initial waveform , while for a larger decrement the state becomes fully delocalized . in analogy to what has been done for the nls equation with periodic potential and quintic nonlinearity @xcite , one can characterize the delocalizing transition in terms of the unstable states which separate localized modes from extended ones . for the parameter used in fig . [ fig10 ] , the critical value in the strength of the nol for the occurrence of a delocalizing transition is found to be @xmath168 . in fig . [ fig11ms ] we show the existence of an unstable stationary state found in correspondence of this value , which has properties similar to the townes solitons of the quintic 1d nls or of cubic multidimensional nls . note that this stationary state corresponds to the unstable branch presented in the bottom panel of fig . [ fig04 ] [ see the exact results for @xmath80 ] . from fig . [ fig12ms ] , it is indeed clear that for slight undercritical variations of the norm ( number of atoms ) the state becomes delocalized , while for slight overcritical variations of the norm it shrinks to a fully localized mode , resembling the behavior of townes solitons . notice that due to the equal number of atoms @xmath170 the modes in the two components have identical chemical potentials and identical profiles . for an undercritical @xmath171 ( top panel ) and overcritical @xmath172 ( bottom panel ) number of atoms . other parameters are fixed as in fig . [ fig11ms].,title="fig : " ] for an undercritical @xmath171 ( top panel ) and overcritical @xmath172 ( bottom panel ) number of atoms . other parameters are fixed as in fig . [ fig11ms].,title="fig : " ] a delocalizing transition is also observed for os - os states with different number of atoms in the two components . in this case the system shows a much rich behavior due to the possibility to use the inter - species interaction to stabilize localized states which in absence of interaction would be extended over the whole system . an example of such inter - species induced localization is given in fig . [ fig14 ] for an os - os symmetric states of fig . [ fig13 ] with an unbalanced number of atoms ( a large difference in the number of atoms in the two components ) . in particular , in absence of the inter - species interactions , the first component has enough atoms to be above the delocalizing threshold , while the second component is taken to be below such a threshold , so that the state delocalizes in absence of interaction . from fig . [ fig14 ] we see , indeed , that the presence of the inter - species interaction prevents the second component to delocalize , while in absence of the inter - species interaction the first component remains localized and the second one delocalizes in a quite short time . due to the many parameters of the problem , a full investigation of the delocalizing transitions of the fundamental os - os mode in binary bec mixtures with nol requires more extensive numerical investigations . we plan to do this in a separated publication . ) with unbalanced number of atoms @xmath173 , and for nol parameters : @xmath174 , @xmath84 , @xmath85 , @xmath86 . the continuous ( dashed ) curve refer to the first ( second ) component . the chemical potentials of the modes are @xmath175 the dashed line refers to the second component.,width=283,height=283 ] in the presence ( top panels ) and in the absence ( bottom panels ) of the inter - species nol of strength @xmath176 . , width=321 ] in the presence ( top panels ) and in the absence ( bottom panels ) of the inter - species nol of strength @xmath176 . , width=321 ] in this paper we have investigated the localized states in two - component bec with periodic modulation in space intra - species and inter - species scattering lengths . the stability regions are analyzed using the variational approach and the vakhitov - kolokolov criterion . the symmetry properties ( with respect to the nol ) of the localized modes in each component were considered and their stability properties investigated . we showed that localized modes of os - os type are always stable and represents the fundamental ground states of the system in the presence of attractive interactions . intersite symmetric modes and mixed symmetry modes also exist but they appear to be metastable under gpe time evolution , decaying into modes of os - os - type . the existence regions in the parameter space of strongly localized modes ( localized on few cells of the nol ) of fundamental type were predicted by mean of the variational ansatz and their stability properties predicted by the vakhitov - kolokolov criterion . localized modes on tops of periodic backgrounds and of bright - dark solitons were also shown to exist in the case of binary mixtures with opposite interactions in the two components . in spite of the quasi 1d nature of the problem we showed that fundamental solitons undergo a delocalizing transition when the strength of the intersite non linear optical lattice is varied . this transition was associated with the existence of an unstable localized solution which extends on many lattice cells of the the nol and which exhibit a shrinking ( decaying ) behavior for slightly overcritical ( undercritical ) variations in the number of atoms . this behavior was shown to exists for fundamental modes both with equal and unequal numbers of atoms in the two components . the existence of the delocalizing transition for the fundamental modes was inferred also from a reduced vector gpe obtained by averaging the original gpe system with respect to the rapid spatial oscillations introduced by the nol . the process of averaging the nol introduces high order nonlinearities ( cubic - quintic ) which make the problem to be effectively equivalent to an higher dimensional vector gpe system for which delocalizing transition , in analogy to single component multidimensional cases , are usually expected . the study of the delocalizing transition for fundamental multi - component solitons in terms of an averaged vector gpe with higher order nonlinearities , as well as the extension of the above analysis to the multidimensional case , appear to be interesting problems which deserve further investigations . fka and ms wish to thank the instituto de fsica terica , universidade estadual paulista ( unesp ) for hospitality . for the financial support , which makes possible to realize this collaboration , we thank fundao de amparo pesquisa do estado de so paulo ( fapesp ) . ms acknowledges partial financial support from the mur through the inter - university project prin-2005 : transport properties of classical and quantum systems " . ag and lt also thank conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) for partial financial support . the numerical methods employed in this paper are described in the following subsections . we solve the nonlinear eigenvalue problem in ( [ gp ] ) by treating the nonlinear part in self consistent manner . this amounts to consider the following linear eigenvalue problem @xmath177 with the effective potentials defined as @xmath178 in the discrete coordinate space representation @xmath179 , @xmath180 , @xmath181 . here @xmath182 denotes the kinetic energy operator @xmath183 , @xmath184 is the length of the system and @xmath185 the number of grid points . by taking as a basis the set of vectors @xmath186 , @xmath187 and noting that @xmath188 is already diagonal in this basis while @xmath182 is diagonal in the momentum representation @xmath189 , we have that the matrix elements of @xmath190 can written as @xmath191 where @xmath192 denotes the fourier ( unitary ) transform of the vector @xmath193 . standard diagonalization routines are then used to find eigenvalues ( chemical potentials ) and eigenfunctions . the nonlinear eigenvalue is then solved in self - consistent manner starting from trial wavefunctions @xmath194 , @xmath195 , calculating the effective potentials , solving the eigenvalue problems ( [ self1 ] ) by diagonalizing the correspondig matrices ( [ matrices ] ) , selecting given eigenstates as new trial functions , and iterating the procedure until convergence is reached ( see refs.@xcite for applications to single and multicomponent bec cases ) . stable states are obtained using standard relaxation algorithm in imaginary time propagation , fixing the normalizations given by number of atoms of the two species , @xmath196 and @xmath197 , and obtaining the chemical potentials @xmath198 and @xmath199 . for the hyperbolic ( unstable ) states we extended to a coupled equation system the method developed in ref.@xcite , scheme c , in which the idea of back renormalization " was used . in this method , it is given the chemical potential to obtain the number of atoms . for a coupled system , the scheme c of ref . @xcite can be generalized , evolving the following equations in imaginary time : @xmath200 where we have normalized @xmath201 and @xmath202 to one , such that @xmath203 and @xmath204 . @xmath205 and @xmath206 are given by eq . ( [ gammas ] ) . where the superscripts ( @xmath210 , @xmath211 , etc ) refer to time steps . @xmath212 is the crank - nicolson evolution operation corresponding to @xmath213 . note that , in this coupled system the back renormalization ( of @xmath214 and @xmath215 ) is done by exchanging the corresponding wavefunctions ( as @xmath196 is associated to @xmath201 and @xmath197 to @xmath202 ) . this procedure is required for stability , as verified in numerical tests . the excited states is - os and is - is depicted in fig . [ fig06 ] can be obtained by relaxing eqs . ( [ ap1])-([ap12 ] ) for @xmath216 and imposing the von neumann boundary conditions in the origin , i.e. , at @xmath217 , @xmath218 and @xmath219 . the present relaxation algorithms are unable to find the state shown in fig . [ fig08 ] , which was obtained by the approach given in subsection a. b.b . baizakov , v.v . konotop , and m. salerno , j. phys . b * 35 * 5105 ( 2002 ) ; b.b . baizakov , b.a . malomed , and m. salerno , europhys . lett . * 63 * , 642 ( 2003 ) ; e.a . ostrovskaya , yu.s . kivshar , phys . * 90 * , 160407 ( 2003 ) . s.inouye et al . nature(london ) * 392 * , 151 ( 1998 ) ; j. stenger et al . lett . * 82 * , 2422 ( 1999 ) ; j.l . roberts et al . , phys . . lett . * 81 * , 5109 ( 1998 ) ; s.l . cornish , n.r . claussen , j.l . roberts , e.a . cornell , and c.e . wieman , phys . lett . * 85 * , 1795 ( 2000 ) ; e.a . donley et al . , nature(london ) * 412 * , 295 ( 2001 ) .

the properties of the localized states of a two component bose - einstein condensate confined in a nonlinear periodic potential [ nonlinear optical lattice ] are investigated . we reveal the existence of new types of solitons and study their stability by means of analytical and numerical approaches . the symmetry properties of the localized states with respect to the nol are also investigated . we show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components , bright localized modes of mixed symmetry type , as well as , dark - bright bound states and bright modes on periodic backgrounds . in spite of the quasi 1d nature of the problem , the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied . this transition is associated with the existence of an unstable solution , which exhibits a shrinking ( decaying ) behavior for slightly overcritical ( undercritical ) variations in the number of atoms .

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Proceed to summarize the following text: the object of this paper is to carry on studying the local potential approximation of the exact renormalization group ( rg ) equation for the scalar theory @xcite . in a previous publication @xcite ( to be considered as the part i of the present work ) , we had already considered this approximation with a view to qualitatively discuss the connection between the standard perturbative renormalization of field theory ( as it can be found in most textbooks on field theory , see for example @xcite ) and the modern view @xcite in which the renormalized parameters of a field theory are introduced as the `` relevant '' directions of a fixed point ( fp ) of a rg transform . actually the local potential approximation , which allows us to consider all the powers of the field @xmath7 on the same footing , is an excellent textbook example of the way infinitely many degrees of freedom are accounted for in ( nonperturbative ) rg theory . almost all the characteristics of the rg theory are involved in this approximation . the only lacking features are related to phenomena highly correlated to the non local parts neglected in the approximation and when the critical exponent @xmath8 is small ( especially for @xmath9 and @xmath10 ) , one expects the approximation to be qualitatively correct on all aspects of the rg theory @xcite . in the following we look at the domains of attraction or of repulsion of fixed points in the @xmath0 scalar theory in three and four dimensions ( @xmath10 and @xmath9 ) . at first sight , one could think that the issue considered is very simple since , with regard to criticality , the @xmath0-symmetric systems in three dimensions are known to belong to the same class of universality ( the ising class ) . now , because the ising class is associated to the domain of attraction of the unique ( non - trivial ) wilson - fisher fixed point @xcite , then by adjusting one parameter ( in order to reach the critical temperature ) any @xmath0 scalar hamiltonian should be driven to the wilson - fisher fixed point under the action of renormalization . consequently there would have only two domains for the @xmath0 scalar theory : the critical subspace @xmath1 ( of codimension 1 ) in the wilson space ( @xmath11 ) of infinite dimensions of the hamiltonian parameters ( in which the rg transforms generate flows ) and the complement to @xmath11 of @xmath1 ( corresponding to noncritical hamiltonians ) . in fact , that is not correct because there is another fixed point in @xmath11 : the gaussian fixed point which , although trivial , controls tricritical behaviors in three dimensions . now each fp has its own basin of attraction in @xmath11 @xcite . the attraction domain of the gaussian fp is the tricritical subspace @xmath4 of codimension 2 ( with no intersection with @xmath1 ) . in addition , we show that there is a second subspace of codimension 1 in @xmath11 , called @xmath2 , which is different from @xmath1 , and thus which is not a domain of attraction to the wilson - fisher fixed point . there is no fp to which a point of @xmath2 is attracted to . @xmath2 is characterized by a negative sign of the @xmath12-hamiltonian parameter @xmath13 and is associated with systems undergoing a first - order phase transition @xcite . we show that an endless attractive rg trajectory is associated to this domain of first - order transitions . it is a renormalized trajectory ( denoted below by t@xmath14 ) that emanates from the gaussian fixed point . the frontier between @xmath15 and @xmath1 corresponds to the tricritical subspace @xmath4 which is the domain of attraction of the gaussian fixed point while @xmath16 and @xmath1 are two distinct domains of repulsion for the gaussian fixed point . actually , the situation is conform to the usual view . considering the famous @xmath5-model [ landau - ginzburg - wilson ( lgw ) hamiltonian ] in which the associated coupling @xmath13 is positive , then the hamiltonian at criticality is attracted exclusively to the wilson - fisher fixed point , but if @xmath13 is negative , a @xmath17-term is required for stability , but then one may get either a tricritical phase transition or a second- or a first - order transition @xcite . in the present study we do not truncate the hamiltonian which involves all the powers of the field @xmath7 . we explicitly show that a system which would correspond to an initial point lying very close to the frontier @xmath4 in the critical side ( in @xmath18 ) would display a retarded classical - to - ising crossover @xcite . this result is interesting with regard to ionic systems ( for example ) in which a classical behavior has been observed while an ising like critical behavior was expected . the eventuality of a retarded crossover from the classical to the ising behavior has previously been mentioned but without theoretical explanation on how this kind of crossover could develop @xcite . in @xcite a calculation suggests that the rpm model for ionic systems would specifically correspond to a scalar hamiltonian with a negative sign for the @xmath5-hamiltonian parameter ( but the order parameter chosen is not the bulk density @xcite ) . that calculation has motivated the present study . we also indicate that the renormalized trajectory t@xmath19 still exists in four dimensions . this makes the gaussian fixed point ultraviolet stable and the scalar field theory _ formally _ asymptotically free . however the associated `` perfect '' action @xcite would have the wrong sign to provide us with an acceptable ( well defined ) field theory . the organisation of the paper is as follows . in section [ lpa - erge ] we briefly present the local potential approximation of the exact rg equation to be studied . we introduce the strategy we have chosen to solve the resulting nonlinear differential equation with a view to show the trajectories of interest in the space @xmath11 of infinite dimension . because the practical approach to the gaussian fixed point is made difficult due to the logarithmic slowness characteristic of a marginally irrelevant direction ( for @xmath10 ) , we found it useful to first test our numerical method with a close approach to the wilson - fisher fixed point . we present the characteristic results of this approach and the various kinds of domains corresponding to @xmath20 ( somewhat a summary of @xcite ) in section [ uneg ] we describe the various kinds of attraction or repulsion domains of the gaussian fixed point ( for a negative value of the @xmath12-hamiltonian parameter ) corresponding to tricritical , critical and first - order subspaces . then we discuss the consequences and especially explicitly show how a retarded crossover from the classical to the ising behavior can be obtained . we then shortly discuss the case @xmath9 when @xmath21 . in two appendices we report on some technical aspects of the numerical treatment of the rg equation studied , in particular on the appearance of spurious nontrivial tricritical fixed points ( appendix [ spurious ] ) . the local potential approximation has been first considered by nicoll et al @xcite from the sharp cutoff version of the exact rg equation of wegner and houghton @xcite , it has been rederived by tokar @xcite by using approximate functional integrations and rediscovered by hasenfratz and hasenfratz @xcite . as in @xcite we adopt the notation of the latter authors and consider the following nonlinear differential equation for the simple function @xmath22 : @xmath23 in which a prime refers to a derivative with respect to the constant dimensionless field @xmath7 ( at constant @xmath24 ) and @xmath25 is the derivative of the dimensionless potential @xmath26 ; @xmath27 stands for @xmath28 in which @xmath24 is the scale parameter defined by @xmath29e@xmath30 and corresponding to the reduction to @xmath31 of an arbitrary initial momentum scale of reference @xmath32 ( the initial sharp momentum cutoff ) . finally , @xmath33 is the surface of the @xmath34-dimensional unit sphere divided by @xmath35 . a fixed point is solution of the equation @xmath36 . the study of the resulting second order differential equation provides the following results : * @xmath37 , no fp is found except the gaussian fixed point . * @xmath38 , one nontrivial fp ( the wilson - fisher fixed point @xcite ) is found @xcite * a new nontrivial fp emanates from the origin ( the gaussian fixed point ) below each dimensional threshold @xmath39 , @xmath40 @xcite . if one represents the function @xmath22 as a sum of monomials of the form : @xmath41 then , for @xmath10 , the wilson - fisher fixed point @xmath42 is located in @xmath11 at @xcite : @xmath43 , @xmath44 , @xmath45 , @xmath46 , etc . once the fp is known , one may study its vicinity which is characterized by orthogonal directions corresponding to the infinite set of eigenvectors , solutions of the differential equation ( [ eq4 ] ) linearized at @xmath42 . the eigenvectors associated to positive eigenvalues are said relevant ; when the eigenvalues are negative they are said irrelevant and marginal otherwise @xcite . the relevant eigenvectors correspond to directions along which the rg trajectories go away from the fp and the irrelevant eigenvectors correspond to directions along which the trajectories go into the fp . a marginal eigenvector may be relevant or irrelevant . our present fp @xmath42 has only one relevant direction and infinitely many irrelevant directions ( no marginal direction , however see @xcite ) . as already explained and shown in @xcite , in order to approach @xmath42 starting from an initial point in @xmath11 , one must adjust one parameter of the initial function @xmath47 . this amounts to fixing the temperature of a system to its critical temperature . starting with a known initial function ( at `` time '' @xmath48 ) say : @xmath49 we adjust @xmath50 to the critical value @xmath51 corresponding to @xmath52 so that @xmath22 [ solution at time @xmath24 of the differential equation ( [ eq4 ] ) ] approaches @xmath42 when @xmath53 . the approach to @xmath54 is characterized by the least negative eigenvalue @xmath55 ( @xmath56 was noted @xmath57 in @xcite ) . this means that , in the vicinity of @xmath42 any parameter @xmath58 evolves as follows ( @xmath53 ) : @xmath59 fig . [ fig1 ] illustrates this feature for the first four @xmath58 s in the approach to @xmath42 . in @xcite the two associated attractive trajectories ( locally tangent to the least irrelevant eigenvector in the vicinity of @xmath42 ) was noted t@xmath60 and t@xmath61 . one may also constrain the trajectory to approach @xmath42 along the second irrelevant direction ( with the associated attractive trajectories noted t@xmath62 or t@xmath63 in @xcite and associated with the second least negative eigenvalue @xmath64 ) . in this case a second parameter of the initial @xmath65 must be adjusted , e.g. , @xmath66 must be adjusted to @xmath67 _ and _ _ simultaneously _ @xmath68 to the corresponding @xmath69 , see @xcite . then , in the vicinity of @xmath42 , any parameter @xmath70 will evolve as follows : @xmath71 looking for this kind of approach to @xmath42 , we have found that @xmath72 and @xmath73 this has allowed us to estimate @xmath74 . although the shooting method is certainly not well adapted to the determination of the eigenvalues ( see the huge number of digits required in the determination of @xmath75 and @xmath67 ) , our estimate is close to @xmath76 found by comellas and travesset @xcite . because @xmath67 can not be perfectly determined , the trajectory leaves the trajectory t@xmath77 before reaching @xmath42 to take one of the two directions t@xmath60 or t@xmath78 ( corresponding to @xmath56 ) . fig [ fig2 ] illustrates this effect with the evolution , for @xmath79 , of the following effective eigenvalue : @xmath80 the definition of which does not refer explicitly to @xmath42 . the evolution of @xmath81 shows a flat extremum ( or a flat inflexion point ) at an rg eigenvalue of @xmath42 each time the rg flow runs along an eigendirection in the vicinity of @xmath42 . similarly to @xmath67 , the value @xmath69 can not be perfectly determined , consequently the trajectory ends up going away from the fixed point . this provides us with the opportunity of determining the only positive ( the relevant ) eigenvalue corresponding to the critical exponent @xmath82 [ @xmath81 shows then a flat extremum at @xmath83 when the flow still runs in the close vicinity of @xmath42 ] . finally , far away from the fixed point , the rg trajectory approaches the trivial high temperature fixed point characterized by a classical eigenvalue ( equal to @xmath85 ) . the global picture summarizing the evolution of @xmath81 along a rg trajectory initialized in such a way as to approach @xmath42 first along t@xmath62 , is drawn on fig . [ fig3 ] . the values we have determined by this shooting method are ( for eigenvalues other than @xmath86 already mentioned ) : @xmath87 which are close to the values found , for example , in @xcite : @xmath88 and @xmath89 . in the preceding section , we have obtained a rg trajectory approaching the wilson - fisher fixed point @xmath42 along t@xmath77 by adjusting two parameters of the initial hamiltonian ( @xmath67 and @xmath69 ) . this is exactly the procedure one must follow to determine a tricritical rg trajectory approaching the gaussian fixed point in three dimensions ( because of its two relevant directions ) . the only difficulty is to discover initial points in @xmath90 which are attracted to the gaussian fixed point . to this end , we again use the shooting method . from usual arguments on the lgw hamiltonian and from the work done by aharony on compressible ferromagnets @xcite , one expects to find the tricritical surface in the sector @xmath21 ( and with @xmath91 ) . thus we have tried to approach the gaussian fixed point starting with initial function @xmath92 of the form : @xmath93 with ( not large ) negative values of @xmath94 , for example @xmath95 . because the gaussian fixed point is twice unstable , we must adjust two parameters to approach it starting with ( [ eq - init - tri1 ] ) . we do that by successive tries ( shooting method ) . for example , if we choose @xmath96 and @xmath97 and determine a value of @xmath98 such as to get a trajectory which does not go immediately towards the trivial high temperature fixed point , the best we obtain is a trajectory which approaches the wilson - fisher fixed point ( thus the corresponding initial point belongs to the attraction domain of @xmath42 although @xmath99 @xcite ) . but if @xmath100 , the adjustment of @xmath50 with a view to counterbalance the effect of the most relevant direction of the gaussian fixed point ( which would drive the trajectory toward the high temperature fp ) yields a runaway rg flow towards larger and larger negative values of @xmath101 . from now on , the target is bracketted : the tricritical trajectory corresponding to @xmath102 can be obtained with a value of @xmath103 in the range @xmath104 2,3\right [ $ ] ( we actually find a rather close approach to the gaussian fixed point for @xmath105 and @xmath106 ) . in order to understand the origin of the direction of runaway in the sector of negative values of @xmath13 , it is worth to study the properties of the gaussian fixed point by linearization of the rg flow equation in the vicinity of the origin . if we request the effective potential to be bounded by polynomials then the linearization of eq . ( [ eq4 ] ) identifies with the differential equation of hermite s polynomials of degree @xmath107 for the set of discrete values of @xmath108 satisfying @xcite : @xmath109 from which it follows that * for @xmath9 : @xmath110 @xmath111 , there are two non - negative eigenvalues : @xmath112 and @xmath113 * for @xmath10 : @xmath114 @xmath111 , there are three non - negative eigenvalues : @xmath112 , @xmath115 et @xmath116 if we denote by @xmath117 the eigenfunctions associated to the eigenvalue @xmath118 , it comes : * @xmath119 , @xmath120 , @xmath121 , @xmath122 , whatever the spatial dimensionality @xmath34 . the upperscript `` @xmath123 '' is just a reminder of the fact that the eigenfunctions are defined up to a global factor and thus the functions @xmath124 are also eigenfunctions with the same eigenvalue @xmath118 . similarly to @xmath125 , the direction provided by @xmath126 in @xmath90 is a direction of instability of the gaussian fixed point . now @xmath127 is associated with the well known renormalized trajectory t@xmath60 on which is defined the usual ( massless ) @xmath128-field theory @xcite , for the same reasons a renormalized trajectory t@xmath129 locally tangent to @xmath126 in the vicinity of the origin of @xmath11 exists with the same properties as t@xmath60 ( see @xcite ) . the difference is that t@xmath19 lies entirely in the sector @xmath21 and is endless ( not ended by a fixed point ) . this endless renormalized trajectory is associated with systems undergoing a first - order phase transition . this is due to the absence of fixed point @xcite , in which case the correlation length @xmath130 can not be made infinite although for some systems lying close to t@xmath19 and attracted to it ( i.e. at the transition temperature ) , @xmath130 may be very large ( because t@xmath19 is endless ) , in which cases one may say that the transition is almost of second order @xcite . of course , a domain of first - order phase transition in @xmath11 was expected from the usual arguments @xcite , we only specify better the conditions of realization , in @xmath11 , of the first - order transition . [ fig4 ] shows the attractive trajectory t@xmath19 together with the attractive tricritical line approaching the gaussian fixed point . the tricritical surface @xmath4 separates the first - order surface @xmath2 from the critical surface @xmath1 . fig . [ fig4 ] shows also that systems lying close to the tricritical surface may still be attracted to the wilson - fisher fixed point . in this case the effective exponents may undergo a very retarded crossover to the asymptotic ising values compared to usual systems corresponding to initial points chosen in the sector @xmath20 of @xmath11 . [ fig5 ] illustrates how minus the inverse of ( [ eq - omegaeff ] ) provides us with different evolutions [ calculated from ( [ eq4 ] ) ] of the effective exponent @xmath131 [ with @xmath132 $ ] according to the initial point chosen in @xmath11 . it is worth to explain how we have defined @xmath133 . we have seen at the end of section [ lpa - erge ] that the quantity ( [ eq - omegaeff ] ) undergoes a flat extremum ( or a flat inflexion point ) at an rg eigenvalue of @xmath42 each time the rg flow runs along an eigendirection in the vicinity of @xmath42 . now it happens that this extremum is less and less flat as one chooses larger and larger values of @xmath134 ( for the eigenvalue @xmath135 ) but still exists . this provides us with a way to express the evolution of an effective exponent @xmath136 when the rg - substitute to @xmath137 , namely @xmath138 , is varied . [ fig6 ] shows such an evolution for some initial hamiltonian ( with @xmath139 ) . notice that for such a hamiltonian , the extremum disappears before @xmath136 reaches the trivial value @xmath85 ( associated with the approach to the trivial high temperature fixed point and to a regular non critical behavior ) while in the case of a hamiltonian initialized close to the tricritical surface , the classical - to - ising crossover is complete ( see fig . [ fig4 ] ) . this is because in the latter case the rg trajectory comes close to the gaussian fixed point ( and @xmath140 has an extremum at @xmath85 ) before approaching to @xmath42 . this reinforces the idea that the so - called classical - to - ising crossover actually exists only between the gaussian and wilson - fisher fixed points @xcite . the same configuration displayed by fig . [ fig4 ] has been obtained also by tetradis and litim @xcite while studying analytical solutions of an exact rg equation in the local potential approximation for the @xmath141-symmetric scalar theory in the large @xmath142 limit . but they were not able to determine `` _ the region in parameter space which results in first - order transitions _ '' @xcite . to decide whether the marginal operator ( associated with the eigenvalue equal to zero , i.e. @xmath143 in four dimensions , or @xmath144 in three dimensions ) is relevant or irrelevant , one must go beyond the linear approximation . the analysis is presented in @xcite for @xmath9 . if one considers a rg flow along @xmath125 such that @xmath145 , then one obtains , for small @xmath146 : @xmath147 $ ] with @xmath148 . hence the marginal parameter decreases as @xmath24 grows . as it is well known , in four dimensions the marginal parameter is irrelevant . however , if one considers the direction opposite to @xmath125 ( i.e. @xmath126 ) then the evolution corresponds to changing @xmath149 . this gives , for small values of @xmath146 : @xmath150 $ ] and the parameter becomes relevant . the parameter @xmath146 is the renormalized @xmath151 coupling constant @xmath152 and it is known that in four dimensions the gaussian fixed point is ir stable for @xmath153 but ir unstable for @xmath154 @xcite . we have verified that the trajectory t@xmath19 survives when @xmath9 ( contrary to t@xmath60 , see @xcite ) . that trajectory t@xmath129 is a renormalized trajectory on which we could define a continuum limit for the @xmath6-field theory and if the corresponding ( perfect ) action was positive for all @xmath7 , one could say that the @xmath155-field theory with a negative coupling is asymptotically free . unfortunately , because the @xmath5-term is dominant for large @xmath7 in the vicinity of the origin of @xmath11 ( due to the relevant direction provided by @xmath126 ) , the negative sign of the renormalized coupling prevents the path integral to be well defined . however , because the action to which one refers in the continuum limit ( the perfect action ) is formal ( because it involves an infinite number of parameters and can not be written down , see @xcite ) we wonder whether the wrong sign of the action is actually a valid argument to reject the @xmath156-field theory with a negative renormalized coupling . it is worth to mention that the asymptotically free scalar field theory which has recently been considered on a lattice @xcite could actually be the @xmath157-field theory with a negative coupling to which we refer here . * acknowledgments * we dedicate this article to professor yukhnovskii in grateful recognition of his efficient and generous help in fostering the ukrainian - french symposium held in lviv in february 1993 , with the hope that in the future the contacts between our two communities will further develop . for technical reasons , instead of studying eq . ( [ eq4 ] ) , we consider the differential equation satisfied by @xmath158 ( i.e. the second derivative of the potential with respect to the field ) : starting with a known initial function ( at `` time '' @xmath48 ) , we follow its evolution in @xmath11 by approximating the differential equation ( [ app4 ] ) by finite differences and a two dimensional grid with the uniform spacings @xmath160 and @xmath161 . the finite difference formulas for the derivatives @xmath162 and @xmath163 have been chosen with the accuracy @xmath164 : @xmath165 + o(dy^{4 } ) \label{app ' } \\ g^{\prime \prime } ( y ) & = & \frac{16}{12dy^{2}}\left [ g(y+dy)+g(y - dy)-30g(y)\right . \nonumber \\ & & \left . -g(y+2dy)-g(y-2dy)\right ] + o(dy^{4 } ) \label{app''}\end{aligned}\ ] ] the evolutionary function @xmath166 is known ( calculated ) at the discret set of points @xmath167 with ( @xmath168 ) and a maximum value @xmath169 . ( this value is large enough to study the approach to the wilson - fisher fixed point with a great accuracy but is too small for studying precisely the approach to the gaussian fixed point . ) at each time @xmath170 , the derivatives are estimated from @xmath171 by using eqs([app],[app ] ) which apply only for @xmath172 . for the marginal points @xmath173 we use the parity of @xmath174 [ by inserting @xmath175 for @xmath176 in eqs([app],[app ] ) ] . for the two other marginal points @xmath177 of the grid , there is no fixed solution and we shall used alternately the two following conditions [ using the obvious abreviation @xmath178 instead of @xmath179 : @xmath181 \label{app2 ' } \\ g^{\prime \prime } ( i ) & = & \frac{1}{dy^{2}}\left [ \frac{915}{244}g(i)-\frac{77% } { 6}g(i-1)+\frac{107}{6}g(i-2)-13g(i-3)\right . \nonumber \\ & & \left . + \frac{61}{12}g(i-4)-\frac{5}{6}g(i-5)\right ] + o(dy^{4 } ) \label{app2''}\end{aligned}\ ] ] condition 2 is more accurate than condition 1 but leads sometimes to strong unstabilities which do not appear when we first use condition 1 and then condition 2 after some finite `` time '' @xmath182 . the validity of the procedure is tested by , for example , trying to approach a given fixed point ( see the main part of the paper ) . [ [ appearance - of - spurious - fixed - points - in - the - approach - to - the - gausian - fixed - pointspurious ] ] appearance of spurious fixed points in the approach to the gausian fixed point[spurious ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ in trying to determine the attractive tricritical trajectory ( approaching the gaussian fixed point ) , we have encountered a spurious twice unstable fixed point lying at some finite and non negligible distance to the gaussian fixed point . to understand the origin of this undesirable numerical effect , it is necessary to discuss a bit the solution of the fixed point equation @xmath183 . from ( [ eq4 ] ) or ( [ app4 ] ) , one sees that the fixed point equation is a second order non linear differential equation and a solution would be parametrized by two arbitrary constants . one of these two constants may easily be determined : since @xmath184 is expected to be an even function of @xmath7 [ o(1 ) symmetry ] then @xmath185 may be imposed . it remains one free parameter , thus a one - parameter family of ( nontrivial ) fixed points are solutions to the differential equation . but there is not an infinity of physically acceptable fixed points ; all but a finite number of the solutions in the family are singular at some @xmath186 @xcite . formally , by requiring the physical fixed point to be defined for all @xmath7 then the acceptable fixed points are limited to the gaussian fixed point and ( for @xmath10 ) to the wilson - fisher fixed point . however , in our study , because we numerically consider the function @xmath187 in some finite range of values of @xmath7 ( see above : @xmath188 ) , it appears that in approaching the origin of @xmath11 , infinitely many pseudo - fixed points exist which have there @xmath186-singularity located outside the finite range explicitly considered and there is at least one of them which looks like a tricritical fixed point . when we enlarge the range of @xmath7 to @xmath189 , the previously observed nontrivial tricritical fixed point disappeared to the benefit of another one located closer to the origin . in conclusion , a larger and larger number of grid - points must be considered as one tries to come closer and closer to the gaussian fixed point . this particularity together with the slowness of the approach along a marginal direction makes it excessively difficult to come very close to the gaussian fixed point . j. f. nicoll , t. s. chang and h. e. stanley , `` _ approximate renormalization group based on the wegner -houghton differential generator _ '' , phys . 33 * , 540 ( 1974 ) ; `` _ exact and approximate differential renormalization - group generators _ '' , phys . rev . * a13 * , 1251 ( 1976 ) . g. felder , `` _ non - trivial renormalization group fixed points _ '' , in `` _ @xmath191 international congress on mathematical physics , marseille _ '' ( 1986 ) ; `` _ renormalization group in the local potential approximation _ '' , com . phys . * 111 * , 101 ( 1987 ) . f. j. wegner , `` _ corrections to scaling laws _ '' , phys . rev . * b5 * , 4529 ( 1972 ) ; `` _ the critical state , general aspects _ '' , in phase transitions and critical phenomena vol . * vi * , p. 7 , _ ed . by _ c. domb and m.s . green ( acad . press , n .- y . , 1976 ) . toledano , l. michel , p. toledano and e. brzin , `` _ renormalization - group study of the fixed points and of their stability for phase transitions with four - component order parameters _ '' , phys . rev . * b11 * , 7171 ( 1985 ) . k. gawedski and a. kupiainen , `` _ non - trivial continuum limit of a @xmath194 model with negative coupling constant _ '' , nucl . phys . * b257 [ fs14 ] * , 474 ( 1985 ) ; `` _ asymptotic freedom beyond perturbation theory _ '' , in `` _ critical phenomena , random systems , gauge theories _ '' , p. 185 , _ ed . by _ k. osterwalder and r. stora ( elsevier science pub . b.v . , 1986 ) . 1 . evolutions for @xmath10 of the first four hamiltonian parameters @xmath195 , @xmath196 , @xmath197 , @xmath198 in a close approach to the wilson - fisher fixed point @xmath42 along t@xmath199 or t@xmath78 . the effective inverse eigenvalue @xmath200 is given by eq . ( [ eq - omegaeff ] ) for @xmath201 . all these quantities reach the same universal value @xmath56 characteristic of the least irrelevant eigendirection of @xmath42 . to get this close approach to @xmath42 from eq . ( [ eq4 ] ) , the initial critical value @xmath69 corresponding to @xmath202 , has been determined with more than twenty digits.[fig1 ] 2 . when a second condition is imposed on the initial hamiltonian parameters , the approach to @xmath42 may be adjusted such as to asymptotically take the second least irrelevant eigendirection . here @xmath203 is given by eq . ( [ eq - omegaeff ] ) for @xmath79 it clearly undergoes ( full line ) a flat inflexion point at the value @xmath204 corresponding to an approach to @xmath42 along t@xmath77 , the greater the critical parameter @xmath67 is accurately determined , the longer is the flat extremum . because @xmath67 is not completely determined [ within the available accuracy in solving eq . ( [ eq4 ] ) ] the trajectory leaves the direction of t@xmath77 to take one of the two directions of approach associated to the least irrelevant inverse eigenvalue @xmath56 ( corresponding to t@xmath60 or t@xmath78 as indicated by dashed curves ) . here , the trajectory corresponding to the full line goes along t@xmath205 . again a flat extemum of @xmath206 indicates the approach along an eigenvector of @xmath42 and requires an accurate determination of the critical value @xmath69 . because this determination is not complete , the trajectory ends up going away from @xmath42 as indicated by the sudden departure of @xmath207 from @xmath56 for the large values of @xmath24.fig2 3 . this figure is a continuation of fig . it shows the various plateaux that @xmath81 undergoes along a rg trajectory first adjusted to approach @xmath42 along the second irrelevant direction ( plateau at @xmath204 ) . because it is not possible to determine exactly the initial conditions , the trajectory always ends up going away the fixed point towards the trivial high temperature fixed point characterized by the classical value @xmath85 ( for minus the inverse of @xmath81 , thus the final plateau at @xmath208 ) . in - between , the rg flow has been influenced by the close vicinity of the least irrelevant eigenvector ( plateau at @xmath56 ) and that of the relevant eigenvector ( plateau at @xmath209 ) . the various regimes of the rg flows are indicated by the vertical arrows on the left ( direction of the flow with respect to the fixed point ) and on the right of the figure ( distance to the fixed point).[fig3 ] 4 . domains of attraction and repulsion of the gaussian fixed point . the figure represents projections onto the plane @xmath210 of various rg trajectories running in the space @xmath11 minus one dimension . the flows have been obtained by solving eq . ( [ eq4 ] ) . black circles represent the gaussian and the wilson - fisher ( w - f fp ) fixed points . the arrows indicate the directions of the rg flows on the trajectories . the ideal trajectory ( dot line ) which interpolates between these two fixed points represents the usual renormalized trajectory t@xmath60 corresponding to the so - called @xmath211 renormalized field theory in three dimensions ( usual rt for @xmath20 ) . white circles represent the projections onto the plane of initial critical hamiltonians . for @xmath212 , the effective hamiltonians run toward the ising fixed point asymptotically along t@xmath60 ( simple fluid ) . instead , for @xmath213 and according to the values of hamiltonian coefficients of higher order ( @xmath214 , @xmath215 , etc . ) , the rg trajectories either ( a ) meet an endless rt emerging from the gaussian fp(dashed curve ) and lying entirely in the sector @xmath21 or ( b ) meet the usual rt t@xmath60 to reach the ising fixed point . the frontier which separates these two very different cases ( a and b ) corresponds to initial hamiltonians lying on the tri - critical subspace @xmath4 ( white square c ) . this is a source of rg trajectories flowing asymptotically toward the gaussian fpalong the tricritical rt . notice that the coincidence of the initial point b with the rg trajectory starting at point a is not real ( it is accidental , due to a projection onto a plane of trajectories lying in a space of infinite dimension ) . the points a or b could correspond to the restricted primitive model of ionic systems ( see @xcite).[fig4 ] 5 . evolutions of an effective exponent @xmath140 [ with @xmath216 $ ] along three different families of rg trajectories ( see text for additional details ) . the full squares indicate the evolution of @xmath140 for a family of trajectories initialized in the sector @xmath20 with @xmath202 and for various values of @xmath217 ( the same system at criticality corresponds to the white circle `` simple fluid '' of fig . [ fig4 ] ) . when @xmath218 the effective exponent approaches the critical exponent value @xmath219 compatible with the present study . one observes that the crossover towards the classical value @xmath85 is not complete because @xmath133 ceases to make sense before @xmath137 becomes large . this is not the case of the evolution represented by the full circles which corresponds to trajectories initialized close to the gaussian fixed point . in this case the complete crossover reproduces the interpolation between the gaussian and the wilson - fisher fixed points and typically corresponds to the usual answer given by field theory @xcite . the third evolution ( full triangles ) corresponds to a family of hamiltonian initialized close to the tricritical surface but still attracted to the wilson - fisher fixed point . one sees that the classical - to - ising crossover is complete but highly retarded compared to the two other cases . this is because at criticality , the rg flow is first attracted to the gaussian fixed point ( showing then an apparent classical value of @xmath135 ) before interpolating between the gaussian and the wilson - fisher fixed point.[fig5 ] 6 . illustration of the evolution of the extrema @xmath220 [ minus the inverse of eq . ( [ eq - omegaeff ] ) ] for various values of @xmath221 and for the family of rg flows initialized at @xmath202 . the extremum ( grey triangle ) disappears at some not very large value of @xmath137 ( about @xmath222 ) and does not reach the classical value @xmath85 . this induces the partial ising - to - classical crossover drawn on fig . [ fig5 ] ( squares).[fig6 ]

using the local potential approximation of the exact renormalization group ( rg ) equation , we show the various domains of values of the parameters of the @xmath0-symmetric scalar hamiltonian . in three dimensions , in addition to the usual critical surface @xmath1 ( attraction domain of the wilson - fisher fixed point ) , we explicitly show the existence of a first - order phase transition domain @xmath2 separated from @xmath3 by the tricritical surface @xmath4 ( attraction domain of the gaussian fixed point ) . @xmath2 and @xmath1 are two distinct domains of repulsion for the gaussian fixed point , but @xmath2 is not the basin of attraction of a fixed point . @xmath2 is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the @xmath5-coupling . this renormalized trajectory exists also in four dimensions making the gaussian fixed point ultra - violet stable ( and the @xmath6 renormalized field theory asymptotically free but with a wrong sign of the perfect action ) . we also show that very retarded classical - to - ising crossover may exist in three dimensions ( in fact below four dimensions ) . this could be an explanation of the unexpected classical critical behavior observed in some ionic systems . _ pacs 05.10.cc , 05.70.jk , 11.10.hi_ -0.7 cm -2 cm

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the main limiting factors in determining the masses of the compact objects in low - mass x - ray binaries is uncertainty in the fraction of the infrared light that is produced by the companion star ( e.g. , hynes , robinson , & bitner 2005 ) . when lmxbs are accreting at low rates , the optical and infrared light from these systems is dominated by the companion stars , so their mass functions can be measured to high accuracy based on the doppler motion of the companion ( e.g. , * ? ? ? the inclination then must be constrained by modeling the modulations in the optical and infrared light curves that are produced by the varying aspect of the distorted , roche - lobe - filling companion . generally , the contribution of the accretion disk to the optical and near - infrared emission is uncertain , and varying the fraction of the light that is assumed to be produced by the accretion disk can lead to differences of a factor of 2 in the derived mass of a compact object ( e.g. , gelino , harrison , & orosz 2001 ) . fortunately , the broad - band spectrum of a multi - temperature accretion disk is significantly flatter than that of a stellar photosphere , so mid - infrared observations could constrain the relative contributions of the two components . however , the spectra of lmxbs in the mid - infrared have not been well studied , and several indirect lines of evidence suggest that these systems might contain circumbinary material that could emit in the mid - infrared . gfirst , some of the white dwarf analogs to lmxbs , cataclysmic variables ( cvs ) , exhibit spectral features that lie at the mean radial velocities of the systems @xcite and excess mid - infrared emission @xcite that could be interpreted as arising in circumbinary material ( see also * ? ? ? * ; * ? ? ? second , the supernovae that produced the compact objects could have left fall - back disks around the binaries . indeed , the first fall - back disk has recently been found around a young , highly - magnetized neutron star ( wang , chakrabarty , & kaplan 2006 ) . third , the planets around the isolated millisecond pulsar psr 1257@xmath412 @xcite could not have survived the supernova that produced the neutron star , and must have formed afterward ( see , e.g. , * ? ? ? millisecond pulsars are usually assumed to have been spun up by accretion as lmxbs , in which case planets could form from material present during the binary phase . therefore , to search for evidence of circumbinary material , we have observed four nearby , quiescent lmxbs with the _ spitzer _ space telescope . we chose the lmxbs in our sample to be detectable with _ spitzer _ if they contained optically - thick circumbinary disks passively illuminated by the mass donor stars ( see * ? ? ? * ; * ? ? ? * and below ) . based on a simple model , we chose sources with : ( 1 ) @xmath5 magnitudes brighter than 17 , ( 2 ) locations more than @xmath6 in projection from the galactic center , and ( 3 ) no 2mass sources within 5 that were brighter than our targets . in table [ tab : targets ] , we list the positions , the orbital periods , estimates of the primary masses , the spectral types of the companions , and the quiescent @xmath5 magnitudes of the four systems in our sample ( see the table notes for references ) . our measurements were taken in the 4.5 and 8.0 @xmath0 m bands with infrared array camera ( irac ) , and in the 24 @xmath0 m band with the multiband imaging photometer ( mips ; tab . [ tab : fluxes ] ) . we used the post - basic - calibration data provided by the _ spitzer _ science center ( ssc ) for most of our analysis . however , the mips image of a 0620@xmath100 contained latent features with a low spatial frequency and a @xmath72% amplitude that were left by a previous observation of a bright , extended source . we corrected the image by creating a flat field from the median of the individual dithered images , dividing each snapshot by the flat , and re - creating the mosaicked image using the script provided by the ssc . three - color images centered on each lmxb are displayed in figure [ fig : img ] . each target is detected at 4.5 and 8.0 @xmath0 m . only a 0620@xmath100 and gs 2023 + 338 are also detected at 24 @xmath0 m . we computed the irac fluxes of each source using the point - spread - function - fitting routine apex from the ssc , and the mips fluxes and upper limits using aperature photometry ( tab . [ tab : fluxes ] and fig . [ fig : sed ] ) . llcccccccc[htp ] a 0620@xmath100 & v616 mon & 95.68561 & @xmath8 & 1.2@xmath90.1 & 7.8 & 8.712.9 & k4v & 1.2 & 14.55(6 ) + gs 2023 + 338 & v404 cyg & 306.01594 & @xmath10 & 2.23.7 & 155.3 & 10.113.4 & k0iii & 4.0 & 12.50(5 ) + xte j1118 + 480 & kv uma & 169.54498 & @xmath11 & 1.8@xmath90.5 & 4.1 & 6.57.2 & k6v & 0.06 & 16.9(2 ) + cen x-4 & v822 cen & 224.59135 & @xmath12 & 1.2 & 15.1 & 1.4 & k5v & 0.3 & 14.66(8 ) lcccccccccc[htb ] a 0620@xmath100 & 2005 mar 25 & 400 & 448(13 ) & 194 & 249(10 ) & 149 & 2005 mar 06 & 180 & 54(18 ) & 43 + gs 2023 + 338 & 2004 oct 09 & 36 & 3020(90 ) & 670 & 1450(40 ) & 500 & 2004 oct 16 & 30 & 153(70 ) & 46 + xte j1118 + 480 & 2004 nov 21 & 400 & 46(1 ) & 17 & 45(7 ) & 34 & 2005 may 13 & 240 & @xmath1316 & @xmath1316 + cen x-4 & 2004 aug 12 & 300 & 199(6 ) & @xmath190 & 95(17 ) & @xmath114 & 2005 aug 28 & 150 & @xmath1330 & @xmath1330 to understand the origin of the mid - infrared emission , in figure [ fig : sed ] we plot for each lmxb the observed and de - reddened fluxes in the infrared and optical bands ( from tab . [ tab : fluxes ] and the references in tab . [ tab : targets ] ) . the optical and near - infrared intensities of quiescent lmxbs often vary by several tenths of a magnitude on time scales of years ( not counting outbursts ; * ? ? ? * ; * ? ? ? * ) , presumably because of changes in the accretion flow . none of the fluxes were obtained simultaneously , so when comparing the flux measurements in figure [ fig : sed ] one can only expect them to be self - consistent to within @xmath1430% . to estimate the contributions of the companions to the spectra , we have obtained model stellar spectra computed with the next - generation phoenix code ( kindly provided by t. barman ; * ? ? ? * ) that correspond to the temperatures and radii of the spectral types in table [ tab : targets ] . we have estimated the contribution of the accretion disk using a standard @xcite model , assuming the disk extends out to the tidal truncation radius ( @xmath1560% of the binary separation ; frank , king , & raine 1992 ) , is inclined by 60@xmath16 to our line of sight , and has the largest accretion rate that is consistent with the optical photometry ( see the caption of fig . [ fig : sed ] for values ) . we have confirmed that our conclusions are robust against changing the assumed spectral types of the companions within the values reported in the literature , and against varying the parameters of the accretion disk so long as the models are consistent with the optical photometry . we find that the spectrum of cen x-4 is consistent with that expected for the companion star , with a possible contribution at long wavelengths from the accretion disk . there are clear excesses of mid - infrared flux above that expected from the companions of a 0620 - 00 , gs 2023 + 338 , and xte j1118 + 480 . in order to conservatively estimate the amounts of the mid - infrared fluxes that are not produced by the companions , we computed the maximum contributions of the companions photospheres by normalizing the model stellar spectra to match the de - reddened @xmath5 fluxes , and reported the difference between the predicted and observed mid - infrared fluxes as the excesses in table [ tab : fluxes ] ( @xmath17 ) . several lmxbs have now been detected in outburst in the mid - infrared ( e.g. , smith , beall , & swain 1990 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , but these are the first detections of quiescent systems . in outburst , the mid - infrared emission could originate from an accretion disk in which @xmath18 g s@xmath19 @xcite , from free - free emission in a strong wind driven from the accretion disk @xcite , or from radio jets with flat spectra extending to infrared wavelengths @xcite . we use these hypotheses as a starting point for understanding the mid - infrared emission from our sample of quiescent lmxbs . the mid - infrared excess from gs 2023 + 338 could originate in an accretion disk , because the 2.224 @xmath0 m fluxes in figure [ fig : sed ] follow the @xmath20 law that one would expect from the rayleigh - jeans tail of a black body . we find that our model for a viscously - heated accretion disk @xcite from 2 adequately describes the excess mid - infrared flux from gs 2023 + 338 in figure [ fig : sed ] . in contrast , the 424 @xmath0 m fluxes from a 0620@xmath100 and xte j1118 + 480 do not follow the @xmath21 law expected for the rayleigh - jeans tail of a blackbody , because there is too much flux at 8 @xmath0 m . a blackbody spectrum that peaks near 8 @xmath0 m would have a temperature of only 640 k. for these two sources , we plot blackbody spectra that match the peak of the mid - infrared excesses ( @xmath22 ) in figure [ fig : sed ] . the sum of the spectra of stellar photospheres and optically - thick blackbodies match the observed mid - infrared fluxes well . however , the inferred emitting areas are large compared to the binary separations . for a planck spectrum , the solid angles of the emitting regions correspond to circular radii of @xmath23 where @xmath24 is the distance to the source . in contrast , the disks will be contained within @xmath1560% of the binary separations @xmath25 , which are given by : @xmath26 where @xmath27 is the orbital period in days , @xmath28 is the mass of the accretor in solar masses , and @xmath29 is the mass ratio of the binary ( e.g. , * ? ? ? * ) . given the fluxes in table [ tab : fluxes ] , we find that the excess mid - infrared emission originates from regions @xmath142 times larger than the orbital separations of gs 2023 + 338 and xte j1118 + 480 , and @xmath304 times larger than the accretion disks . therefore , the mid - infrared emission is produced in a region that extends beyond the binary orbit . jets could produce an emitting region larger than the binary orbit , but given the spectra of the mid - infrared excesses we suggest that jets are only minor contributors . two lmxbs in our sample have recently been detected as radio sources in quiescence : gs 2023 + 338 with flat - spectrum radio emission with an intensity of 350@xmath0jy over the frequency range 1.48.4 ghz @xcite , and a 0620@xmath100 with a radio flux of 50 @xmath0jy at 8.4 ghz @xcite . the 24@xmath0 m excesses from a 0620@xmath100 is equal to the flux in the radio , so it could be produced by a jet with a flat ( @xmath31 , with @xmath32 ) spectrum between the radio and mid - infrared . for gs 2023 + 338 , the 24 @xmath0 m flux is slightly lower than the radio flux , which would imply that its jet has a steeper @xmath33 spectrum . however , the 8 @xmath0 m excesses from a 0620@xmath100 , gs 2023 + 338 , and xte j1118 + 480 are not consistent with a flat - spectrum radio jet , because for xte j1118 + 480 and a 0262@xmath100 they are 4 times larger than the excesses at 24 @xmath0 m , and for gs 2023 + 338 it is 10 times larger . if we assume the 24 @xmath0 m emission is from a flat - spectrum jet , then it contributes @xmath13 25% to the 8 @xmath0 m flux from xte j1118 + 480 and a 0620@xmath100 , and @xmath1310% from gs 2023 + 338 . we suggest that the excesses from a 0620@xmath100 and xte j1118 + 480 originate from circumbinary dust that re - processes the light of the companions . to estimate the masses of the dust , we assume it is contained in optically thick disks . the disks could lie as close to the center of masses of the binaries as 1.7@xmath25 , at which point they would be tidally truncated @xcite . we can estimate the temperature profile of such disks by assuming they are flat , in which case @xmath34 where @xmath35 and @xmath36 are the temperatures and radii of the disks , and @xmath37 and @xmath38 are the temperatures and radii of the companions ( e.g. , * ? ? ? * ; * ? ? ? using equation [ eq : orbit_sep ] and the parameters in table [ tab : targets ] , we would expect the inner edges of the circumbinary disks to have temperatures of @xmath14600 k , which is consistent with our detection of excesses that peak near 8 @xmath0 m . the lack of excess flux at 24@xmath0 m implies that material does not re - process much stellar light beyond radii @xmath143 times larger than the binary separations . if we assume the disks are composed of dust with an opacity @xmath39@xmath7300 @xmath40 g@xmath19 at 8 @xmath0 m ( e.g. , * ? ? ? * ) , then for them to have an optical depth of @xmath41@xmath71 , the disks would only need to contain @xmath7@xmath2 g of dust . if we assume the emitting material is optically thin , we find dust masses that are similar within a factor of a few ( e.g. , eq . 3 in * ? ? ? if the gas - to - dust ratio is similar to that of the interstellar medium , @xmath7100 , then the total mass of the circumbinary material could be @xmath7@xmath3 g. the detection of excess mid - infrared flux from a 0620@xmath100 and xte j1118 + 480 ( fig . [ fig : sed ] ) provides evidence that circumbinary material is present around some lmxbs , but its spectrum suggests that it will only be detectable in the mid - infrared . the excess emission peaks at 8 @xmath0 m , and if we model it as a single - temperature blackbody , we predict that the circumbinary material produced negligible flux shortward of @xmath143 @xmath0 m . moreover , the lack excess mid - infrared emission from cen x-4 demonstrates that such material is not ubiquitous . finally , for gs 2023 + 338 , the mid - infrared emission appears to originate from a hot accretion disk , and the contribution of the accretion disk to the mid - infrared light can be predicted based on the optical measurements ( the dotted line in fig . [ fig : sed ] ) . these results provide reassurance that efforts to determine the inclinations of lmxbs by modeling the ellipsoidal modulations of their infrared and optical light curves are not compromised by the presence of circumbinary material . for a 0620@xmath100 and xte j1118 + 480 , we suggest that the circumbinary disks are either the remains of fall - back disks produced in the supernovae that formed the compact objects ( e.g. , * ? ? ? * ; * ? ? ? * ) , or material injected into circumbinary orbits during the process of mass loss by the roche - lobe filling companions ( e.g. , * ? ? ? these circumbinary disks contain @xmath7@xmath2 g of dust , which for a standard dust - to - mass ratio of @xmath7100 implies a total mass of @xmath7@xmath3 g. if the disks are produced by matter ejected into circumbinary orbits during the process of mass transfer , then it represents only a tiny fraction of the @xmath7@xmath42 g ( @xmath70.1 @xmath43 ) that will be lost by the companions over the lifetimes of these lmxbs . alternatively , @xcite have proposed that fall - back disks containing @xmath30@xmath3 g of material could form asteroids and inject them into the magnetospheres of pulsars with a high enough rate to explain the the observed intermittency in their radio pulses . asteroids might not survive the outbursts of lmxbs @xcite , but the mid - infrared excesses that we have identified could be the remnants of similar fall - back disks . however , the circumbinary material is not massive enough either to affect the evolution of the orbital angular momentum of lmxbs ( e.g. , @xmath7@xmath44 g in * ? ? ? * ) , or to form planetesimals ( e.g. , @xmath7@xmath45 g in * ? ? ? as the formation and evolution of circumbinary matter around lmxbs and cvs are considered further , it may be that the paucity of matter around a 0620@xmath100 and xte j1118 + 480 ( and the apparent lack of circumbinary matter around the one neutron star lmxb in our sample , cen x-4 ) may be the most lasting aspect of this result . we are grateful to t. barman for providing the model stellar spectra , to e. gallo and s. migliari for alerting us to a mistake in our original photometry , to m. jura for conversations about circumstellar dust , to j. mcclintock for discussions about the contribution of the accretion disk to the mid - infrared light , and to the referee for comments that clarified the text . support for this work was provided by nasa through an award issued by jpl / caltech . 0 casares , j. , charles , p. a. , naylor , t. , & pavlenko , e. p. 1993 , , 265 , 834 chevalier , c. , ilovaisky , s. a. , van paradijs , j. , pedersen , j. , & van der klis , m. 1989 , , 210 , 114 cordes , j. m. & shannon , r. m. 2006 , astro - ph/0605145 draine , b. t. & lee , h. m. 1984 , , 285 , 89 dubus , g. , campbell , r. , kern , b. , taam , r. e. , & spruit , h. c. 2004 , , 349 , 869 evans , a. , geballe , t. r. , rawlings , j. m. c. , eyres , s. p. s. , & davies , j. k. 1997 , , 292 , 192 fender , r. p. 2001 , , 322 , 31 frank , j. , king , a. , & raine , d. 1992 , `` accretion power in astrophysics '' , cambridge university press gallo , e. , fender , r. p. & hynes , r. i. 2005 , , 356 , 1017 gallo , e. , fender , r. p. , miller - jones , j. c. a. , merloni , a. , jonker , p. g. , heinz , s. , maccarone , t. j. , & van der klis , m. 2006 , submitted to , astro - ph/0605376 gelino , d. m. , harrison , t. e. , & orosz , j. a. 2001 , , 122 , 2668 harrison , t. e. & gehrz , r. d. 1991 , , 101 , 587 harrison , t. e. & gehrz , r. d. 1994 , , 108 , 1899 hauschildt , p. h. , lowenthal , d. k. , & baron , e. 2001 , , 134 , 323 homan , j. , buxton , m. , markoff , s. , bailyn , c. d. , nespoli , e. , & belloni , t. 2005 , , 624 , 295 howell , s. b. 2006 , in press , astro - ph/0606202 hynes , r. i. , robinson , e. l. , & bitner , m. 2005 , , 630 , 405 jura , m. 2003 , , 584 , l91 mcclintock , j. e. , narayan , r. , garcia , m. r. , orosz , j. a. , remillard , r. a. , murray , s. s. 2003 , , 593 , 435 mcclintock , j. e. , & remillard , r. a. 2006 , to appear in `` compact stellar x - ray sources , '' eds . w. h. g. lewin & m. van der klis , cambridge university press , astro - ph/0306213 migliari , s. , tomsick , j. a. , maccarone , t. j. , gallo , e. , fender , r. p. , nelemands , g. , & russell , d. m. 2006 , astro - ph/0605051 mikoajewsa , k. , ruthowski , a. , gonalves , d. r. , & szostek , a. 2005 , , 362 , l13 miller , m. c. & hamilton , d. p. 2001 , , 550 , 863 ruden , s. p. & pollack , j. b. 1991 , , 375 , 740 shahbaz , t. , naylor , t. , & charles , p. a. 1993 , , 265 , 655 shakura , n. i. & sunyaev , r. a. 1973 , , 24 , 337 smith , h. a. , beall , j. h. & swain , m. r. 1990 , , 99 , 273 solheim , s .- e . & sion , e. m. 1994 , , 287 , 503 taam , r. e. , sandquist , e. l. , & dubus , g. 2003 , , 592 , 1124 thorsett , s. e. & chakrabarty , d. 1999 , , 512 , 288 torres , m. a. p. , callanan , p. j. , garcia , m. r. , zhao , p. , laycock , s. , & kong , a. k. h. 2004 , torrres , m. a. p. , casares , j. , martnez - pais , i. g. , & charles , p. a. 2002 , , 334 , 233 van paradijs , j. , telesco , c. m. , kouveliotou , c. , & fishman , g. j. 1994 , , 429 , 19 wang , z. , chakrabarty , d. , & kaplan , d. l. 2006 , , 440 , 772 wolszczan , a. & frail , d. a. 1992 , _ nature _ , 355 , 145 zurita , c. , casares , j , martinez - pais , i. g. , piccioni , a. , bernabei , s. , bartolini , c. , & guarnieri , a. 2002 , iauc 7868

we report the discovery of excess 4.5 and 8@xmath0 m emission from three quiescent black hole low - mass x - ray binaries , a 0620@xmath100 , gs 2023 + 338 , and xte j1118 + 480 , and the lack of similar excess emission from cen x-4 . the mid - infrared emission from gs 2023 + 338 probably originates in the accretion disk . however , the excess emission from a 0620@xmath100 and xte j1118 + 480 is brighter and peaks at longer wavelengths , and most likely originates from circumbinary dust that is heated by the light of the secondary star . for these two sources , we find that the inner edges of the dust distributions lie near 1.7 times the binary separations , which are the minimum radie at which circumbinary disks would be stable against tidal disruption . the excesses are weak at 24 @xmath0 m , which implies that the dust does not extend beyond about 3 times the binary separations . the total masses of circumbinary material are between @xmath2 and @xmath3 g. the material could be the remains of fall - back disks produced in supernovae , or material from the companions injected into circumbinary orbits during mass transfer .

You are an expert at summarizing long articles. Proceed to summarize the following text: markus bttiker was certainly one of the most influential scientists in the field of mesoscopic physics . among all his important contributions , time in quantum mechanics has a peculiar flavor since it occupied his mind at the right beginning and at the end of his carrier . intrigued at first by the traversal time of an electron through a tunnel barrier @xcite , he came back to this topic after the emergence of `` on - demand single electron sources '' @xcite , which he greatly contributed to develop @xcite , via the concept of waiting time distribution ( wtd ) @xcite . charge transport at the nanoscale is known to be stochastic due to the quantum nature of particles @xcite . therefore , going beyond the knowledge of average quantities , such as the average electronic current , appears to be unavoidable and extremely fruitful at the same time . a deep physical insight can indeed be inferred from the fluctuations of the signal and extracted from various observables . noise @xcite and full counting statistics ( fcs ) @xcite , namely the second moment of current fluctuations and the statistics of charges transferred during a long time interval , are among the most popular quantities and have been proved to be powerful tools . with the development of electron quantum optics @xcite and the progress in single electron detection at high frequencies @xcite , it is now relevant and possible to consider electron dynamics and time resolved quantities at quantum mechanical time scales ( typically nano - seconds and below ) . therefore , new theoretical tools have been developed to describe the current fluctuations at such time scales , such as finite frequency noise @xcite and fcs @xcite , wigner functions @xcite , or the wtd @xcite . the latter , describes the statistical distribution of time intervals between the detection of two electrons and therefore gives accurate information about correlations between subsequent electrons . is set to an energy @xmath0 above the fermi energy of the normal part and the gap @xmath1 is much larger than the potential difference @xmath0 . ] the wtd has been studied for particularly simple systems like single and multiple electronic quantum channels connected to two normal leads via a quantum point contact ( qpc ) @xcite , a quantum capacitor @xcite , a double quantum dot @xcite , a train of lorentzian pulses @xcite or a quantum dot connected to a normal and a superconducting lead @xcite , among others . in this paper we revisit the physics of normal - superconducting ( ns ) junction through the point of view of waiting times in order to illustrate the effect of superconducting correlations and entanglement @xcite on their distribution . indeed , as we will discuss later , such a system may emit entangled electrons in the normal part , and leads to interesting features in the wtd . the paper is organized as follows . in sec . [ sec : model ] , we describe the model used for the ns junction and the formalism needed for computing the wtd . in sec . [ sec : onespin ] , we discuss the effect of the transparency of the barrier ( the energy dependence of the andreev reflection ) when the detection process is sensitive to only one electronic spin species and a certain range of energy . section [ sec : twospins ] is devoted to the effect of entanglement between spin up and spin down electrons emitted from the superconducting part , on the wtds . we finally conclude and discuss some perspectives in sec . [ sec : conclusion ] . moreover , for the sake of clarity , technical details are moved to the appendices . [ app : a ] demonstrates the formal analogy between our setup and a single quantum channel conductor for a specific detection process whereas important steps for the numerical and analytical calculations of the wtd in the entangled case are explained in [ app : b ] . one very important consequence of superconductivity is the existence of andreev reflection . such a phenomenon arises because the superconducting device can not accommodate any single particle excitation with energy below the gap @xmath1 . therefore , if a single particle like an electron or a hole flows from the normal part to the superconducting part with an energy below this threshold it can only be scattered back at the interface . however , there are now two possibilities . an electron ( a hole ) can be either normally reflected ( specular reflection ) , that is to say , reflected as an electron ( a hole ) or converted to a hole ( an electron ) . this is the so called andreev reflection which originates from the fact that the incoming electron finds a partner to create a cooper pair which can enter in the superconductor and leave a hole behind . to be more specific , the system of interest is a polarized ns junction ( with an s - wave superconductor ) , at zero temperature , as presented on fig . [ fig : fig1 ] . the superconductor chemical potential @xmath2 is set to be at a potential @xmath0 above the fermi level @xmath3 of the normal metal . in such a situation , there is an incident hole , coming from the metal , that can be either normally reflected or andreev reflected as an electron . another way of picturing the andreev effect is to think about the inverse configuration where a cooper pair in the superconductor ( at energy @xmath2 and zero momentum for an s - wave superconductor ) splits at the interface and gives birth to an entangled pair of electrons . from now on , we will take @xmath0 much smaller than the superconducting gap @xmath1 in order to focus on this sub - gap phenomenon . this also has the benefit to make the andreev time @xmath4 ( the typical time needed for an andreev event ) much smaller than @xmath5 ( the typical time separation of two single particle wave packets emitted in the normal metal @xcite ) . this allows us to assume that andreev events are instantaneous and make use of scattering theory . in addition , this assumption allows one to linearize the dispersion relation around @xmath2 as @xmath6 , with @xmath7 and @xmath8 measured from @xmath2 and its corresponding momentum ( or around the fermi level since @xmath9 and @xmath2 ) . at the interface , the scattering is in general not perfect and both normal and andreev reflection will play a role . in order to describe this effect , we use the standard blonder - tinkham - klapwijk ( btk ) model @xcite which has been widely used in the literature . the junction is modeled by a point - like barrier potential @xmath10 , where @xmath11 is the fermi wavelength and @xmath12 is a parameter measuring the strength of the barrier . it is then possible to compute the scattering matrix of this setup exactly and obtain the normal and andreev transmission / reflection coefficients @xcite . we do not reproduce these results in the present paper but give the corresponding numerical values of the coefficients when necessary . figure [ fig : fig1 ] illustrates the scattering processes that we are now going to describe mathematically . the incident holes of energies @xmath13 lying between @xmath3 and @xmath14 , arriving from the left and propagating to the right will be either normally reflected as holes of the same energies with amplitude @xmath15 or andreev reflected as electrons of energies @xmath16 with amplitude @xmath17 . the incoming scattering state is therefore a slater determinant of holes of the form @xcite @xmath18 where @xmath19 stands for the filled fermi sea up to @xmath2 in the normal part . however , in the electron language , this state is just the fermi sea @xmath20 filled up to @xmath3 instead of @xmath2 . in the following , we will rather use the electronic picture to simplify the notation but both pictures are equivalent @xcite . due to scattering at the interface , the outgoing state is therefore a superposition of reflected holes , entangled electrons and non - entangled electrons @xcite @xmath21 indeed , it is pretty straightforward to see that the previous equation , for a given energy , gives birth to three kinds of term with different levels of complexity . the terms @xmath22 and @xmath23 correspond to non - entangled contributions whereas @xmath24 describes fully entangled electrons originating from the splitting of a cooper pair at the interface . when andreev reflection is absent ( @xmath25 ) , the fermi sea is unperturbed by the interface and nothing interesting happens . counter - intuitively , perfect andreev reflection does not lead to perfect entanglement . on the contrary , the state is a slater determinant of non - entangled electrons and the ns junction acts as a conventionnal electron source @xcite . it appears that the maximally entangled situation arises when andreev and normal reflection probabilities are both equal to one half . nevertheless , the wtd of a fully entangled state has never been studied to our knowledge and we will take the opportunity to study it in this paper before considering the general and more realistic state emitted at the interface . in order to conclude this section , we recall a few definitions about wtds . as mentioned in the introduction , the waiting time @xmath26 is defined as the time delay between the detection of two single particles . due to scattering and the quantum nature of particles , this time is a random variable , which distribution ( the wtd ) brings an elegant and instructive picture of the physics . for stationary systems , namely when there is no explicit time dependence , the wtd @xmath27 depends on @xmath26 only ( and not on absolute time ) and is closely related to the idle time probability ( itp ) @xmath28 , the probability to detect no electron during a time interval @xmath26 @xmath29 where @xmath30^{-1}_{\tau=0}$ ] is the mean waiting time @xcite . to go further , we must now specify the detection procedure to compute the wtd . in what follows we will assume perfect single electron projective measurement but will consider two different situations as described below . such perfect detection process is still theoretical and extremely challenging at very short time scales ( nano - second and below ) but recent experiments @xcite are very promising about this issue . in order to compute the itp , we regularize the scattering problem as usual @xcite . we discretize energy , ranging from @xmath31 to @xmath0 , in @xmath32 slices and wave vectors as @xmath33 and consider the limit @xmath34 to mimic a stationary process . the first application of this setup will be to measure the wtd of holes via the detection of andreev reflected electrons or in other words the wtd of andreev events @xcite . a single electron detector is located at a position @xmath35 far away from the interface . in this section we assume that the detector measures only one spin orientation that we will choose upward for concreteness . moreover , we make the additional assumption that it is only sensitive to energies above @xmath2 , using a quantum dot for instance @xcite . as a consequence , this allows us to avoid complications due to entanglement @xcite , which will be the subject of the next section . following @xcite , the itp is defined as @xmath36 with the explicit condition that only electrons with energies above @xmath2 contribute . here @xmath37 and @xmath38 stands for the normal ordering . in principle , the itp can not be reduced to a single determinant since the many body state is not a slater determinant . however , according to the assumptions introduced above , only one term survives and the final result can be cast as a single determinant @xcite . under these conditions , entanglement no longer plays a role and the problem thus boils down to a single quantum channel with energy dependent transmission @xcite , where the role of the energy - dependent transparency @xmath39 of the qpc is played here by @xmath40 , the andreev reflection amplitude ( see [ app : a ] for details ) . is increased , @xmath41 decreases from one to zero as shown in the inset . for @xmath42 , @xmath43 and the wtd is described by the wigner distribution ( blue dots , eq . ( [ eq : wigner ] ) ) . close to perfect specular reflection ( @xmath44 ) the distribution approaches an exponential law except for very small times ( black squares ) . for intermediate @xmath12 , small oscillations with period @xmath45 are superimposed to an exponential decay . together , with the dip at @xmath46 , they are manifestations of pauli s exclusion principle @xcite . ] in fig . [ fig : fig2 ] , we have plotted the spin up electronic wtd for different barriers strengths @xmath42 ( perfect andreev reflection ) , @xmath47 and @xmath48 ( strong barrier ) as a function of @xmath49 . for information we show as an inset the corresponding andreev reflection coefficient @xmath50 as a function of energy . however , the energy dependence is very weak since the energy window @xmath0 is supposed to be much smaller than the superconducting gap @xmath1 . as expected , the wtd for @xmath42 is approximately given by the wigner surmise @xcite @xmath51\,.\ ] ] indeed , in that case the train of free holes is perfectly converted at the interface into free electrons which are described by random matrix theory @xcite . as @xmath12 is increased , the distribution is broadened since @xmath41 is no longer equal to one and therefore not all holes are converted into electrons . the situation is exactly equivalent to free electrons injected into a single quantum channel and partitioned by a quantum point contact with transmission probability @xmath41 due to the electron - hole symmetry in the system . the difference is only conceptual since here the detector is measuring indirectly the statistics of holes converted into electrons by the ns junction that acts as an andreev mirror . finally , for large @xmath12 , namely , small andreev reflection , most of the holes are normally reflected and the detector collects rare events which are almost uncorrelated and the wtd is exponential ( except for very short times ) . indeed , following @xcite we derive the asymptotic behavior of the wtd in the long time limit . for @xmath43 the decay is gaussian with algebraic corrections well described by the wigner surmise and for partial andreev reflection ( @xmath52 ) it is exponential with a rate given by the geometrical mean of the logarithm of @xmath53 over energy in @xmath54 $ ] , namely @xmath55\ , g(\tau/\overline\tau),\ ] ] where @xmath56 is an oscillatory function that decays as @xmath57 and depends on @xmath58 . to conclude this section , we note that the role of electrons and holes may be interchanged by inverting the polarization ( @xmath59 ) due to electron - hole symmetry . we now move a step forward and discuss the effect of entanglement on waiting times . as mentioned before , andreev reflection might be thought in terms of splitting of a cooper pairs in the vicinity of the interface , leading to the injection of two entangled electrons with opposite spins and energy ( with respect to @xmath2 ) in the normal part as shown in fig . [ fig : fig3 ] ( left part ) . however , we have seen that the scattering state ( eq . ( [ eq : psiout ] ) ) is a mixture of entangled and non - entangled components which makes the effect of entanglement hard to separate from other physical ingredients . therefore , we focus here on the wtd of the fully entangled state of the form @xmath60 \vert f \rangle\,,\ ] ] even if this is not the real quantum state emitted at the interface . indeed , this work on ns junction rises a fundamental question of the effect of entanglement on waiting times that has never been considered to our knowledge and deserves to be investigated with proper care . at the end of this section we will give a few hints on how the additional terms included in the full state ( eq . ( [ eq : psiout ] ) ) modify the picture but the common thread of this section will be the study of the fully entangled quantum state ( eq . ( [ eq : psife ] ) ) . to point out the specific features of entanglement , we will compare this situation to the one of two independent electrons which would correspond to the emission of two electrons with opposite spins from a normal metal ( see fig . [ fig : fig3 ] right ) @xcite . this time , the single electron detector is sensitive to both spins and energies between @xmath61 and @xmath62 . using the energy discretization mentioned before , the entangled state reads @xmath63 \ , \vert f \rangle\ , , \ ] ] whereas non - entangled electrons emitted from a normal lead in an energy window @xmath0 ( see fig . [ fig : fig3 ] right ) would be described by @xmath64 . according to the detection process mentioned above , the itp is formally given by @xcite @xmath65 where the quantum average is taken over the state @xmath66 or @xmath67 leading to @xmath68 and @xmath69 respectively . without any further assumption it is clear that the final result can not be expressed as a single determinant but rather as a sum of @xmath70 terms except in the non - entangled case where the itp factorizes to @xmath71 where the averages are taken over the two spin sectors separately @xcite . indeed , each term in the product of eq . ( [ eq : psie ] ) can be split into two parts : @xmath72 and @xmath73 . @xmath74 is then a sum of @xmath75 terms of the form @xmath76 , where @xmath77 can be either @xmath78 or @xmath79 . we shall denote a particular term by a string made of a succession of @xmath80 s and @xmath81 s defined as follows : @xmath82 . this is a slater determinant , while @xmath83 can not generally be cast as a simple slater determinant . @xmath84 will then be the sum of @xmath70 terms of the form , typically , @xmath85 , with @xmath86 and @xmath87 two generally different slater determinants . as previously shown in refs . @xcite , each @xmath88 is also a determinant . details about the procedure to calculate the itp are given in [ app : b ] . we now discuss our results obtained from a direct and exact enumeration of the @xmath70 terms in the itp . owing to the exponentially growing number of terms , this approach is limited to relatively small values of @xmath32 . figure [ fig : fig4]a ) presents the wtd for increasing values of @xmath32 up to @xmath89 . as can be seen , the curves reasonably converge to a limiting distribution that would be obtained for @xmath90 . to insure this , we have computed several finite size corrections that we will discuss later on . figure [ fig : fig4]b ) compares the wtds of entangled and non - entangled electrons @xcite which are qualitatively similar . however , as we will discuss in more detail in the next subsection , the maximum of the curve in the entangled case is more pronounced and closer to zero than in the independent case . the presence of such a peak in the wtd is the hallmark of pair rigidity due to entanglement . to be more quantitative we can evaluate the probability that the waiting time is smaller than the average waiting time @xmath91 ( which is the same in both situations ) . we find that this probability is about ten percent larger in the entangled case , demonstrating that the entangled electrons are more correlated than the non - entangled ones . the short time behavior is one of the most interesting characteristics of the wtd as it reflects the short time correlations encoded in the many body state and not the ones due to scattering . for free electrons it is universal since it is the expression of the pauli s principle . indeed , two spinless electrons can not be emitted in the same state which enforces the wtd to start from zero ( with a quadratic behavior ) . however , if the electrons have other degrees of freedom like spin or if the mesoscopic conductor supports several channels , the wtd may start from a non - zero value but correlations are still visible and universal @xcite . however , these correlations only originate from electrons of the same channel and not between different channels . they eventually disappear in the limit of large number of independent channels @xcite . . the inset shows how @xmath92 converges to the asymptotic value @xmath93 as predicted by eq . [ eq : stimewtd ] ( full black line ) . b ) comparison between the wtd of entangled electrons and non - entangled electrons ( see text ) . inset : short time behavior in the entangled case for @xmath89 ( full red line and green dashed line ( see eq . [ eq : stimewtd ] ) ) and @xmath94 ( black long dashed lines ) . ] in order to get the short time expansion of the wtd up to second order in time , we need to expand the itp ( eq . ( [ eq : idle2 ] ) ) to fourth order in terms of moments of @xmath95 , @xmath96 and their products . this is a straightforward but somehow cumbersome calculation that we do not reproduce in detail here . after some algebra we obtain the short time expansion of the wtd , including finite size corrections . the final expression for @xmath97 reads , to second order in @xmath26 @xmath98\ ] ] where @xmath99 in the entangled case and one in the absence of ns junction . the insets of fig.[fig : fig4]a ) and fig.[fig : fig4]b ) show how this prediction is confirmed by our numerical evaluation of the wtd in the entangled case . in both situations , the wtd starts from the same initial value ( in the @xmath34 limit ) with a quadratic behavior but the coefficient is four times larger in the entangled case . again , this reflects the pair rigidity due to entanglement . differences between entangled and non - entangled electrons can also be inferred from the tail of the wtd . following the approach developed in @xcite we obtain the asymptotic behaviors of the wtd in the limit @xmath100 ( for details see [ app : b ] ) . in both cases we find a gaussian decay @xmath101\,,\ ] ] with @xmath102 a constant which is equal to one in the entangled case and two in the non - entangled case . therefore , the itp or equivalently the wtd decays more slowly in the entangled case , meaning that the two electrons are not at all independent . as mentionned before , entangled pairs are only one component of the real out - going scattering state which appears to be much more complex . for abritrary value of the reflection coefficients @xmath103 and @xmath41 , numerical evaluation of the wtd is pretty challenging since the number of terms grows as @xmath104 . in that case , direct numerical evaluations are restricted to very small values of @xmath32 ( typically 4 or 5 ) which is not sufficient to mimic a stationary situation . it is then necessary to resort to more sophisticated approaches like monte - carlo sampling of the itp for instance . however , there is a simple but rather interesting limit which is amenable to analytical calculations , namely the tunneling limit for @xmath44 @xmath105 . in that case , the full many - body state simplifies to @xcite @xmath106|f\rangle\,,\ ] ] namely , to first non - trivial order in @xmath41 , it is made of scarce singlet pairs . following the same approach as before we obtain the small time behavior @xmath107\ ] ] and the long time asymptotics @xmath108\,.\ ] ] at short waiting times , the distribution starts from a constant value and the physics is dominated by correlations within a single cooper pair . indeed , simple considerations show that pairs are roughly separated in time by @xmath109 whereas electrons from the same pair are rather in a span of time of the order of @xmath110 . the situation is then akin to a single pair which has been studied by hassler et al . @xcite and confirms our predictions . at large time , the decay is exponential with a rate twice smaller than in the non - entangled case , which again is an hallmark of pair rigidity . we have presented a theory of waiting times in polarized ns junction at zero temperature . in this setup , andreev reflection brings new characteristic features in the wtd such as entanglement . if a detector is sensitive to only one type of spin and to energies above the superconductor chemical potential , the situation is reminiscent of a single quantum channel connected to two normal leads via a qpc with energy - dependent transmission . the interface acts as an andreev mirror and allows to measure the wtd of holes converted to electrons . however , if the detector measures electrons with both spins in the whole energy window above the fermi sea of the normal conductor , entanglement between electrons leaves fingerprints in the wtd . although still academical , we have focused on a fully entangled state which is only one component of the full many - body state flowing out of the ns interface . in that case , such signatures are visible for both small and large waiting times but the most important feature is the existence of a peak in the wtd centered before the average waiting time . when taking into account all components of the scattering state , we have shown that some charateristics are still visible in the wtd in the tunneling limit . in the near future , it would then be useful to extend this work beyond the ideal situation and the tunneling limit and evaluate the effect of entanglement for arbitrary values of reflection coefficients . in addition , a natural extensions would be to study correlations between waiting times of different spin species in the spirit of @xcite . this would probably yield an even clearer signature of entanglement than the wtd itself . among other future investigations , it would be possible to study the effect of cross - andreev reflection in a superconducting - normal - superconducting junction or the physics brought by exotic states like majorana modes created by majorana guns @xcite . the paper is dedicated to the memory of markus bttiker who was very enthusiastic about the quantum theory of waiting times . m. a. , will always be grateful to him for his kindness , trust and freedom he gave to him when m. a. was a post - doc in his group . markus was a mentor to him and a source of endless inspiration . we thank d. dasenbrook , c. flindt , g. haack , p. p. hofer and m. moskalets for useful discussions and remarks . the research of d. c. was supported by the foundation for fundamental research on matter ( fom ) , the netherlands organization for scientific research ( nwo / ocw ) , and an erc synergy grant . this appendix is dedicated to prove the formal correspondence between the wtd of a single channel normal conductor and the andreev mirror defined in section [ sec : onespin ] . this happens because the detection process is only sensitive to spin up electrons and energies above @xmath2 , the superconducting chemical potential . to begin the proof we recall the definition of the itp in terms of the scattering state @xmath111 with @xmath112 where @xmath32 corresponds to the number of slices due to the energy discretization between @xmath2 and @xmath62 ( which has to be taken equal to infinity at the end of the calculation ) and @xmath8 is measured with respect to @xmath113 the fermi momentum at energy @xmath2 there are two important steps in the derivation . the first one consists in expanding the products and rearranging the terms in order make the relevant terms appear ( the ones with spin up and positive energy electrons ) . if we restrict ourselves to these terms we build terms like @xmath114 . then , we include the other terms of the product and show that they are irrelevant . indeed , in the following example @xmath115 the two off - diagonal terms vanish since the operator @xmath116 conserves the number of particles . concerning the two diagonal ones , they leave us with @xmath117 then , we can repeat the same argumentation for all the other irrelevant terms from @xmath118 to @xmath32 . finally , we are left with @xmath119 with @xmath120 . as a second step , we rearrange the fock space by moving all the spin @xmath121 to the left . spin @xmath122 are invariant under @xmath123 , thus we only have to compute @xmath124 we then recognize the expression for the itp of a spinless quantum channel with bias @xmath0 and transmission amplitude @xmath17 @xcite . in this appendix we give some details on the construction on @xmath28 . starting from the n - body states reads @xmath125 \ , \vert f \rangle,\ ] ] with @xmath126 is the fermi sea defined in the main text and @xmath127 the probability to detect nothing in a range of time @xmath26 with a detector sensitive to all the energies and all spins can be written as @xmath128 where the average is taken over @xmath129 . from here , it is convenient to separate eq . ( [ wave_function ] ) in two parts @xmath130 and @xmath131 in order to get a sum of @xmath75 terms where each term is a slater determinant . to simplify , it is useful to change the notation : now each configuration is associated to a specific ket ( bra ) and each element of this configuration can be mapped to an ising classical spin . namely , @xmath130 ( @xmath131 ) is associated to @xmath132 . 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( 2004 ) 1249 .

we consider a normal - superconducting junction in order to investigate the effect of new physical ingredients on waiting times . first , we study the interplay between andreev and specular scattering at the interface on the distribution of waiting times of electrons or holes separately . in that case the distribution is not altered dramatically compared to the case of a single quantum channel with a quantum point contact since the interface acts as an andreev mirror for holes . we then consider a fully entangled state originating from spliting of cooper pairs at the interface and demonstrate a significant enhancement of the probability to detect two consecutive electrons in a short time interval . finally , we discuss the electronic waiting time distribution in the more realistic situation of partial entanglement . normal - superconducting junction , andreev reflection , waiting time distribution , entanglement 02.50.ey , 72.70.+m , 73.23.hk

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Proceed to summarize the following text: the emerging field of cold molecules is a very vibrant topic in physics and physical chemistry . the considerable interest in this topic is related to the properties of cold molecules and their many potential applications . cold molecules have been identified as attractive systems for ultrahigh - resolution spectroscopy @xcite , quantum information processing @xcite , for developing new time standards and testing fundamental physical theories such as the time variation of physical constants @xcite , the existence of a dipole moment of the electron @xcite , and for the measurement of parity violation @xcite . in chemistry , cold molecules are essential tools to explore quantum - mechanical effects in chemical reactions . in contrast to atoms , molecules have a very complicated level structure that consists of vibrational and rotational states as well as electronic levels . this abundance of states is the main obstacle for direct laser cooling of molecules . usually , molecules do not provide the closed transitions required for cooling and non - destructive state - selective detection . this makes it impossible to perform direct spectroscopic measurements on _ single _ molecules a standard technique in atomic physics . additional complications result from the small energy differences between the rotational levels , leading to a thermal distribution of the population over the molecules ro - vibrational states . despite important achievements @xcite , the control of molecular states never caught up with that of atomic systems . however , there has been remarkable progress in the synthesis of ultra - cold alkali dimers from samples of ultra - cold atoms ; see e.g. , refs . @xcite . furthermore , methods which enable the preparation of more diverse ( e.g. polyatomic ) cold molecular species in their vibrational ground - states have been successfully demonstrated . these methods include : supersonic beam expansion followed by stark deceleration @xcite , optical stark deceleration @xcite , electrostatic velocity selection @xcite , collisional cooling in crossed molecular beams @xcite and buffer gas cooling @xcite . the vibrationally cold ( but rotationally hot ) states that result will be taken to be the starting point for the schemes described in this paper . the experimental advances which have enabled the production of these cold molecular states have inspired theoretical investigations of the cooling of molecules by laser pulses @xcite or even by coupling molecules to an optical cavity @xcite . bartana et al . @xcite used the electronic excited - state as a heat reservoir in order to cool the vibrational states of the electronic ground - state by means of short , shaped , laser pulses . in later work @xcite they employed state selective optical pumping , hiding the target state in a dynamically trapped state . through this , bartana et al . achieved a vibrational ground state population of 97% after only 25 vibration periods . a related scheme , investigated by schirmer @xcite , increased the vibrational ground - state population to a similar level . these efforts to cool the internal degrees of freedom focus on the widely spaced vibrational states of molecules . in ref . @xcite bartana et al . investigated the possibility of cooling the _ rotational _ degrees of freedom in a simplified model employing the same techniques . however , even though the results of their calculations are very promising , the model has limited application as it neglects the vibrational degree of freedom of the molecule . in contrast to neutral molecules , ionized molecules can be sympathetically cooled by trapping them alongside atomic ions in a paul trap . under these conditions , state sensitive ultra - cold chemical reactions have been measured @xcite and high resolution spectroscopy has been demonstrated on small ensembles @xcite . despite these achievements , the rotational degrees of freedom could not be controlled , but they led to new laser cooling schemes for the internal states of molecules @xcite exploiting the unique properties of these systems . in the present paper , we show that by means of purely coherent manipulations of internal states ( i.e. rotational states ) , and by using sympathetic cooling of motional states , single molecular ions can be cooled close to their motional and rotational ground - state . the internal vibration of the molecule needs to be initially cold , in order for the final state to be cold in _ all _ its degrees of freedom . this initial state can be achieved using the existing methods described above . then the cooling of the internal molecular state is achieved in three steps : first a laser cooled atomic ion is trapped alongside the molecule and a common mode of vibration is used to prepare the molecule in its motional ground - state . next , using adiabatic passage methods @xcite , the thermal internal state of the molecule can be mapped onto the molecule s motion in an external trapping field with high fidelity . during this mapping process the internal ( rotational ) states are transferred to the ground - state whereas the molecule s motion is excited . finally , the molecule s motion is sympathetically cooled back into its ground - state using the common mode with a cooled atomic ion . by doing this without exciting other degrees of freedom , a molecule which is vibrationally , rotationally , and translationally cold can be obtained . the overall process is a kind of molecular `` heat pump '' : the heat energy in the rotational degree of freedom is transferred by the coherent processes to the motional degree of freedom . this heat energy is in turn transferred to the environment by means of conventional cooling techniques , e.g. side - band cooling : this ensures that the process is uni - directional . vogelius et al . @xcite investigated the cooling of molecular ions by coupling a single rotational state to the motion of the ion . by means of black - body radiation the population is pumped into the rotational ground - state resulting in a ground - state population of about 80% after a cooling time of the order of minutes . in this paper we focus on the cooling process by employing techniques from coherent control providing much faster and more efficient cooling . the result is a robust and highly efficient cooling process which enables the deterministic manipulation of the internal states of molecules . we will examine adiabatic passage schemes which exhibit high fidelity in conjunction with relaxed requirements on the experimental parameters compared to direct raman transitions . with cooling times of the order of milliseconds , and final ground - state populations of more than 92% , the proposed scheme provides a fast and efficient method for preparing molecules in their ro - vibrational ground - state . even though we focus here on the cooling of molecular rotational levels , the technique is also applicable to other multi - level systems . so the technique can also be employed in atoms with complicated level schemes , or more general ro - vibrational states of molecules . the paper is organized as follows : in section [ sec:2 ] , we present the model used for our calculations . to find the best adiabatic passage process for our application we compare the results of numerical simulations for stimulated raman adiabatic passage ( stirap ) , stark chirped raman adiabatic passage ( scrap ) and chirped adiabatic rapid passage ( carp ) in a @xmath0-type level system in section [ sec:3 ] . we also choose parameter ranges which are relevant for a possible experimental implementation . section [ sec:4 ] contains the results for an extended level scheme , and in section [ sec : conclusion ] we present our conclusions . the model we employ in this paper is based on the states of a quantum mechanical rigid rotator which is a good approximation for the rotational states of small diatomic molecules . however , the techniques described here are applicable to most other level structures with the sole requirement that allowed raman transitions between the states involved exist . in general , the method can be applied to ro - vibrational states of molecules as well as zeeman and hyperfine levels . in order to simplify the discussion , and to avoid specializing to molecules with specific symmetries , we will not apply specific selection rules to the raman transitions involved . nevertheless , the results presented in this paper can be applied to a particular system by imposing the specific selection rules for that case with an appropriate relabelling of states . for example , we could utilize @xmath1 for linear molecules . we take the energy @xmath2 of the rotational levels of the electronic ground state to be @xmath3 , with the rotational quantum number @xmath4 and a rotational constant @xmath5 . in order to limit the number of levels in our calculations , only rotational levels up to a cut - off are considered , i.e.@xmath6 . for typical , light , diatomic molecules at room temperature only rotational states with @xmath7 are significantly populated . this decreases to below 10 states for rotational temperatures lower than about 50k . this kind of temperature can be easily achieved by supersonic beam expansion @xcite . in our scheme the levels @xmath4 are coupled by laser pulses to an electronically excited state @xmath8 ( see fig.[fig : levelsystem ] ) . and the rotational states @xmath9 . the excited state decays towards the states @xmath10 at a rate @xmath11 and towards the uncoupled state @xmath12 at a rate @xmath13 . the laser pulses have time - dependent rabi frequencies @xmath14 and are detuned from the relevant transition frequency by @xmath15 . ] from the excited state the molecule can spontaneously decay back into the electronic ground - state with a rate @xmath16 . to represent the decay of the excited state into levels outside the system , e.g. rotational states with @xmath17 or vibrationally excited states , an additional level @xmath12 is also included in the model . the excited state can decay into this uncoupled state with a rate @xmath18 . the single molecule is trapped in a harmonic potential as provided by rf - traps for molecular ions . the quantized motion of the molecule in the trapping potential gives rise to an equally spaced ladder of motional states in addition to the internal states of the molecule . to take the molecule s motion into account , we use the standard notation @xmath19 with @xmath20 representing the states @xmath10 , @xmath21 and @xmath12 . the quantum number describing the motional state is @xmath22 . as discussed in the introduction , in order to prepare the molecule s motion in its ground - state , it is trapped alongside one or more atomic ions which can be directly laser cooled . the two types of ion form a crystal - like structure due to their mutual coulomb repulsion , which enables sympathetic cooling of the molecule . in experiments with two types of atomic ion , ground - state populations of better than 95% have been achieved @xcite . in the molecular case , the minimal system would be one trapped molecular ion and one trapped atomic ion . in this case , we find that due to the frequency splitting of the motional modes , one mode ( com or stretch mode ) can be singled out and used for the proposed scheme . the other mode only imposes an additional limit on the laser pulse length as we will discuss later in this paper . the laser - molecule interaction , in the rotating wave approximation @xcite , is described by the following hamiltonian @xcite : @xmath23\left\vert j-1\right\rangle\left\langle j-1\right\vert \\ & + \sum_{k = s , p}\left[\hat{h}^k_{c}(t)+\hat{h}^k_{r}(t ) + \hat{h}^k_{b}(t)\right ] , \end{split}\ ] ] where @xmath24 represents the carrier resonance @xcite for either the pump @xmath25 , or the stokes - pulse @xmath26 , with @xmath27 the corresponding red sideband transitions in eq . are the @xmath28 , which are given by @xmath29 and for the first blue sideband transitions the @xmath30 are given by @xmath31 in eqs . ( [ eq:1]-[eq:4 ] ) , the secular frequency of the molecule in the external trapping potential is @xmath32 and @xmath33 are the corresponding lamb - dicke parameters @xcite . the detuning between the @xmath34-th laser frequency @xmath35 and the transition frequency @xmath36 of the @xmath37 transition is @xmath38 and @xmath39 . the raising operator for the internal states is @xmath40 and the lowering operator is @xmath41 . the creation and annihilation operators for the motional number states @xmath42 are @xmath43 , and @xmath44 respectively . note that we work in an interaction representation with explicit time - dependence removed , and keep both the resonant and non - resonant couplings . the reason for this is that in pursuing the adiabatic limit in section [ sec:3 ] we will consider `` strong '' rabi frequencies ( @xmath45 ) which do not allow us to make a second rwa on the hamiltonians for the sideband transitions . we start our calculations with a molecule in an internal thermal state such that it is already cooled in its internal vibrational mode ( e.g. by the methods mentioned in the introduction ) and such that the excited state @xmath8 is not populated . we also assume that the molecule s vibrational motion in the trap has been cooled ( e.g. by sympathetic sideband cooling @xcite ) so that only the manifold of rotational states @xmath46 are populated . the density matrix of this initial state is given by : @xmath47 where @xmath48 is the cut - off introduced for the numerical calculations . the normalization factor @xmath49 is given by @xmath50 with @xmath51 , and with @xmath52 as the internal rotational temperature of the molecule . starting from such an initial distribution , we will apply coherent control techniques to transfer population between the different states . we aim to have a state mapping of the form @xmath53 where @xmath54 are the populations of the initial states @xmath55 and the final target states @xmath56 respectively . a sequence of pulses will be used to map population in each @xmath4-state to a corresponding @xmath22-state with @xmath57 . throughout this work , we assume that the system is initially prepared in a state given by eq . ( [ eq:5 ] ) , and derive the requirements for achieving the state mapping in eq . , i.e. after completion of a number of the coherent pulse sequences . thus we obtain a superposition of just the motional states which can then be cooled to the motional ground - state @xmath58 by applying the sympathetic cooling @xcite . [ the state mapping can also be employed for non - destructive state detection : e.g. by measuring the initial thermal distribution ( [ eq:5 ] ) . by coupling the electronic state of an atom trapped alongside the molecules to its motion , the mapped state can be read out . ] for the coherent mapping the key idea is to use pairs of pulses , @xmath59 and @xmath60 , to induce population transfer between the states @xmath61 and @xmath62 , see fig.[fig : levelsystem ] . for this step it is important to have a resonance , so that additional states do not get strongly involved and disturb the mapping . here , the resonance is arranged so that the quantity @xmath63 is conserved at each step ( although this is the simplest way to make the mapping , it would not be the only way , as we only require the transfer of population between unique pairs of states . ) repeating the @xmath64 step @xmath48 times , where in each step @xmath4 is decreased by one , the distribution of population can be moved to the @xmath57 motional states as in eq . ( [ eq:6 ] ) . since states with @xmath65 are involved in the intermediate steps , it is clear that if we do not start in the motional ground - state the final state need not be entirely @xmath57 . however , when we start in the motional ground - state ( i.e. @xmath66 ) , the population transfer is uni - directional . hence , the population is transferred solely to the lower lying rotational states . in our analysis we numerically integrate the master equation for the density matrix @xmath67 : @xmath68+\hat{\mathcal{l}}(\rho(t))+\hat{\mathcal{l}}_{{\scriptscriptstyle}u}(\rho(t)).\ ] ] the hamiltonian @xmath69 is given in eq . . the two liouville terms @xmath70 and @xmath71 describe the irreversible decay of the excited electronic state @xmath72 to the rotational levels @xmath73 of the electronic and vibrational ground - state @xmath74 and the decay out of the system , i.e. to the uncoupled state @xmath12 , @xmath75 in eq . the raising and lowering operators @xmath76 have the same form as @xmath77 , with the substitution @xmath78 . we will assume that the laser pulses are gaussian , with a fixed width @xmath52 and rabi frequencies @xmath79 where @xmath80 . the delay between the two pulses @xmath81 and @xmath82 that drive the transition @xmath83 is @xmath84 , whereas the delay between the @xmath4 and @xmath85 pulse pair is @xmath86 . the corresponding detunings will be either constants or have the form of time - dependent frequency chirps . we take all the lamb - dicke parameters to be the same , i.e. @xmath87 , and for simplicity we assume that the decay rates are the same , i.e.@xmath88 . in order to compare the three methods ( stirap , scrap and carp ) in section [ sec:3 ] we characterize the population transfer efficiency by a parameter @xmath89 representing the total population of the rotational ground - state after the transfer : @xmath90 where @xmath91 is the population of the @xmath56 state after the transfer . for an efficient state mapping , the ground - state population ( @xmath57 ) will have increased . however , if the initial state also has some population in the @xmath92 state , the efficiency @xmath89 may also decrease due to laser - induced transfer out of @xmath93 . thus the definition is not only a measure for the transfer efficiency , but also for the uni - directionality of the mapping process . for ideal state mapping the efficiency measure reaches the limit @xmath94 . in order to test the various passage methods in the next section , the initial state is taken to be a mixture of 70% rotationally excited states ( @xmath95 ) and 30% ground - state ( @xmath57 ) population for each adiabatic method . this approach will test how uni - directional the scheme is . to transfer the population from the state @xmath9 to @xmath96 , various coherent processes can be employed . here we focus on stimulated raman adiabatic passage ( stirap ) @xcite , stark chirped raman adiabatic passage ( scrap or siarp ) @xcite , and chirped adiabatic rapid passage ( carp ) @xcite which offer highly efficient population transfer in combination with robustness against variations of the pulse parameters . in order to simplify the numerical simulations we first investigate these processes in a system with just two rotational levels . that is , we examine in detail a single step in our multi - pulse coherent transfer scheme . in this case the most important states are the two lowest motional states for @xmath57 and the lowest ( @xmath66 ) motional state for @xmath97 ( see fig . [ fig : lambdasystem ] ) . -configuration used to investigate the transfer efficiency of various adiabatic passage schemes in section [ sec:3 ] . the pump and stokes pulses , @xmath98 and @xmath99 respectively , will induce a two - photon raman transition ( which may be chirped ) from state @xmath100 to state @xmath101 . the best results are obtained if the initial population in @xmath102 remains there . the off - resonant intermediate states @xmath103 decays towards the two rotational states and towards the uncoupled state @xmath12 at a rate @xmath18 . the decay rate @xmath16 represents decay from the levels @xmath104 to the levels @xmath105 . other off - resonant states are included as shown . ] however , to include the off - resonant effects all nine of the states shown in fig . [ fig : lambdasystem ] are included in the numerical calculation . we choose parameter ranges which are relevant for an experimental implementation . the electronically excited state can decay to both rotational states as well as into the uncoupled state @xmath12 . stirap is widely used in the optical control of molecules @xcite where a stokes pulse and a pump pulse are used in a `` counter - intuitive '' order to transfer the population between two states . the main requirement for stirap is the two - photon resonance condition which corresponds to @xmath106 for a transition on a motional sideband ( see fig . [ fig : levelsystem ] ) . in addition to this the pulse area must be large , @xmath107 , and the delay between the stokes and pump pulse must be of the order of the pulse width , @xmath108 , to ensure the adiabatic evolution of the system . another constraint arises from the necessity to address particular motional sidebands . the narrow splitting of the motional states imposes the use of laser pulses with narrow bandwidth @xmath109 @xcite . fast pulses will inevitably result in the coupling of the target state to other , close - lying states . this in turn will reduce the transfer efficiency . furthermore , the resolved sideband condition @xcite requires that the rabi frequencies are small compared to the trap frequency , i.e.@xmath110 . thus , only slow and weak pulses can be used which , as we will see , substantially reduces the efficiency of stirap . however , this can be overcome by employing other adiabatic passage schemes as described below . + in fig . [ fig : stirapa ] the transfer efficiency @xmath89 is plotted for different pulse widths @xmath52 and delay times @xmath111 . as mentioned above the initial mixed state is described by the populations @xmath112 and @xmath113 . the chosen decay rate of @xmath114 is in the typical range of values for the decay rate of an electronically excited molecule , when compared to a typical trap frequency @xmath32 ( of the order of several mhz ) . the results in fig . [ fig : stirapa ] show that the efficiency @xmath89 is below 76% . the performance of stirap for short pulses is relatively poor due to the violation of the adiabaticity requirement ( @xmath115 ) and the limitation on the rabi frequency @xmath116 posed by the resolved sideband condition ( @xmath117 ) . for long pulses the efficiency of stirap is compromised by the spontaneous decay @xmath118 , so the compensation of small rabi frequencies by long laser pulses is not an option . therefore , the efficiency of stirap in this parameter range is low and the population transfer is governed by optical pumping rather than coherent evolution . this is particularly visible in the plateau region for @xmath119 . in this regime the pulse delay is too large to sustain the adiabatic evolution of the system , which results in a net loss of the ground state population . highly efficient , fast stirap between motional states requires large motional frequencies which are beyond current ion - trap technology . we can try to suppress the excited - state population by detuning the raman transition from the excited state . however , it has been known for some time that detuning adversely affects the stirap process in the absence of decay @xcite . with decay present one has to consider the balance of the adverse effect of detuning against a possible reduction in spontaneous emission from the excited state of a model @xmath0 system . studies with such systems support the suggestion that the minimal losses ( for moderate decay rates ) are found by remaining on resonance @xcite . figure [ fig : stirap - detuning ] shows how the efficiency of stirap drops , for our model system , as we detune from resonance . the parameters are those for fig . [ fig : stirapb ] , with the rabi frequency @xmath116 chosen to be at the peak of efficiency in fig . [ fig : stirapb ] . we see that both with , and without , decay processes it is best to be resonant . in the case @xmath120 the resonance is much sharper , however . ( @xmath121 ) . we show both the case with decay , @xmath122 ( solid line ) , and without decay , @xmath123 ( dashed line ) . the other parameters are as in fig . [ fig : stirap ] with the best values taken for @xmath124 , @xmath125 , and @xmath126 . ] to exploit the adiabatic evolution of the system and to suppress the excited - state population by far detuning the raman transition leads us to stark chirped raman adiabatic passage ( scrap ) and chirped adiabatic rapid passage ( carp ) , which will be discussed in sections [ sec:32 ] and [ sec:33 ] . in the limit of far detuning , the excited state population is strongly suppressed and the system s dynamics are effectively that of a two - level system @xcite . the effective raman coupling between the states @xmath9 and @xmath127 is @xmath128 and the effective splitting of the coupled rotational levels is @xmath129 with the effective two - photon raman detuning @xmath130 , and the two stark shifts @xmath131 ^ 2/4\delta^s_{{\scriptscriptstyle}j}(t)$ ] and @xmath132 ^ 2/4\delta^p_{{\scriptscriptstyle}j}(t)$ ] induced by the stokes and pump pulse respectively . within this effective two - level model , adiabatic rapid passage techniques ( arp ) @xcite can be applied . the main idea behind arp is to drive the system through the resonance ( @xmath133 ) adiabatically , to achieve a complete population transfer . the technique of stark chirped raman adiabatic passage ( scrap ) @xcite takes advantage of the stark shifts whereas the chirped adiabatic rapid passage ( carp ) @xcite uses overlapping laser pulses along with frequency chirps to transfer the population . we turn to these methods in the next sections . for the case of scrap ( earlier known as self - induced adiabatic passage , or siarp @xcite ) the laser pulses are engineered so that the system undergoes an avoided level crossing ( @xmath134 ) near a maximum of the effective coupling @xmath135 induced by the stark shifts due to the delay of the pump and stokes pulses . in order to obtain an efficient population transfer the system needs to evolve adiabatically in the crossing region @xcite . for gaussian pulses this leads to the condition : @xmath136 + figure [ fig : siarp_a ] shows the efficiency @xmath89 for various rabi frequencies and pulse delays for a decay rate of @xmath114 and a fixed pulse length of @xmath137 . the behavior described by the adiabaticity requirement is clearly visible . for a fixed rabi frequency the efficiency decreases with increasing pulse delay @xmath111 as predicted by eq . ( [ eq:14 ] ) . the improvement due to increased rabi frequency is also evident . however , for large rabi frequencies the pulse violates the resolved sideband condition @xmath138 , leading to a sudden deterioration of the efficiency @xmath89 at large intensities , see fig . [ fig : siarp_b ] . for small rabi frequencies the efficiency @xmath89 strongly depends on the delay times . for @xmath139 the method is not robust . as the effective decay rate increases with increasing rabi frequencies , the efficiency slowly degrades for greater laser intensity . this leads to a ridge in the @xmath111-@xmath140 diagram . because the efficiency depends on the pulse delay and the rabi frequency , accurate knowledge of these pulse parameters is required . this can be moderated by increasing the detuning of the laser pulses . because the constraints on the pulse length are less severe than for stirap the population transfer can be faster with scrap . together with the far detuning this leads to a improved robustness against the detrimental effect of spontaneous decay @xcite . another way of achieving an adiabatic population transfer is the application of simultaneous pump and stokes pulses ( @xmath141 ) with one laser having a frequency chirp . this is a raman chirped adiabatic passage , sometimes called rcap @xcite , though here we refer to it as chirped adiabatic rapid passage ( in a @xmath0-system ) or carp . with this system the stark shifts are eliminated , and the detuning @xmath142 reads @xmath143 in order to ensure the adiabatic evolution the chirp rate @xmath144 needs to fulfill @xmath145 and @xmath146 [ with the two - photon rabi frequency @xmath147 given by eq . ] . these conditions arise from a landau - zener adiabaticity and from requiring completion of a landau - zener transfer within the time - scale of the pulse . because there is no limit on the pulse duration arising directly from the adiabaticity requirements the transition can be fast : it is only limited by the narrow bandwidth condition @xmath148 @xcite . this in turn reduces the susceptibility to spontaneous emission . the resolved sideband condition @xcite requires that @xmath149 , which is easy to satisfy since the system is in the far - detuned limit @xmath150 . under these conditions the population transferred to the target state can be estimated with the landau - zener formula @xcite @xmath151 + where @xmath152 . this behavior is confirmed by our numerical simulation [ see fig.[fig : carp_a ] ] . it shows the simulated efficiency for different rabi frequencies and chirp rates for a system initially in a state with @xmath153 and @xmath113 . the efficiency increases rapidly with increasing rabi frequency @xmath116 until it reaches a plateau . in this region the transfer efficiency is well above 98% . for small rabi frequencies the efficiency @xmath89 deteriorates with increasing chirp rate . however , this effect diminishes for large pump intensities . similarly to scrap , the increase in the effective decay rate for high rabi frequencies leads to a slow degradation of the transfer efficiency for large pulse intensities . for very high laser intensities the efficiency rapidly drops due to the violation of the resolved sideband condition @xmath154 . in a different parameter regime , that of small chirp rates @xmath155 , the efficiency oscillates with changing rabi frequency , see fig . [ fig : carp_b ] . here the evolution is governed by rabi oscillations between the two rotational states which together with the finite pulse length leads to large fluctuations in the efficiency . in this regime a very precise control of the laser pulses is required , so carp is not robust for very small chirp rates . for large rabi frequencies , the efficiency is essentially independent of the chirp rate and carp provides the best robustness against uncertainties in the pulse parameters . both of the adiabatic rapid passage methods , scrap and carp , provide fast population transfer . together with the large detuning of the raman transition from the excited state they provide robustness against the adverse effect of spontaneous emission . this is clearly evident in fig.[fig : comparison ] where the efficiency of the three methods is plotted against the decay rate . -configuration of fig . [ fig : lambdasystem ] , for different decay rates @xmath156 and the three methods we consider . the parameters are optimised at @xmath122 for each of the three methods and are listed separately in the following . for stirap ( solid line ) : @xmath157 , @xmath124 , @xmath125 , and @xmath158 . for scrap ( dashed line ) : @xmath159 , @xmath137 , @xmath160 , and @xmath161 . for carp ( dotted line ) : @xmath162 , @xmath137 , @xmath163 , @xmath164 , @xmath165 with @xmath166 . other parameters which are fixed for all three cases are : @xmath167 , @xmath168 , @xmath169 , @xmath112 , and @xmath113 . ] in this figure , the values of the parameters @xmath116 , @xmath52 and @xmath111 are optimised for @xmath170 and the detunings given . ( in the case of carp , the value of @xmath144 is also optimal . ) the same optimal values have been used for fixed parameters in figs . [ fig : stirap][fig : carp ] . the parameter @xmath171 has been fixed to a reasonable value for diatomic molecules like n@xmath172 or co@xmath173 . this optimisation gives a fairly wide range of parameters in fig.[fig : comparison ] . [ fig : comparison ] also shows that , even though stirap is efficient for small decay rates , it decreases rapidly for larger decay rates . scrap and carp are far more efficient and their suppression of the excited - state population exceeds that of stirap within the limits imposed by the experimental requirements . even though scrap and carp exhibit similar efficiencies carp is more robust against uncertainties in the laser pulse parameters . hence , we use carp for our investigation of the population transfer in a multilevel system with @xmath174 in the next section . + having derived the conditions for efficient population transfer with chirped two - photon raman transitions , we turn now to systems where we include a larger number of rotational states in the calculation ( @xmath176 ) . consequently the mapping process involves multiple ( @xmath48 ) pairs of laser pulses . we investigate the state mapping given in eq.([eq:6 ] ) , for a thermal initial state distribution with @xmath177 . this corresponds to a system where approximately six rotational levels are significantly populated and consequently we will take @xmath174 . in our simulation we use a lamb - dicke parameter of @xmath169 and a decay rate of @xmath170 which for realistic trap frequencies ( of the order of a few mhz ) corresponds to typical decay rates of electronically excited states of diatomic molecules . the pump laser is detuned by @xmath178 , and the stokes laser is chirped with the rate @xmath179 . both lasers have a peak rabi frequency of @xmath180 , a pulse length of @xmath137 and a delay between successive pulse pairs of @xmath181 . these parameters were chosen on the basis of the simulations with two rotational levels ( section [ sec:3 ] ) and realistic experimental parameters . in fig . [ fig : multilevel1 ] the initial population distribution over all states @xmath9 is plotted , along with the final distribution in fig . [ fig : multilevel2 ] . apart from some small population loss , see fig . [ fig : losses ] , and some weak scattering of population into states other than the states @xmath182 , the two distributions agree very well . the total population transferred into the rotational ground - state is @xmath183 , while the total population loss into the uncoupled states is only @xmath184 . , see fig . [ fig : multilevel ] . the detuning for the pump pulse is equal to @xmath164 and the rabi frequency is @xmath185 . other parameters are as in fig . [ fig : multilevel ] . ] the remaining 6.5% of the total population is mostly left in the initial states @xmath186 [ see fig . [ fig : multilevel2 ] ] . losses due to spontaneous emission can be further reduced by increasing the detuning . the population which remained in the higher , coupled rotational states @xmath187 can be transferred into the ground - state by sympathetically cooling the molecule s motion and reapplying the cooling pulse sequence . using this approach the population of the ground - state can be even further increased from 92% to above 98.4% with a total loss into the uncoupled state of only 1.6% . additional simulations with @xmath188 and @xmath189 , resulted in higher efficiencies , @xmath190 and @xmath191 , respectively . the corresponding losses are less than @xmath192 for @xmath188 and @xmath193 for @xmath194 . for large numbers of populated states the transfer efficiency can be estimated with the following equation : @xmath195 where @xmath89 is the transfer efficiency for the simple @xmath0-system , and @xmath196 is the initial population distribution ( [ eq:5 ] ) . this agrees well with our numerical simulation of six rotational levels . starting from a vibrationally cold system , these values of the efficiency @xmath89 show that we can reach the motional and rotational ground - state of molecules @xcite . for a co@xmath173 ion with a rotational temperature of @xmath197k , the first 15 rotational levels are significantly occupied initially with a ground - state population of only 3% . by applying the carp state mapping with a detuning of @xmath178 and taking the selection rule @xmath198 into account the population of the two lowest lying states can be can be increased to 85% . using a detuning of @xmath194 this can be improved to 97.8% for a single cooling cycle . for a trap frequency of 4 mhz this cooling cycle will be completed within 10 ms . in this work we presented an efficient method to cool the internal states of molecules by means of coherent processes ( and sympathetic cooling ) thus suppressing the problematic spontaneous decay into uncoupled states . by coupling the internal molecular state to the motion of the molecule , that internal state can be mapped onto a motional state . utilizing this , the _ internal _ state is cooled close to its ground - state if the molecule s motion was initially reduced to the motional ground - state through sympathetic cooling . ultimately all the degrees of freedom of the molecule can be cooled by the application of sympathetic cooling to the final motional excitation . due to its high efficiency the method presented here is not only useful to cool the internal state , but can also be employed to detect the internal state of the molecule by measuring its motional state with an atom which is trapped alongside . we have studied various adiabatic methods for a range of laser pulse parameters which are relevant for an experimental implementation of this cooling scheme . the motion of the ion imposes restrictions on the dynamics of the population transfer process which severely limit the possible parameter range for the laser pulses . for the near - resonant method ( stirap ) , population transfer efficiency is very low accompanied with a large population of the excited state . population losses can be suppressed , if far - detuned chirped adiabatic two - photon raman passage methods are employed . schemes that use chirped laser pulses ( carp ) , or self - induced adiabatic passage ( scrap / siarp ) by stark shifting the transition frequencies , turned out to be very efficient . when it comes to the comparison of carp and scrap the former method has the advantage of easy optimization since it has no dependence with respect to pulse shape . furthermore , for both methods , and unlike stirap , the resolved sideband condition imposes less severe constraints on the useful parameter space . the requirements for all three methods were derived with simulations for a @xmath0-system . using the results from this simple model , we were able to demonstrate the applicability of carp in systems with more than two rotational states . for far - detuned transitions , a high - fidelity population mapping from the internal to the motional degrees of freedom is possible . losses were very low and our simulations indicate that the fidelity can be further improved by detuning the laser pulses further from the transition . in the scheme we propose here , each rotational level is coupled to the excited state by a laser . due to the large rotational level splitting of light molecules this means in turn that multiple lasers are required . the number of levels @xmath199 which have a population larger than the cut - off population @xmath200 , and therefore the number of required lasers , can be estimated as @xmath201 for small @xmath200 . even though the number of levels @xmath199 for molecules at room temperature can be of the order of 25 , for temperatures of a few kelvin this reduces to well below 10 states . in many experiments molecular ions can be prepared in low lying rotational states by employing photo - association or state selective photo - ionisation in conjunction with supersonic beam expansion or buffer - gas cooling . however , due to the interaction with black - body radiation and collisions the internal temperature quickly thermalises . by applying the scheme proposed here , this thermalisation can be suppressed to maintain the ground - state population . additionally , the state mapping can be employed to detect the internal states of the molecule in a non - destructive manner which is beneficial for high - resolution spectroscopy of molecules . in conclusion , we have developed a fast scheme for cooling the internal states of single molecules by employing adiabatic passage methods which provide a high efficiency in conjunction with robustness against variations in the parameters of the involved laser pulses . see n. v. vitanov , m. fleischhauer , b. w. shore , and k. bergmann , in _ advances in atomic , molecular , and optical physics _ , edited by h. walther and b. bederson ( academic press , 2001 ) , vol . 46 , pp . 55190 , and references therein . m. d. barrett , b. demarco , t. schaetz , v. meyer , d. leibfried , j. britton , j. chiaverini , w. m. itano , b. jelenkovi , j. d. jost , c. langer , t. rosenband , and d. j. wineland , phys . rev . a * 68 * , 042302 ( 2003 ) . in the conext of ion traps , see , e.g. , d. j. wineland , j. j. bollinger , w. m. itano , and d. j. heinzen , phys . a * 50 * , 67 ( 1994 ) , see section viii ; j. steinbach , j. twamley , and p. l. knight , phys . a * 56 * , 4815 ( 1997 ) ; d. j. wineland , c. monroe , w. m. itano , d. leibfried , w. e. king , and d. m. meekhof , j. res . inst . stand . technol . * 103 * , 259 ( 1998 ) . as explained in ref . @xcite , the far off - resonant stirap process can be seen as a limit of scrap / siarp in the absence of decay . in the presence of decay the stirap process is affected more due to the involvement of the excited state in the dynamics taking place over a long time . due to the generally different rotational constants , any population in vibrationally excited states is basically unaltered by the mapping process we have presented here . thus only the vibrational ground - state of the molecule is transferred into its rotational ground - state .

in this work we investigate the theory for three different uni - directional population transfer schemes in trapped multilevel systems which can be utilized to cool molecular ions . the approach we use exploits the laser - induced coupling between the internal and motional degrees of freedom so that the internal state of a molecule can be mapped onto the motion of that molecule in an external trapping potential . by sympathetically cooling the translational motion back into its ground state the mapping process can be employed as part of a cooling scheme for molecular rotational levels . this step is achieved through a common mode involving a laser - cooled atom trapped alongside the molecule . for the coherent mapping we will focus on adiabatic passage techniques which may be expected to provide robust and efficient population transfers . by applying far - detuned chirped adiabatic rapid passage pulses we are able to achieve an efficiency of better than 98% for realistic parameters and including spontaneous emission . even though our main focus is on cooling molecular states , the analysis of the different adiabatic methods has general features which can be applied to atomic systems .

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Proceed to summarize the following text: floquet topological insulators , a new kind of topological phase , have been attracting much attention in recent years @xcite . these phases are produced by periodically time - dependent hamiltonians . by tuning the amplitude , phase , and frequency of the periodic drive myriad topological phases may be realized . among floquet topological insulators , effective time - reversal breaking floquet chern insulators ( fcis ) have even been experimentally realized in cold - atoms @xcite and optical wave - guides @xcite . these states contain unique @xmath0 edge states that have no equivalent in a static system @xcite . however , characterizing a driven system as topological is non - trivial , as the usual adiabatic arguments @xcite for topological properties can not be applied . naively , the topological nature of these systems could be deduced by analyzing the ground state of the floquet hamiltonian @xcite , which we call the `` floquet ground state '' ( fgs ) . however the fgs has no general connection to the non - equilibrium state realized in experiment , and therefore it can not be used to determine the existence of topological properties . in this work , we study the topological properties of fcis using the entanglement spectrum ( es ) @xcite . entanglement statistics have already been demonstrated to detect ground states @xcite , critical states @xcite , topological states @xcite , and universal exponents @xcite . the es in particular is known to have edge states for topological states that mimic the edge states appearing in the physical boundaries of the same system @xcite . as the es is a function of the state of the system at a given time , it may be used to detect topological properties for a time - evolving state . we calculate the es for four different phases of the fci generated by a sudden turn on of the driving laser both numerically and analytically , and we complement these results with that for a slow turn on of the laser . we label states obtained this way , i.e. , via unitary time evolution , as physical or true states , and we compare the es of these states with the es of the fgs . we present three main findings . first , the es of the fgs correctly detects both the usual and @xmath0-edge states of the system . second , in the case of a resonant laser , the fgs is qualitatively different from the physical state obtained from unitary time - evolution . third , the es in the physical resonant state does not show all the expected edge states . therefore , the system driven by a resonant laser does not have the naively expected topological properties . these results do not depend on how rapidly the laser was switched because , unlike for an off - resonant laser , for a resonant laser there is no adiabatic limit @xcite . we also explicitly show this lack of adiabaticity of the resonant laser via the properties of the es . in ref . it was shown that a chern number constructed out of unitarily evolved states is conserved . thus if the initial state before the periodic drive was turned on had a zero chern number , it would stay zero always . however , this chern number is a property of the full density matrix and is not related to physical observables directly . rather , local physical observables probe a local spatial region and therefore a reduced density matrix for which the arguments of ref . do not hold . evidence that the chern number of the full density matrix is not relevant also comes from the fact that that unitarily evolved states show a non - zero dc and ac hall conductivity even for an initial state with zero hall conductivity @xcite , although this conductivity is smaller in magnitude than @xmath1 , as would be expected for the fgs . thus our result that the es of a unitarily evolved state reflects some of the topological properties of the fgs is consistent . the paper is organized as follows . in section [ sec1 ] the model is presented , and the topological properties of the fcis of interest to us are summarized . in section [ sec2 ] the methods for obtaining the es are described , while the results are presented and discussed in section [ sec3 ] . many details are relegated to the appendices . other than figure [ fig : slowfastturnon ] , all other plots are for the sudden quench . appendix [ app1 ] contains results for the es for the slow turn on of the laser . appendix [ app2 ] gives the bulk occupation probabilities for the quench switch on protocol of the laser , as well as the projections of these occupation probabilities on the translationally invariant axis in order to highlight their relation to the bulk states of the es . plots showing edge states , their chiralities , and decay lengths in the es for fgs and the unitarily evolved states are given in appendix [ app3 ] . a key equation whose solution yields the es is derived in appendix [ app4 ] , while analytic solutions for the edge states in the es of the dirac model are given in appendix [ app5 ] , and they help to provide physical intuition for the more complex es structure of the unitarily evolved state in the presence of the laser . we study the quench from the ground state of graphene at half - filling to a time - periodic hamiltonian corresponding to driving by a circularly polarized laser . the transport and optical properties following such a quench has been extensively studied @xcite . the hamiltonian before the quench is that of a half - filled infinite sheet of graphene , @xmath2 where @xmath3 is the n.n . spacing , and @xmath4 . at @xmath5 , the hamiltonian is changed by substituting @xmath6 , so that the @xmath7 is now periodically time - dependent . a state @xmath8 of momentum @xmath9 evolves under this hamiltonian for @xmath10 according to @xmath11 where @xmath12 is the initial state and @xmath13 , @xmath14 are periodic in time and given by the floquet - bloch equation @xmath15|a_k(t)\rangle = 0 ; \label{eq : floquetbloch}\ ] ] and likewise for @xmath16 . the @xmath17 are the two quasi - energies , which are only defined up to integer multiples of @xmath18 . we take them to lie between @xmath19 calling this range of @xmath17 and @xmath9 the floquet brillouin zone ( fbz ) . we take @xmath20 . the information of the initial state is encoded in the overlaps with @xmath3 and @xmath21 and may be quantified by the excitation density @xmath22 where @xmath23 is the occupation probability of the lower floquet band and likewise for @xmath24 . _ topology _ the wavefunctions at fixed @xmath25 and quasi - energies @xmath17 may be interpreted as the band - structure of some underlying hamiltonian , @xmath26 and we may define a ground state " of this hamiltonian by completely filling the lower band @xmath27 it is found that for some choices of parameters , the corresponding hamiltonian has a non - trivial chern number and is therefore topological . as in the static case , the chern number is related to the existence of edge bands @xcite . in the fci , there are two kinds of edge states : `` conventional '' edges that disperse through @xmath28 and `` anomalous '' @xmath0 edges that disperse through @xmath29 . the chern number is given by @xmath30 where @xmath31 is the difference between the number of left - moving and right - moving conventional ( anomalous ) edge modes . however , the natural physical question is whether the many - body state generically produced by experiment actually displays any topological properties . note that despite appearances , there is no reason for the fgs to be produced in a generic experiment . as @xmath32 is periodic , there is no notion of a lowest - energy state and hence no notion of relaxation to a ground state . in fact , it is believed that generic disorder and interactions cause a driven closed system to reach infinite temperature @xcite . therefore we must consider the physical state as determined by time evolution under a reasonable experimental protocol , here a quench , both sudden and slow . having produced the physical state , we now must decide if it is topological . here several standard arguments fail . as the system is time - dependent , there is no good notion of adiabatic flux threading @xcite . the application of a physical edge will qualitatively change the evolution of the system and therefore does not provide good information about the system in the absence of such an edge . as the application of generic disorder drives the system to infinite temperature , there is no notion of static localization . these problems are solved by using the entanglement spectrum ( es ) @xcite . the es is calculated by imposing a fictitious boundary on the density matrix at a particular time . the degrees of freedom outside the boundary are traced out , and the spectrum of the reduced density matrix is the es . the fictitious entanglement boundary functions similarly to a physical boundary , and it may host chiral edge states that can be used as evidence of topological properties . for a free system @xcite the entanglement spectrum may be derived from the eigenvalues of the correlation matrix @xmath33,\ ] ] where @xmath34 is the operator that annihilates an electron at site @xmath35 , and the lattice sites @xmath35 , @xmath36 are restricted to lie in the sub - system of interest . the eigenvalues @xmath32 of this matrix lie between @xmath37 and @xmath38 with a value of @xmath37 or @xmath38 representing an unentangled or pure state mode and @xmath39 representing a maximally entangled mode . .[tab : phases]summary of the four phases analyzed . columns from left to right : name of the phase , @xmath40 the amplitude of the driving laser in units of the inverse lattice spacing , @xmath41 laser frequency in units of the hopping strength , whether or not the laser is resonant with the static spectrum , @xmath42 number of usual edge modes of the fgs , @xmath43 number of anomalous @xmath0 edge modes , @xmath44 the total chern number . [ cols="^,^,^,^,^,^,^",options="header " , ] we consider four representative phases ( @xmath45 ) of the fci summarized in table [ tab : phases ] . the first phase @xmath46 is the simplest case with an off - resonant laser @xmath47 and a single conventional edge mode . the other three ( @xmath48 ) are for resonant laser frequencies of @xmath49 , where @xmath0 edge states may appear . @xmath50 has one conventional edge mode and two counter - propagating @xmath0 modes so that the total chern number is 3 . @xmath51 has a single conventional mode like @xmath46 but is produced by a high - amplitude resonant laser . finally , @xmath52 is an unusual topological phase where there are two conventional and two @xmath0 modes so that the total chern number is zero . the occupation of these edge states following the laser quench in a system with boundaries was recently discussed in ref . . the occupation probabilities for the infinite system are discussed and plotted in appendix [ app2 ] . c we compute the long - time behavior of the es and ee for the four phases @xmath53 . the time is fixed to be @xmath54 periods for the sudden quench , whereas the results are at non - stroboscopic times for the slow quench ( see appendix [ app1 ] ) . the floquet - bloch wave - vectors are computed by expanding eq . in a finite number of fourier modes and solving the linear system . the result is used to construct @xmath55 . we take the entanglement boundary to be the zigzag edge of a strip of width @xmath56 sites so that the momentum @xmath57 along the strip remains a good quantum number . thus , @xmath55 may be diagonalized at fixed @xmath57 , and the results are plotted in figure [ fig4 ] . the results are compared with the es of the fgs a single statistic that may be calculated from the es is the entanglement entropy , given by @xmath58,\ ] ] the sum being taken over all eigenvalues . the results for the entropy are shown in figure [ fig3 ] for the fgs and the physical state . the fgs state shows area - law scaling , @xmath59 . this is because the floquet state is the ground state of @xmath60 , which is local and gapped , and such states are known to show area law@xcite . however the physical state shows volume - law scaling ( @xmath61 ) as is expected of a generic state with ballistically propagating excitations . this behavior , therefore , does not provide detailed behavior about the phases . the full es for both states , namely the fgs and the physical state , for phases @xmath53 is shown in figure [ fig4 ] for the sudden quench and figure [ fig : slowfastturnon ] in appendix [ app1 ] for the slow quench . the figure is symmetric around @xmath62 as a consequence of particle - hole symmetry . let us first discuss the es for the fgs . the es shows a bulk of bands clustered near @xmath37 and @xmath38 , with a gap between the two bands . additionally , there are a small number of bands that disperse through @xmath62 that appear suggestively like chiral edge states . on comparing with table [ tab : phases ] , one finds that the number of edge states that cross the gap in the es of the fgs is the same as the chern number . this means that when anomalous edge states appear , as in phases @xmath63 , the chiralities of the edge states in the es reverses in such a way that the chern number as calculated from the es agrees with the chern number of the phase . thus for @xmath50 , for example , there are 3 chiral right - movers in the es even though the anomalous modes are left moving at the physical edge . a similar reversal of chiralities is observed in @xmath52 . therefore , the es `` understands '' that the @xmath0 modes contribute to the chern number with opposite sign . while the es is a snapshot at a particular time , at other times , the location of the edge states changes , but their number and chirality are maintained so as to preserve the chern number . thus the phase @xmath52 can show odd behavior due to the fact that it corresponds to @xmath64 . this is highlighted in figure [ fig : p4ent ] , where within a laser period , the edge - modes can appear with canceling chiralities ( @xmath5 in figure ) , and totally disappear ( @xmath65 in figure ) , both of these cases being consistent with @xmath64 . next we turn to the es of the physical state . in the off - resonant case @xmath46 , the floquet and quench es generally agree . in the phases @xmath48 , the two appear drastically different . while the central crossing at @xmath66 and the gap remain intact , a continuum of eigenvalues that pass through @xmath39 appears in the quench case . these continua of states cover the region where the two edge states of @xmath50 located at @xmath67 appear in the fgs spectrum and completely cover the gap in @xmath52 . this picture qualitatively holds for a slow quench . as shown in figure [ fig : slowfastturnon ] , the edge modes that are absent for the sudden quench continue to be absent for a slower quench . in the next section , we explain these results . this behavior of the es may be explained as follows . in the long - time limit , the contribution to the correlation function @xmath55 of the overlap between the upper and lower bands oscillates as @xmath68 with @xmath69 . as @xmath70 , @xmath71 changes rapidly with @xmath9 when @xmath72 and the overlap terms may be neglected . straightforward manipulation ( see appendix [ app4 ] ) then gives @xmath73 where , @xmath74 if the size @xmath56 of the sub - region @xmath75 is taken to infinity , @xmath76 may be diagonalized by fourier transformation , and its eigenvalues may be read off from eq . as @xmath77 with eigenvectors @xmath78 and @xmath16 . therefore we may think of @xmath79 as a hamiltonian with two bands , where the gap between the two bands is @xmath80 , and the wavefunctions have some topological properties . note that the smaller the quench - induced excitation density is , the larger is the gap @xmath80 in the es . the problem of finding the spectrum of @xmath79 at finite but large @xmath56 is then akin to the problem of finding the spectrum of a hamiltonian in a finite geometry . we expect two kinds of states : bulk states and edge states . the bulk states should have the `` energies '' , @xmath81 , but where @xmath9 is quantized in the finite direction . therefore , the `` bulk '' states can be produced by projecting the occupation number onto the @xmath82 plane ( the translationally invariant direction ) ; see figure [ fig : proj ] in appendix [ app2 ] . in addition to these bulk states , there may be edge states . if the floquet bands are topological , that is , if the chern number is non - zero , _ and _ @xmath81 is non - zero , then the topological protection of edge modes of @xmath79 applies and there will necessarily be edge modes in the es . this property is topological in the sense that a small deformation of the state , which is equivalent to a deformation of @xmath81 and @xmath83 , can not change the net number of edge modes . for the fgs , where @xmath84 by definition , there will exist edge modes that agree with the chern number of the bands . this also appears to be the case in @xmath46 , where @xmath85 only at isolated points , and the edge modes of the fgs and physical states agree , although perhaps the issue is more delicate when the entanglement cut is very rough so that momentum @xmath57 is not conserved . that these states are indeed topological can be seen in the numerics . their number does not scale with @xmath56 and their crossings are not lifted by varying the phase of the driving laser . moreover , a direct analysis of the wavefunctions associated with the eigenvalues shows that they are chiral and located on the edge . figures [ fig : edgef12 ] and [ fig : edgef34 ] plot the wavefunctions for the edge states for the fgs , while figures [ fig : edgeq12 ] and [ fig : edgeq34 ] plot the same for the true state , and figure [ fig : edgedecay ] highlights the decay length of the edge states . if @xmath86 , then this is equivalent to a gap closing in the hamiltonian , and topological protection arguments no longer hold . indeed , we find that the edge states do not coexist with the bulk states in the numerics . since a resonant process will co - occur with a population inversion ( see figure [ fig13 ] ) , we do not expect edge states to be protected in the resonant case . further , since @xmath85 precisely where the resonant laser causes band crossings , the resulting edge modes are not robust at all . indeed , this is what is seen in the phase @xmath50 where the two edge modes seen in the floquet bands at the boundary of the fbz do not appear in the es of the true state . this is not an artifact of the quench protocol , and similar population inversions appear in the slow turn on of the resonant laser ( see appendix [ app1 ] ) . although the edge modes lose topological protection when @xmath86 , there may still be non - protected edge states . since we study a perfectly clean system with translational invariance in the @xmath87 direction , an edge mode will appear under the much weaker condition that @xmath88 for the same @xmath57 at which the edge mode appears . therefore , we find that the central mode appears in @xmath50 and @xmath51 even though @xmath81 is zero over a large arc in the bz . we expect that this edge mode would disappear for other geometries . for @xmath52 , the high - amplitude laser produces population inversions throughout the bz , and there are no edge states whatsoever in the physical state . for phase @xmath46 , on the other hand , one has a robust gap , with gap closings only at some special points ( rather than large arcs ) . these excitations can be made even smaller for a slow laser switch on , as for the off - resonant case , an adiabatic theorem can hold . this provides a complete description of the entanglement spectrum at long times after a floquet quench . the key quantity that determines the applicability of the chern number calculation to the edge modes of the quench is the population difference @xmath81 . it is interesting to ask how the results of the es compare with physical observables . in ref . , the occupation of the edge states in a system with boundaries was studied , and it was found that the edge state for the off - resonant case @xmath46 was occupied at a low effective temperature . in contrast , only one set of edge modes in the phase @xmath50 was occupied at a low effective temperature , while the two new sets of @xmath0-edge modes appearing due to resonant band crossings were occupied at a high effective temperature and coexisted with bulk excitations . the dc conductivity for these phases was studied in ref . , where it was found that the conductance was very close to the maximum , @xmath89 , for @xmath46 , and it persisted at this value for @xmath50 even though new edge modes appear for @xmath50 . this observation implies that the high effective temperature of the resonant edge modes prevent them from contributing much to transport , in contrast to the off - resonant edge mode . thus our results show that the study of the es is a useful way to understand the topological properties of a periodically driven system . an important direction of research would be to extend this analysis to interacting floquet topological insulators . _ acknowledgements : _ this work was supported by the us department of energy , office of science , basic energy sciences , under award no . de - sc0010821 . to generate occupation probabilities for the slowly turned on laser , the schrdinger equation at fixed @xmath57 is solved by implementing the commutator free exponential time method following ref . . the occupation probabilities and entanglement spectrum are calculated from these wavefunctions . due to the computational cost of this procedure , the system width was taken to be @xmath90 sites . the results for the es shown in figure [ fig : slowfastturnon ] , are in qualitative agreement with the quench protocol discussed in the body of the paper . band inversions appear near momenta where the laser is resonant with the band gap . these band inversions are accompanied by a high bulk excitation density , leading to a closing of the entanglement gap , and ruining the topological protection of edge states . figure [ fig : slowfastturnon ] shows data for two different laser switch - on rates for the two resonant phases @xmath91 . @xmath50 has three pairs of chiral edge states in the ground state ( fgs ) of the floquet hamiltonian . however , we find that for all switch - on rates of the laser , two of the three pairs of edge states ( located at @xmath92 ) are absent from the es of the unitarily evolved states . only the central edge state located at @xmath66 survives . for phase @xmath51 , there is only one pair of chiral edge states in the floquet ground state , and this is visible in the es of all the unitarily evolved states as well . however this edge state coexists with bulk excitations that survive both for a fast and slow laser quench . paradoxically , a slower laser switch on creates more bulk excitations when the laser frequency is resonant . we understand this as follows . a large - amplitude laser can modify the band structure considerably , making the bands flatter . thus a resonant laser can become effectively off - resonant when the amplitude of the laser becomes too large . thus when a laser amplitude is switched on slowly , ( while its frequency is kept unchanged ) , since for a longer period the effective amplitude of the laser is smaller for a slower switch on than a faster switch on , more resonant excitations are created for the slower switch on as the resonance condition is obeyed for a longer duration of time . that is the reason why in figure [ fig : slowfastturnon ] , there are more bulk excitations for the slower switch on than for the faster switch on . the only reason why the central edge state at @xmath66 is unaffected by these bulk excitations is because our entanglement cut is smooth and momentum @xmath57 is conserved . this prevents hybridization between the central edge state and the bulk excitations , since the latter are located at other values of @xmath57 . however a rough entanglement cut will wipe out even this central edge state . the occupation probabilities of the floquet bands for phases @xmath94 are shown in figure [ fig : drhop12 ] , and those for the phases @xmath95 are shown in figure [ fig : drhop34 ] . except for @xmath46 , all the phases correspond to a resonant laser . for @xmath46 , the quasi - energy of the floquet hamiltonian is similar in appearance to static graphene with the important modification of becoming gapped at k and k. thus , the occupation probabilities follow intuitively from the static picture , and the sharp spikes in figure [ fig : drhop12 ] correspond to excitations around the k and k points . the laser being off - resonant is equivalent to the statement that the periodicity of the quasi - energy does not play a role since corresponding boundaries in the fbz are far from the maximum and minimum of the bands . both @xmath91 have lasers of the same frequency , but the amplitude of the laser for @xmath51 is larger than that for @xmath50 . in @xmath50 there is a population inversion at @xmath96 in addition to the excitations at k and k. this is simply the resonance condition , as now the frequency of the laser is such that it connects the peak and trough of the static graphene bands . in the quasi - energy picture , this translates to the original static bands extending beyond the fbz and thus forming quasi - energy bands that avoid one another and bend away at the boundary of the fbz . a large amplitude laser modifies the effective band structure to such a degree that a resonant laser becomes effectively off - resonant . the bands are drawn toward the center of the fbz , away from the boundaries . it is for this reason that @xmath51 has fewer bulk excitations than @xmath50 , despite the two phases sharing the same laser frequency . the flattening of the bands explains the broadening of the excitations at the k and k points . for phase @xmath51 , even though the laser is resonant , the excitation density around @xmath66 , where the central edge mode in the es appears , is still fairly low . the laser creates pockets of excitations in regions symmetrically located around the central edge mode . momentum conservation prevents the central edge mode from mixing with these pockets of bulk excitations . the amplitude of the laser for @xmath97 is much larger than all the other phases , and its frequency is much smaller as well . as a consequence , the @xmath52 phase is highly excited , with the two floquet bands being almost equally occupied . figure [ fig : proj ] is the projection of the occupation probabilities onto the @xmath93 axis . these projections are generated by selecting constant @xmath98 slices of the occupation probabilities that correspond to the modes that satisfy the boundary conditions of the strip . these slices are then superimposed on top of one another to give the projection image . as explained in the main text , these projected plots reproduce the bulk states of the es . there are some small discrepancies between the projection plot and the true spectrum . as seen in figure [ fig4 ] , the bulk excitations on either side of the central edge state in the quenched @xmath50 phase are smooth , but the analogous bands constructed from figure [ fig : proj ] appear ragged and cross @xmath99 , where they do not in the former . we explain this difference below . as the width is increased , we first expect the bulk excitation bands of the es to take on the rough appearance as predicted in the projection plot . the larger width will also lead to more bands coalescing towards @xmath39 . thus in the very large limit , we expect a continua of excitations on both sides of the central edge state of the quenched @xmath50 phase . the reason for the differing appearances in the large , but not extreme width limit is due to the bulk bands having residual knowledge of the edges and thus undergoing a smoothening and repulsion procedure , as one expects for perturbations . thus as the width is increased , this smoothening will diminish and bands from both figures [ fig4 ] and [ fig : proj ] will agree . the reason the repulsion argument does not extend to the edge states is precisely due to their local nature and exponentially small overlap . in the @xmath51 phase , @xmath98 projections originate from occupation probabilities that have a relatively smooth structure throughout the fbz , as seen in figure [ fig : drhop34 ] . thus we do not run into this problem for the @xmath51 phase because the projection already creates a continuum that agrees with the spectrum in figure [ fig4 ] . however , for the @xmath50 phase , the probability occupations have sharp , nearly vertical transitions that lead to discrete bands that single out and amplify this otherwise small feature . the continuum of excitations will only be created by the projection plot for @xmath50 if the strip width is so large that the spacing between @xmath98 states is narrow enough to allow for many states to be selected along the cliff faces ( see figure [ fig : drhop12 ] ) found in the occupation probability of the @xmath50 phase . the entanglement spectrum for phase @xmath52 is shown in figure [ fig : p4ent ] . the floquet state shows the behavior expected of a @xmath64 floquet chern insulator . the `` bulk '' bands are clustered around @xmath28 and @xmath100 . there are edge states seen around @xmath39 , however these do not have a net chirality . this can be seen either from direct inspection of the wavefunctions in figure [ fig : edgef34 ] , or by noting that the edge states do not connect with the bulk bands . therefore , these edge states are not topologically protected , and they should disappear under disorder . in fact figure [ fig : p4ent ] shows that the edge states appear at certain times in the laser cycle , and they disappear for certain other times . this does not happen for phases @xmath46 , @xmath50 , and @xmath51 ; when probed at times away from an integer number of laser periods , the precise location of the edge states is shifted in the es , but the results for the number and chirality of the edge states that are visible remain the same . the true quench state for phase @xmath52 does not show any edge states , and the bulk bands approximately fill the space ( see figure in main text ) . this is in agreement with the occupation number difference shown in figure [ fig : drhop34](b ) , which is found to vary widely between @xmath101 and @xmath38 and crosses @xmath37 on large arcs through the bz . from the perspective of the entanglement and occupation number , therefore , the true quench state for @xmath52 appears similar to a thermal state . the amplitudes of the edge states are shown in figures [ fig : edgef12 ] , [ fig : edgef34 ] for the floquet ground state , and in figures [ fig : edgeq12 ] , [ fig : edgeq34 ] for the true state . to be clear , by state " we mean the eigenvector of the correlation matrix @xmath102 . as discussed in the main text , we expect these edge states , at least for the floquet ground state , to behave as the usual edge states of a topological hamiltonian . in fact , the states are highly localized to the edge with a decay length on the order of one lattice site . moreover , we can directly verify the chirality of the edge states by analyzing how the wavefunctions vary with momentum @xmath57 . in figures [ fig : edgef12 ] , [ fig : edgef34 ] , [ fig : edgeq12 ] , [ fig : edgeq34 ] the two states with @xmath32 closest to @xmath62 are shown . the higher @xmath103 state - analogous to the higher energy state - is colored dark while the lower state ( @xmath104 ) is colored light . as can be seen at @xmath57 near the crossing point , the edge states are well defined and localized on opposite edges . moreover , at the crossing , the high @xmath32 state switches with its counterpart . this is expected in a chiral edge because the two bands cross as @xmath57 is varied . therefore , by analyzing the pattern of switching , the chirality of the bands stated in the main text can be verified . as an example of this analysis we consider the @xmath105 edge in figure [ fig : edgef12 ] for the fgs of phase @xmath50 . we see a pattern of black to gray for all three crossings , which indicates that all the downward arching bands in figure [ fig4 ] correspond to chiral edges that reside on the @xmath106 edge of the strip . the edge states corresponding to the downward trending bands at each crossing will be termed left movers " and the upward arching bands will be termed right - movers " . thus , by the same analysis , the @xmath107 edge is found to contain all right movers at each crossing . the same can be done for the fgs of the @xmath52 phase to verify the assignment of @xmath108 . both sides contain two crossings of both types , black to gray and gray to black . thus each side contains an equal number of left movers and right movers . note that these edge states vanish altogether at other times during the laser period , as pointed out in figure [ fig : p4ent ] . in contrast , for all other phases with @xmath109 , the edge - states slightly shift their @xmath57 positioning in a periodic manner . figures [ fig : edgeq12 ] , [ fig : edgeq34 ] contain states corresponding to @xmath32 near @xmath62 for the unitarily evolved system . these figures show that unlike the fgs , not all the states at this eigenvalue are edge states . while phase @xmath46 is similar to the fgs , for phases @xmath91 only one pair of edge modes exist , with a large fraction of states around @xmath110 being bulk states . the decay of the edge states is shown in figure [ fig : edgedecay ] . the decay lengths of the edge states are determined by the inverse of the entanglement gap , which is controlled by the bulk excitation density . thus @xmath46 which has few bulk excitations in the quenched state has a similar decay length to the fgs . in contrast , since the quenched state of @xmath51 has more bulk excitations , it shows a longer decay length than the edge state of the fgs . the surviving edge state of @xmath50 in the quenched state has a similar decay length as the fgs because , as seen in fig . [ fig4 ] , in the vicinity of this edge state the entanglement gap of the quenched state is almost the same as that of the fgs . c + + c + c + + c c + c + c + + c generally , the correlation matrix is given by @xmath111,\end{aligned}\ ] ] where @xmath112 and likewise for @xmath113 , and @xmath114 is the initial state . we make the assumption that the third and fourth off - diagonal terms proportional to the exponential vanish at long times because of the rapidly varying phase . we emphasize that this is an acceptable assumption because we are considering @xmath35 to be restricted to a sub region of the lattice . for general @xmath35 the stationary phase approximation implies that the exponentials will contribute in a region where @xmath115 where @xmath116 is the group velocity of the floquet dispersion . therefore , as long as we consider times that are much larger than @xmath117 , where @xmath118 is some characteristic velocity , we may neglect the oscillating terms . we have also checked the validity of our argument from the numerical results . we should mention that the order of limits here does not commute . the limit @xmath119 must be taken before width @xmath120 . physically this corresponds to only analyzing the interior of the light - cone . if we neglect these off - diagonal terms , we obtain @xmath121.\end{aligned}\ ] ] now employing @xmath122 from the completeness of the basis , and the fact that @xmath123 , we obtain the stated equation where we identify @xmath124 . before we discuss the es , let us review how edge states appear in the physical boundaries of the dirac hamiltonian . the dirac hamiltonian in a semi - infinite geometry with a boundary at @xmath106 is , @xmath125 + m(x)\sigma_z.\end{aligned}\ ] ] the boundary is controlled by the behavior of @xmath126 . we may write the eigenstate as @xmath127 , which should obey @xmath128 + m(x)\sigma_z\biggr)\psi_k(x , y ) \nonumber\\ & & = e_k\psi_k(x , y),\end{aligned}\ ] ] and implies the eigenvalue equation , @xmath129\chi + v_f k_y \sigma_y \chi = e_k \chi.\end{aligned}\ ] ] the solution is @xmath130 , @xmath131 . thus the eigenstate is , @xmath132 the above is an edge state with a decay length controlled by @xmath133 . now let us consider how edge states emerge in the entanglement spectrum . for this , we will consider a spatially uniform system ( @xmath134 ) with eigenstates , @xmath135 it is convenient to define @xmath136 where the correlation function in real space is @xmath137 for any generic state , be it one generated at long times after a quench when dephasing has set in , or a mixed state at finite temperature , we may write , @xmath138 we will assume the system to be at half - filling . note that since , @xmath139 , a filled band will give , @xmath140 the above corresponds to an unentangled product state with eigenvalues of the correlation matrix being @xmath38 . it is useful to note that @xmath141 is a matrix with eigenvalues @xmath142 . we write @xmath143 , where @xmath144 are pure states . since we are interested in the es of a semi - infinite geometry ( @xmath145 ) , we may study the @xmath57 fourier component of @xmath44 in the translationally invariant @xmath87 direction , @xmath146 note that the above correlation matrix is @xmath147 added to the fourier transform of @xmath148 times a matrix with eigenvalues @xmath142 . therefore , in the infinite system limit it has eigenvalues @xmath149 corresponding to plane - waves eigenvectors . in the presence of a boundary we expect some linear combinations of these plane waves to form extended scattering states . the boundary is now determined by the entanglement cut at @xmath106 , rather than @xmath126 . if @xmath152 and smoothly varying around @xmath93 , we may approximate it by it s value at @xmath82 . this may then be pulled out of the integral . then using the fact that @xmath153 and since @xmath154 decays exponentially away from @xmath106 , this term can be replaced by a delta - function , @xmath155 . thus , @xmath156,\end{aligned}\ ] ] up to terms which are higher order in derivatives . note that if @xmath157 the second term vanishes and @xmath44 is controlled by the higher order terms which will not generically have edge states . this is solved by the same ansatz as for the usual physical edge mode of the dirac hamiltonian , @xmath160 where , @xmath161 and the step function to explicitly impose the boundary conditions . the entanglement eigenvalues are , @xmath162 in particular this shows that the edge state with @xmath39 exists at @xmath163 , and it disperses linearly with @xmath57 about this point . now we return to the assumption we made that @xmath164 . if @xmath165 were to hold at some point in @xmath98 , then there would be an extended state with @xmath166 . as this extended state , and edge mode , both have equal @xmath57 momentum , and the same eigenvalue under @xmath44 , we expect them to mix . therefore , in this case expanding around @xmath82 is not valid , the argument above does not hold , and we do not expect edge states . while this analytic argument was shown for the dirac model , and the floquet chern insulator is far more complicated , especially for large chern numbers , yet we find that the same principles apply . for example figure [ fig13 ] shows how @xmath167 varies along @xmath98 for the @xmath50 phase . we have chosen @xmath168 which coincides with the edge mode in the es of the floquet eigenstate that is located to the left of the central edge mode ( see figure [ fig4 ] ( b ) in main text ) . this particular edge mode is absent in the true wavefunction . the reason now is clear . figure [ fig13 ] shows that @xmath169 vanishes at two points in @xmath98 , implying extended states with the same energy as the edge state . these extended and edge states can now mix , resulting in the absence of the edge state at @xmath170 . it is interesting to note that the resonant laser results in a rather precise location of the vanishing of @xmath81 relative to the avoided band crossings of the upper and lower floquet bands . 33ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrevb.79.081406 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.017401 [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.235108 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.113.236803 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.113.266801 [ * * , ( ) ] link:\doibase 10.1103/physrevb.91.155422 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevx.3.031005 [ * * , ( ) ] link:\doibase 10.1103/physrevb.23.5632 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.010504 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.aop.2010.09.012 [ * * , ( ) ] , link:\doibase 10.1103/revmodphys.82.277 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.170505 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.nuclphysb.2006.12.012 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.115122 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.110405 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.96.110404 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.170605 [ * * , ( ) ] link:\doibase 10.1103/physrevb.94.024306 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.130502 [ * * , ( ) ] link:\doibase 10.1103/physrevb.93.241406 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.92.165111 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.195429 [ * * , ( ) ] link:\doibase 10.1103/physrevb.93.205437 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.115.030402 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.140401 [ * * , ( ) ] link:\doibase 10.1103/physrevx.4.041048 [ * * , ( ) ] http://stacks.iop.org/1751-8121/42/i=50/a=504003 [ * * , ( ) ] @noop ( ) http://stacks.iop.org/1367-2630/14/i=10/a=105008 [ * * , ( ) ]

results are presented for the entanglement entropy and spectrum of half - filled graphene following the switch on of a circularly polarized laser . the laser parameters are chosen to correspond to several different floquet chern insulator phases . the entanglement properties of the unitarily evolved wavefunctions are compared with the state where one of the floquet bands is completely occupied . the true states show a volume law for the entanglement , whereas the floquet states show an area law . qualitative differences are found in the entanglement properties of the off - resonant and on - resonant laser . edge states are found in the entanglement spectrum corresponding to certain physical edge states expected in a chern insulator . however , some edge states that would be expected from the floquet band structure are missing from the entanglement spectrum . an analytic theory is developed for the long time structure of the entanglement spectrum . it is argued that only edge states corresponding to off - resonant processes appear in the entanglement spectrum .

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Proceed to summarize the following text: the phoenix dwarf galaxy , discovered by schuster & west ( 1976 ) , has a morphological type of dirr / dsph ( mateo 1998 ) , meaning that although this galaxy has a low surface brightness and a mass and morphology like a dwarf spheroidal ( dsph ) galaxy , there is evidence of recent star formation ( canterna & flower 1977 ) . color - magnitude diagrams obtained from ground - based images have been described by ortolani & gratton ( 1988 ) , van de rydt , demers , & kunkel ( 1991 ) , held , saviane , & momany ( 1999 ) , and martinez - delgado , gallart , & aparicio ( 1999 ) . these studies find a broadly distributed metal - poor population of evolved giants , as well as a concentration of young stars in the central region of the galaxy . several studies have found h i in the direction of phoenix ( carignan , demers , & cote 1991 ; young & lo 1997 ) ; most recently , st - germain , carignan , cote , & oosterloo ( 1999 ) have found that a mass of @xmath0 @xmath1 of h i gas is likely associated with phoenix . studying star formation in dwarf galaxies has the potential to provide clues about the mechanisms which govern star formation , and the relative formation epochs for different types of galaxies . key questions include the age of the oldest stellar populations are dwarf galaxies the oldest stellar systems and the building blocks for larger galaxies ? and the degree to which star formation is episodic . the latter issue has implications for the visibilities of different types of galaxies at high redshifts ; if star formation is strongly episodic , then even small galaxies can appear bright when observed at redshifts for which star formation rates were greatest . we initiated a hst / wfpc2 study of phoenix to investigate its stellar population in greater detail . in particular , our project was motivated by the desire to determine whether an ancient population of stars exists by looking for a horizontal branch population . during the period between proposal submission and the time the observations were taken , however , several groups detected such a horizontal branch from ground - based observations ( held _ et al . _ 1999 , martinez - delgado _ et al . _ 1999 , tolstoy _ et al . _ 2000 ) . nonetheless , the deeper hst data still provide several pieces of valuable information about the star formation history of phoenix , which we discuss in this paper : 1 ) wfpc2 photometry of the horizontal branch is more precise than can be obtained from the ground , leading to the ability to study its morphology and determine its magnitude with higher precision , 2 ) the deeper observations allow us to observe stars down to the level of a turnoff for an ancient population , providing information on the distribution of ages of stars in phoenix , and 3 ) accurate photometry of the younger main sequence stars allows us to assess the degree to which more recent star formation has been episodic and also to get some indication of the relative metallicities of the young population compared with the older one . in section 2 we discuss our observations and data reduction . section 3 discusses the morphology of the color - magnitude diagram . in section 4 we use numerical simulations of the distribution of stars in the color - magnitude diagram to infer the star formation history of phoenix , and discuss the uncertainties which attend such an analysis . some implications of our phoenix results for the evolution of dwarf galaxies in general are discussed in section 5 . one field in phoenix centered on @xmath2 ( j2000 ) = 01:51:06.3 , @xmath3 ( j2000 ) = -44:26:40.9 was imaged with wfpc2 on hst on 15 january 1999 . while appearing to be close to the center of the galaxy as seen on the digitized poss , this field was chosen to avoid bright ( field ) stars . our program was specifically intended to concentrate on the old background stellar population of phoenix . however , given that various authors ( held _ et al . _ 1999 , martinez - delgado _ et al . _ 1999 , ortolani & gratton 1988 , canterna & flowers 1977 ) find the youngest stars in phoenix to be concentrated towards the center of the galaxy , our wfpc2 field does sample these stars as well . table 1 shows the relevant parameters for the hst observations . the wfpc2 observations were obtained over a total of 4 orbits . six exposures were made through each of the f555w and f814w filters . half of the exposures were made at one telescope pointing , while the remaining half were slightly offset by @xmath4 arcsec along the axes of the wfpc2 ccds . data were processed through the stsci pipeline . the combined exposures in f555w at one of the pointings are shown in figure [ fig : phoenix ] . photometry was obtained using simultaneous psf - fitting on the entire stack of frames , as described in holtzman _ et al . _ ( 1997 ) , using custom software within the xvista data reduction package that was developed / modified based on the routines of stetson ( 1987 ) ; this software has been extensively used to reduce hst / wfpc2 data . in brief , all frames were constrained to have the same lists of stars with the same relative positions ( allowing for slight scale changes between the two different filters ) . the stars in different exposures through the same filter were constrained to have the same relative brightnesses , with frame - to - frame scaling constrained by the relative exposure times . model psfs were derived individually for each frame using phase retrieval of a few bright stars to derive the mean focus for each frame , individual exposure jitter information , and the most recent estimates for the variations of aberrations and pupil functions across the wfpc2 field of view . prior to the psf fitting , frames at each of the two pointings were compared with each other to flag cosmic rays in each frame , and these were ignored during the fitting process . aperture corrections were determined by comparing the psf magnitudes with measurements of stars through @xmath5 radius apertures made on frames in which all neighbors had been subtracted using the psf - fitting results . both psf - fitting and aperture photometry results were corrected for the effect of charge transfer efficiency problems in the wfpc2 detectors using the prescription of dolphin ( 2000 ) ; results presented in this paper were largely unaffected by this correction , however . zeropoints from holtzman _ et al . _ ( 1995 ) were used to place the instrumental magnitudes on the synthetic wfpc2 photometric system ( which we use for comparison with isochrones ) , and transformations to johnson / cousins v and i from holtzman _ et al . _ ( 1995 ) were used for comparison with previous results on the tip of the red giant branch and giant branch colors . the v vs. ( v - i ) and i vs. ( v - i ) color - magnitude diagrams ( cmds ) for the full wfpc2 field observed in phoenix are shown in figure [ fig : cmd ] . photometric errors as estimated from the psf - fitting are plotted versus @xmath6 magnitude in figure [ fig : err ] and demonstrate the high accuracy of the wfpc2 photometry ; typical 1@xmath7 photometric errors at the horizontal branch are @xmath8 mag . more detailed photometric errors were derived using a set of twenty artificial star tests . in each test , over a thousand simulated stars of a given brightness were added to images in a grid pattern ( to prevent additional crowding ) and photometry was rederived for all stars on the frame . the 20 tests covered brightnesses between @xmath9 , with finer sampling at the faint end to measure the completeness of the photometry ; at each brightness , the artificial stars were chosen to have the median color of the observed stars at that brightness . photometry for the simulated stars at each magnitude were extracted to provide error histograms . these were used in the construction of artificial cmds discussed below . in general , the true errors ( difference between observed and simulated brightnesses ) are comparable to those estimated from the psf - fitting routines for the bulk of the simulated stars , but there is always a subset of stars which have significantly larger errors . these are generally the product of systematic errors caused by crowding , and as a result , are correlated in the two bandpasses . in addition , the errors for the artificial stars are smaller than those estimated by the psf - fitting routines for the brighter stars , which is most likely a result of using the exact same psf to create and reduce the artificial stars . an indication of these effects is shown in the bottom panel of figure [ fig : err ] ; the various vertical bands are the measured errors for the simulated stars . the distribution of these errors is non - gaussian since a small subset of stars have relatively large errors ; still , it is apparent that the bulk of the artificial stars have observed errors consistent with the rms errors which are output from the psf - fitting routine ( and shown as the continuous set of points ) . we emphasize , however , that we use the observed error distribution from the artifical star tests in our detailed comparison of simulated cmds with the data discussed below . error bars plotted in figure [ fig : cmd ] show the rms errors from the artificial star tests as a function of magnitude . since the true distribution of errors is non - gaussian , these do not fully convey all the information we have used about the error distribution ; the error bars shown in figure [ fig : cmd ] were computed by determining an rms from the measured error distribution with rejection of @xmath10 outliers . the cmd of phoenix provides examples of stars in many different stages of stellar evolution . the cmd exhibits the types of sequences , most notably the near - vertical blue and red `` plumes , '' which are typical of dwarf irregular galaxies ( _ e.g. _ , cole _ et al . _ the main stellar sequences which constitute such plumes are illustrated by dohm - palmer _ et al . _ ( 1997 ) and gallart _ et al . _ ( 1996 ) . the vertical blue sequence is the main sequence of a young stellar population in phoenix . the red sequence is a combination of giant branch and asymptotic giant branch stars which can have a wide range of ages and metallicities . in addition , there is a clear detection of a horizontal branch in phoenix , with both blue and red components . there is a red clump of stars found near the intersection of the horizontal branch and the giant branch that extends more than 0.5 mag in v and i brighter than the red horizontal branch ( rhb ) . the stars in this clump that are brighter than the rhb are intermediate - age ( 1 - 10 gyr ) core helium - burning stars . finally , there are indications of two sequences extending upwards from the red clump ; the most populated one is nearly vertical , and is particularly pronounced in the magnitude range @xmath11 , while a second very sparsely populated sequence gets bluer at brighter magnitudes ; these ( especially the bluer one ) can perhaps be seen more clearly in the larger area ground - based observations of held _ et al . _ ( 1999 ) . these two sequences correspond to core helium burning stars at either end of the blue loops through which relatively young massive stars evolve . the red giant branch ( rgb ) is located between @xmath12 and 1.6 , and extends brightward to @xmath13 . the color of the rgb can be used to place constraints on the metallicity of stars in phoenix , and the brightness of the tip can be used as a distance indicator . if we adopt a distance modulus of @xmath14 ( held _ et al . _ 1999 , and see below ) , then the mean color of the red giant branch at @xmath15 ( @xmath16 ) is @xmath17 , or @xmath18 . using the calibration of da costa & armandroff ( 1990 ) between @xmath19 at @xmath20 and [ fe / h ] , gives a mean metallicity for the phoenix red giants of [ fe / h ] @xmath21 . our estimate of the mean metallicity of phoenix is in good agreement with the result of held _ et al . _ ( 1999 ) , @xmath22 dex lower than that of martinez - delgado _ et al . _ ( 1999 ) , and 0.13 dex higher than that of van de rydt _ et al . _ ( 1991 ) . with an absolute visual magnitude of @xmath23 ( pritchett & van den bergh 1999 ) , phoenix falls very close to the [ fe / h ] versus @xmath24 relation obtained by caldwell _ et al . _ ( 1992 ) for local group dwarf elliptical and spheroidal galaxies . the giant branch at @xmath15 is @xmath25 mag wide in @xmath26 . if this is assumed to be entirely due to a metallicity spread within an ancient stellar population then the calibration of da costa & armandroff ( 1990 ) indicates a metallicity spread within phoenix of [ fe / h ] @xmath27 to @xmath28 however , two effects complicate the above determination of the mean metallicity and metallicity spread of phoenix from giant branch colors . first , the blue edge of the giant branch is likely to include some asymptotic giant branch stars . second , and probably more important , the giant branch color is a function of age as well as metallicity , in that younger giants are bluer than older ones . at fixed metallicity but mixed ages , the giant branch color derived from assuming an exclusively old population will underestimate the mean metallicity , and the giant branch width will overestimate the metallicity spread . however , if ages and metallicity are correlated , as would occur if younger populations are enhanced in metals , the true metallicity spread can be significantly _ larger _ than that inferred from the giant branch width assuming an exclusively old population . although there is some uncertainty about the metallicity depending on the age , there is still an upper limit which can be placed on the metallicity of any component of the population . even if the entire population was young ( less than a few gyr ) , the color of the giant branch would still require a metallicity less than [ fe / h ] @xmath29 -0.7 based on recent padova isochrones ( girardi _ et al . _ 2000 , girardi _ et al . _ this is demonstrated in figure [ fig : isorgb ] , which shows the phoenix cmd with isochrones for ages 1.6 , 3.2 , 6.4 , and 12.8 gyr at @xmath30 and @xmath31 . in order for the rgb to be consistent with a metallicity of z=0.004 ( [ fe / h ] = -0.7 ) , the padova isochrones indicate that the ages of rgb stars would have to be less than 2 gyr . the deep hst data allow us to constrain the population further , since the red giant branch of phoenix is found to be well populated right down to @xmath32 , _ i.e. _ , down to the luminosities expected for the base of the rgb of a 13 gyr @xmath30 stellar population . this demonstrates conclusively that not all the stars are young . within the allowed range of metallicity , giants which are this faint must be older than @xmath29 6 gyr , based on comparisons with the padova isochrones ; for such ages , the dependence of giant branch color on age is relatively weak . since a significant fraction of the red giants must be relatively old , the isochrones indicate that they must also be relatively metal - poor , _ i.e. _ [ fe / h ] @xmath33 -1.7 . a more quantitative discussion of the metallicity spread within phoenix requires analysis of the relative numbers of younger and older stars . this is discussed further below . however , in terms of both the mean metallicity and the extension to faint magnitudes , the rgb of phoenix is consistent with this galaxy containing a substantial component of stars analogous to population ii of the milky way . the magnitude of the tip of the red giant branch can be used as a distance indicator for metal - poor populations . the accuracy of this technique in the current data set may be limited by the relatively small number of stars ; one might expect that the tip magnitude derived from the relatively smaller number of stars in the wfpc2 frame would be fainter than a true tip magnitude , since our tip is essentially defined by the single brightest red giant . still , we find an apparent tip magnitude of @xmath34 , which is essentially identical to the tip magnitude derived by held _ et al . _ ( 1999 ) and just @xmath29 0.1 mag fainter than the rgb tip determined by martinez - delgado _ et al . _ ( 1999 ) , both from ground - based photometry . we adopt an absolute magnitude of @xmath35 for the tip ( lee _ et al . _ 1993 ) and a reddening of @xmath36 ( burstein & heiles 1982 ) , giving e(v - i ) = 0.026 ( @xmath37 ) . this yields a distance modulus for phoenix based on the tip of the rgb of @xmath38 ; the error in this quantity is several percent based on uncertainties in the adopted extinction . if the true tip is brighter than our brightest star , the distance modulus would be smaller . our distance provides additional support for the derived trgb distances of held _ et al . _ ( @xmath39 ) and martinez - delgado _ et al . _ ( @xmath40 ) . a horizontal branch ( hb ) can be discerned at @xmath41 and @xmath42 which merges with the red giant branch at @xmath43 . there are well - defined blue ( bhb ) and red ( rhb ) components of the horizontal branch , separated by a gap which likely contains rr lyrae stars . at first glance the occurrence of a rhb in phoenix , which has a mean metallicity as low as [ fe / h ] @xmath44 ( _ i.e. _ , comparable to galactic globular clusters such as m13 and m3 ) , might indicate that phoenix is an example of a second - parameter stellar system . however , if there is a metallicity spread , at least some of these rhb stars may have evolved from the most metal - rich old red giants in phoenix . given that there is likely both an age and metallicity ( see below ) spread in phoenix , interpreting the morphology of the horizontal branch is a complex problem . nevertheless , the presence of a distinct blue hb clearly indicates that there is an ancient population of stars in phoenix , as noted by tolstoy _ et al . _ ( 2000 ) . the accurate wfpc2 photometry allows a measurement of the horizontal branch magnitude at @xmath45 . from ground - based photometry , et al . _ ( 1999 ) estimated the horizontal branch to be at @xmath46 . using the lee _ _ ( 1990 ) relation between the absolute visual hb magnitude and metallicity , our measurement gives a distance of @xmath47 , larger than the trgb method , but perhaps not in bad agreement given uncertainties in the hb calibration . a young main sequence locus extends brightward to @xmath48 ( with a few even brighter stars ) or @xmath49 . a similar blue plume with @xmath50 was found by martinez - delgado _ et al . _ ( 1999 ) at magnitudes brighter than @xmath51 ( which corresponds to the faint limit of their photometry at @xmath52 ) , and is evident in the region @xmath53 and @xmath54 of fig . 12 of held _ et al . _ ( 1999 ) . in these ground - based cmds the main sequence population does not appear to be as prominent as seen in the wfpc2 cmd ; this may be an effect of larger photometric errors smearing out the ground - based observations of the main sequence and increasing incompleteness for bluer stars in the ground - based observations . the accuracy of the wfpc2 photometry shows that the main sequence is well populated , and hence that the young star contribution to the total population of phoenix is significant , as will be shown quantitatively below . figure [ fig : iso ] shows the padova isochrones for two ages ( 100 myr and 3.2 gyr ) and metallicities of @xmath30 and 0.004 . the extension of the main sequence up to @xmath55 suggests that stars as young as 100 myr exist in phoenix , in good agreement with previous works ( martinez - delgado _ et al . _ 1999 et al . _ main sequence turnoff stars this bright would have masses of @xmath56 @xmath57 . several points are apparent from figure [ fig : iso ] . first , there are many stars which fall significantly redward of any of the zams isochrones . the observed width ( in color ) of the upper main sequence is significantly larger than expected from observational errors ; from artificial star tests we find that magnitudes for the bulk of the stars are measured to an accuracy of a few percent down to @xmath58 . the color spread of main sequence stars almost certainly implies that stars with a range of ages exist in phoenix ; it is highly unlikely that the color spread results entirely from metallicity spread , since very metal - rich stars would be required . such a metal - rich population would produce giants significantly redder than any that are observed . stars which fall redward of the zams are stars which are evolving off of the main sequence . despite the accurate photometry , there are no pronounced main sequence turnoffs in the cmd between ages of 100 myr and 4 gyr , suggesting that the young population in phoenix is _ not _ the result of one or several distinct bursts of star formation , but rather is the result of a more continuous star formation history over the past several gyr . this is consistent with the analysis of martinez - delgado _ et al . _ ( 1999 ) , who estimate that the recent ( @xmath59 gyr ) star formation rate is comparable to the average star formation rate for the age interval 1 - 15 gyr . it appears that the @xmath30 isochrone ( [ fe / h ] = @xmath60 , _ i.e. _ the metallicity inferred for the giants ) falls blueward of the main body of the main sequence , suggesting that the younger stars in phoenix have significantly higher metallicities than the older population . this would imply that phoenix has undergone chemical enrichment over its lifetime . this suggestion is supported by the giant branch colors as well ; if all of the stars in phoenix had [ fe / h ] @xmath29 -1.7 , the younger giants would be bluer than any of the observed giants . a clear red clump of stars exists near the intersection of the horizontal branch with the red giant branch . these are intermediate age ( several gyr ) core helium burning stars , and confirm the existence of an intermediate age population suggested by the width of the main sequence . a sequence of red helium - burning stars ( with ages of around 0.4 - 0.8 gyr ) extends vertically upwards from the red end of the horizontal branch to an @xmath61 magnitude comparable to the tip of the red giant branch . this sequence is located about 0.2 mag blueward of the @xmath30 rgb isochrones . these are younger stars in the reddest stage of the so - called `` blue loops '' of later stellar evolution . there is a hint of a sequence corresponding to the blue end of the blue loops ; this sequence slopes towards the main sequence and although it has only a few stars , it is defined by the observation that there are gaps around it in the cmd , suggesting that these stars are not foreground stars . the exact details of the ages of these stars is difficult to ascertain without knowledge of their metallicities . intermediate - age ( 1 - 10 gyr ) asymptotic giant branch stars are seen in cmds of dwarf irregular galaxies such as ic 1613 ( cole _ et al . _ 1999 ) , sextans a ( dohm - palmer _ et al . _ 1997 ) , and ngc 6822 ( gallart _ et al . _ 1996 ) , where they extend redward of @xmath62 from the region of the tip of the first ascent giant branch . there are few such stars apparent in the phoenix cmd of figure [ fig : cmd ] . however , held _ et al . _ ( 1999 ) found a significant number of such stars in their ground - based cmd and they concluded on this basis that intermediate - age stars constitute a substantial component of phoenix ; additional evidence for the presence of agb stars is discussed by martinez - delgado _ et al . _ ( 1999 ) . given that stars of many different ages are present in phoenix , it is of interest to know the relative number of stars at different ages , or alternatively , the history of the star formation rate . to some extent , the answer to this question depends on where one looks in phoenix , since it is apparent from larger field studies ( held _ et al . _ 1999 , martinez - delgado _ et al . _ 1999 ) that the star formation history varies across the galaxy , with younger stars being more centrally concentrated . here we consider somewhat more quantitatively the approximate star formation history that is required to match the populations seen in the central region of phoenix imaged by hst / wfpc2 . we can also try to quantitatively address the relative numbers of stars of different metallicities . there are many complications to extracting star formation histories from the distribution of stars in a cmd , and several groups have developed different methods for doing so ( _ e.g. _ , tolstoy & saha 1996 ; dolphin 1997 ; hernandez , valls - gabaud , & gilmore 1999 ; ng 1998 ; gallart _ et al . _ 1999 ; holtzman _ et al . _ 1999 ) . in the current paper , we analyze our cmd based on the methods of holtzman _ et al . _ ( 1999 ) , in which the number of stars in many regions of the cmd are fit using least - squares techniques to a combination of basis functions of stellar populations of different ages and metallicities . the best fit was derived using the @xmath63 estimator of mighell ( 1999 ) . the interpretation of derived star formation histories , and in particular an understanding of the uncertainties in the results , is a complex issue . detailed comparisons of results obtained by using different methods have not yet been performed , and the sensitivity of the results to systematic errors , _ e.g. _ , in the input stellar models or assumed initial mass function ( imf ) , is difficult to assess . consequently , the results presented here should be viewed as preliminary ; we attempt to assess uncertainties by showing results not only for different parameters , but also using different subsets of the observed data to derive star formation histories . for the current purpose , we derive a star formation history assuming an imf with the salpeter ( 1955 ) slope . since we do not sample the lower main sequence of completely unevolved stars , independent constraints on the imf slope are difficult . we also assume that unresolved binaries are not important in the observed cmd of phoenix , which will be the case regardless of the true binary fraction in the galaxy if the masses of binary stars are drawn independently from the same initial mass function . this is because the most likely companion masses in this case are significantly lower than the masses of the stars which are observed and thus do not strongly affect the system luminosity or color . the issue of whether masses in binary star pairs are uncorrelated or not is still the matter of significant debate . we derive star formation histories for different age bins , where the width of the age bins increases logarithmically with lookback time . this is necessitated by the fact that it becomes increasingly difficult to resolve ( in a temporal sense ) star formation events as one goes to older ages , since isochrones become more and more similar for older ages . as a result , the derived star formation histories can not provide information about the variation of the star formation rate on time scales shorter than the width of the age bins , roughly 25% of the age of any given population . within each age bin , the star formation rate is assumed to be constant . star formation histories were extracted from the phoenix cmd for five different choices of distance modulus : @xmath64 , 23.0 , 23.1 , 23.2 , and 23.3 . using @xmath65 , the distance as derived from our trgb , gave the best fits ( lowest @xmath66 ) , although they were only marginally better than those using @xmath40 . for the reasons discussed above , it was not possible to find good fits to the cmd using an exclusively metal - poor population , or for that matter , using any star formation history in which the metallicity does not change over time . this is because the giant branch requires a metal - poor population , while the main sequence requires a population of higher metallicity . as a result , we imposed no constraints on an age - metallicity relation to fit the phoenix population ; the constituent basis stellar populations cover all combinations of age and metallicity , although only seven discrete metallicities ( corresponding to those for which we have isochrones from the padova group ) were used . results for our best - fitting model are shown in figure [ fig : sfh ] . the upper left panel gives the best fitting star formation rate as a function of lookback time . since we only cover a small region of the galaxy , this star formation history is given in a relative sense only ; the total number of stars formed is normalized to sum to unity . the quality of the fit can be judged by looking at the residual hess diagram shown in the lower right . this panel shows the difference between the number of observed and the number of stars in the best - fitting model , in units of the expected @xmath67 errors from counting statistics ; the diagram is scaled so that the range of black to white covers from @xmath68 to @xmath69 , where @xmath7 is defined based on the suggestions made by mighell ( 1999 ) . other panels show the derived cumulative age and metallicity distribution functions , and a comparison of the observed and model luminosity functions . in general , results are more uncertain for the youngest populations ( @xmath70 gyr ) , since the stars that are uniquely contributed by these populations ( upper main sequence stars ) comprise a relatively small number of the stars in the color - magnitude diagram . the lower left panel gives a representation of the `` population box '' ( hodge 1989 ) for the central regions of phoenix . this combines information about the derived formation rate as a function of age and metallicity . remarkably , without imposing any constraints on the age - metallicity relation , we recover a rough relation in which metallicity increases with time ; a similar result also emerged from a similar analysis of several fields in the lmc ( holtzman _ et al . _ this lends quantitative support to our suggestion that the metallicity in phoenix has evolved over time ; the derived metallicity distribution is shown in the upper right panel . the population box may also highlight some problems with the models as well ; the observed color - magnitude diagram appears to require a small population of young metal - poor stars . the presence of this feature comes from the existence of a few relatively blue upper main sequence stars , so it is difficult to know for sure how seriously to consider them ; accurate photometry over a larger field is required . a comprehensive understanding of the uncertainties in the results presented in figure [ fig : sfh ] is difficult . the formal errors on the derived star formation rates are relatively small , usually 5 - 10% for most of the age bins ( larger for the youngest ages ) . however , the true uncertainties are probably significantly larger and systematic , depending on the choice of constituent basis stellar populations , distance , reddening , imf , stellar models , the use of discrete metallicities , and the details of how the best fit is defined . our experience with allowing these to vary suggests that while the star formation rate in individual age bins can vary significantly ( factor of two ) , the variations are correlated such that changes in derived star formation rate for one stellar population are generally accompanied by nearly opposite changes in other similiar ( _ i.e. _ , in age or metallicity ) populations . as a result , the cumulative age distribution tends to be relatively robust against systematic errors . this is demonstrated in figure [ fig : sfhcum ] , which shows cumulative age distributions for a variety of models . the solid curves all use the same distance modulus , but are determined using different subsets of the observed data : results were derived using two different faint magnitude cutoffs ( @xmath71 and @xmath72 ) for the region of the cmd which was fit and using both the complete data set as well as using randomly selected subsamples of half the observed stars . the dotted curves show the systematic changes resulting from using different distance moduli , although the extremes of these choices ( @xmath73 and 23.3 ) provide noticeably poorer fits . our overall conclusion is that the cmd of phoenix requires star formation to have been ongoing at a roughly constant rate over a hubble time . the fits suggest that the bulk of the stars even in the central region of phoenix are older than @xmath29 5 gyr . the details of fluctuations in the overall star formation rate are significantly more uncertain . although the fit in figure [ fig : sfh ] suggests some fluctuations in the star formation rate , we feel that the actual evidence for strong variations is weak , although they are certainly not ruled out . different choices of population parameters ( _ e.g. _ , distance ) or what regions of the cmd are fit can change the timing and amplitude of these fluctuations . as a result , it is less clear as to what the current results show about whether the star formation rate in phoenix is episodic . however , it is clear from an inspection of the cmd that there are no distinct main sequence turnoffs as can be seen , for example , in the cmds of the carina dwarf spheroidal ( smecker - hane _ et al . _ 1996 ) . one of the main reasons for uncertainties in the details of the star formation history comes from the age - metallicity degeneracy of the location of stars around the turnoff in a cmd ; different combinations of age and metallicity can place stars at the same location in the cmd . without the presence of the unevolved lower main sequence in the current data or independent measurement of metallicities , it becomes difficult to accurately separate these different populations . the situation would be improved with significantly deeper photometry and/or with the measurement of independent metallicities for individual stars in phoenix . deep hst / wfpc2 imaging has provided a color - magnitude diagram ( cmd ) of the central regions of the phoenix dwarf galaxy at unprecedented accuracy . the presence of stars at all ages is apparent in this diagram . a distinct horizontal branch and the presence of a red giant branch which extends all the way down to a turnoff expected for an old population indicates that this system has a substantial component of metal - poor stars with metallicities comparable to the mean of the galactic halo ; phoenix appears to have an underlying component of ancient population ii stars . in this respect it is similar to dwarf spheroidal galaxies like ursa minor , draco , and sculptor , as previously concluded by held _ et al . _ ( 1999 ) , martinez - delgado _ et al . _ ( 1999 ) , and tolstoy _ et al . _ ( 2000 ) . however , the existence of red clump stars and main sequence stars confirm that phoenix has experienced considerable star formation over the past few gyr , with star formation possibly as recent as 100 myr ago . the location of the main sequence suggests that the metallicity in phoenix has evolved with time ; old blue horizontal branch stars and the giant branch stars require low metallicities , comparable to those in the halo of the milky way , while the younger main sequence stars suggest somewhat higher metallicities . fits to the observed cmd suggest that star formation has been roughly continuous over the lifetime of phoenix . there is no obvious evidence for strongly episodic star formation , although the formal fits for the star formation history suggest that a mildly varying star formation rate can certainly fit the data . one caveat to this statement is that it applies only over relatively long ( _ e.g. _ , gyr ) timescales ; burstiness on very short timescales would be essentially impossible to distinguish from the cmd for all but the most recent past . another caveat is that the details of the derived star formation rate are suspect due to a variety of reasons including the age - metallicity degeneracy ; independent measurements of metallicities of stars in phoenix should improve the situation considerably . the implications are that a dwarf galaxy can have star formation which extends over nearly a hubble time . with roughly continuous star formation , phoenix appears to resemble the magellanic clouds ( _ e.g. _ , holtzman _ et al . _ 1999 ) . however , phoenix is a significantly less massive system in which it might be expected that the loss of interstellar gas and interruptions in star formation would be much more important . in fact , the apparent star formation history of phoenix stands in contrast to the strongly episodic star formation of carina ( smecker - hane _ et al . _ 1994 , 1996 , hurley - keller _ et al . _ 1998 ) , a dwarf galaxy that is only 0.8 mag fainter than phoenix ( mateo 1998 ) . it appears that the mechanisms governing star formation in dwarf galaxies may be rather complex . the hst / wfpc2 data on phoenix add support to the growing evidence in the literature that dwarf galaxies , both dirrs and dsphs , can have very varied star formation histories , particularly in recent epochs . by contrast , it may be possible that there is at least some commonality between dwarf galaxies in regard to their early chemical evolution . as noted in section 3.1 , phoenix appears to follow the same integrated - magnitude versus metallicity relation as dwarf spheroidal galaxies in the local group , some of which , like draco and ursa minor , do not appear to have supported recent star formation such as that of phoenix . nonetheless the degree of chemical enrichment in these systems , as indicated by the color of the old giant branch , would appear to be correlated with the overall ( luminous ) mass of the galaxy . this suggests that the amount of metal enrichment during the earliest phases of star formation within dwarf galaxies ( perhaps more than @xmath74 gyr ago ) does vary in a way that depends to some extent on basic galaxy properties such as mass . in classical closed box models of chemical evolution the average metallicity of all stars formed up to a given time is proportional to the fraction by mass of the galaxy that has turned into stars ( _ e.g. _ , smith 1985 ) . within the context of such a model , the observations indicate that the fraction of a galaxy which is turned into old metal - poor stars , and the attendant early chemical evolution history , is governed by similar mechanisms in both dsph and dirr galaxies . support for this work was provided by nasa through grant number go-0698.02 - 95a from the space telescope science institute , which is operated by the association of universities for research in astronomy , incorporated , under nasa contract nas5 - 26555 . burstein , d. & heiles , c. 1982 , aj , 87 , 1165 caldwell , n. , armandroff , t. e. , seitzer , p. , & da costa , g. s. 1992 , aj , 103 , 840 canterna , r. , & flower , p. j. 1977 , apj , 212 , l57 carignan , c. , demers , s. , & cote , s. 1991 , apj 381 , l13 cole , a. a. , _ et al . _ 1999 , aj , 118 , 1657 da costa , g. s. , & armandroff , t. e. 1990 , aj , 100 , 162 dohm - palmer , r. c. , skillman , e. d. , saha , a. , tolstoy , e. , mateo , m. , gallagher , j. , hoessel , j. , chiosi , c. , & dufour , r. j. 1997 , aj , 114 , 2514 dolphin , a. 1997 , newa , 2 , 397 dolphin , a. 2000 , pasp , in press gallart , c. , aparicio , a. , bertelli , g. , & chiosi , c. 1996 , aj , 112 , 1950 gallart , c. , freedman , w. l. , aparicio , a. , bertelli , g. , & chiosi , c. 1999 , aj , 118 , 2245 girardi , l. , bressan , a. , bertelli , g. , & chiosi , c. 2000 , a&as 141 , 371 girardi , l. , bressan , a. , chiosi , c. , bertelli , g. , & nasi , e. 1996 , a&as 117 , 113 held , e. v. , saviane , i. , & momany , y. 1999 , a&a , 345 , 747 hernandez , x. , valls - gabaud , d. , & gilmore , g. 1999 , mnras , 304 , 705 hodge , p. 1989 , ara&a , 27 , 139 holtzman , j. a. , burrows , c. j. , casertano , s. , hester , j.j . , trauger , j. t. , watson , a. m. , worthey , g. , & the wfpc2 idt 1995 , pasp , 107 , 1065 holtzman , j. a. , mould , j. r. , gallagher , j.s . , watson , a. m. , grillmair , c. j. , & the wfpc2 idt 1995 , aj , 113 , 656 holtzman , j. a. , gallagher , j.s . , cole , a.a . , mould , j.r . , grillmair , c.j . , & the wfpc2 idt 1999 , aj , 118 , 2262 hurley - keller , d. , mateo , m. , & nemec , j. , 1998 , aj , 115 , 1840 lee , y.w . , demarque , p. , & zinn , r. 1990 , apj 350 , 155 lee , m. g. , freedman , w. l. , & madore , b. f. 1993 , apj , 417 , 553 martinez - delgado , d. , gallart , c. , & aparicio , a. 1999 , aj , 118 , 862 mateo , m. 1998 , ara&a , 36 , 435 mighell , k.j . , 1999 , apj , 518 , 380 ng , y. k. 1998 , a&as , 132 , 133 ortolani , s. , & gratton , r. g. 1988 , pasp , 100 , 1405 pritchett , c. j. , & van den bergh , s. 1999 , aj , 118 , 883 salpeter , e. e. 1955 , apj , 121 , 161 schuster , h. e. , & west , r. m. 1976 , a&a , 49 , 129 smecker - hane , t. a. , stetson , p. b. , hesser , j. e. , & lehnert , m. d. 1994 , aj , 108 , 507 smecker - hane , t. a. , stetson , p. b. , hesser , j. e. , & vandenberg , d. a. 1996 , in from stars to galaxies : the impact of stellar physics on galaxy evolution , asp conf . 98 , eds . c. leitherer , u. fritze - von - alvensleben , & j. huchra ( san francisco : asp ) , p. 328 smith , g. h. 1985 , pasp , 97 , 1058 stetson , p.b . 1987 , pasp 99 , 191 st - germain , j. , carignan , c. , cote , s. , & oosterloo , t. 1999 , aj , 118 , 1235 tolstoy , e. , gallagher , j. , greggio , l. , tosi , m. , de marchi , g. , romaniello , m. , minniti , d. , & zijlstra , a. 2000 , the messenger , 99 , 16 tolstoy , e. , & saha , a. 1996 , apj , 462 , 672 van de rydt , f. , demers , s. , & kunkel , w. e. 1991 , aj , 102 , 130 young , l. , & lo , k.y . 1997 , apj 490 , 710 c c c + + dataset&filter&exposure time ( s ) + u48i0201r&f814w&100 + u48i0202m&f814w&1300 + u48i0203r&f814w&1000 + u48i0204r&f555w&100 + u48i0205r&f555w&1300 + u48i0206r&f555w&1100 + u48i0207r&f555w&100 + u48i0208r&f555w&1300 + u48i0209r&f555w&1000 + u48i020ar&f814w&100 + u48i020br&f814w&1300 + u48i020cr&f814w&1100 +

we present hst / wfpc2 photometry of the central regions of the phoenix dwarf . accurate photometry allows us to : 1 ) confirm the existence of the horizontal branch previously detected by ground - based observations , and use it to determine a distance to phoenix , 2 ) clearly detect the existence of multiple ages in the stellar population of phoenix , 3 ) determine a mean metallicity of the old red giant branch stars in phoenix , and suggest that phoenix has evolved chemically over its lifetime , 4 ) extract a rough star formation history for the central regions which suggests that phoenix has been forming stars roughly continuously over its entire lifetime .

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Proceed to summarize the following text: in these lines i intend to show that there exists consistent evidence pointing to the need of revision and further study of what seem at present a settled issue , namely the independence of the speed of electromagnetic radiation on the motion of its source . the main point in the evidence is the range disagreement during the earth flyby of the spacecraft near in 1998 . its range was measured near the point of closest approach using two radar stations of the space surveillance network ( ssn ) , and compared with the trajectory obtained from the deep space network ( dsn)@xcite . as for the range , the two measurements should match within a meter - level accuracy ( the resolution is 5 m for millstone and 25 m for altair ) , but actual data showed a difference that varies linearly with time ( with different slopes for the two radar stations ) up to a maximum difference of about 1 km , i.e. more than 100 times larger than the accuracy of the equipment used ( see figure 10 of @xcite ) . further , when near crossed the orbits of gps satellites ( orbital radius 26,600 km ) the measured range difference was 650 m , that is , a time difference of 2 @xmath0s . is it reasonable that any standard gps receiver performs better than dsn or ssn ? there has not been a complete explanation for the range discrepancy . it is very difficult to find any physical reason that may produce this anomaly , for any physical disturbance of the path of the spacecraft should manifest equally in ssn and dsn measurements . guruprasad@xcite proposed an explanation that points to a time lag in the dsn signals , but the model is , at best , within 10% of the measured data and , more important , it fails to explain an important feature , that is , the different slope for the two radars . if we assume that systems are working properly , then the measured range difference ( time lag ) could be due to different propagation time of the employed signals . additional points in the evidence come from anomalies related to the tracking of spacecrafts , present in both doppler and ranging data . the pioneer anomaly@xcite and the flyby anomaly@xcite refer to small residuals of the differences between measured and modeled doppler frequencies of the radio signals emitted by the spacecrafts . although these residuals are very small ( less than 1 hz on ghz signals ) the problem is that they follow a non - random pattern , indicating failures of the model . according to the temporal variation of those residuals the pioneer anomaly exhibits a main term , an annual term , a diurnal term and a term that appears during planetary encounters . it should be clarified that a few years ago an explanation of the pioneer anomaly was published@xcite . however , it is a very specific solution that applies only to the main term of the pioneer spacecraft anomaly , but left unresolved many other anomalies , including those of the spaceships cassini , ulysses and galileo ; the annual term ; the diurnal term ; the increases of the anomaly during planetary encounters ; the flyby anomaly ; and the possible link between all them ( it is hard to think that there are so many different causes for the mentioned anomalies ) . for all this , i believe that the issue can not be closed as it stands . as a matter of fact , the range difference between ssn and dsn , @xmath1 , is perfectly fitted with @xmath2 where @xmath3 is a vector range pointing from the spacecraft to the radar , @xmath4 the spacecraft velocity relative to the radar , and @xmath5 the speed of light . figure [ range ] shows this fit and its comparison with measured data . the orbital and measured data were taken from@xcite . although the exact location of the radar stations are unknown , the fit is statistically significant for both radar stations ( @xmath6 ) including the first outliers points . it reproduces the ( almost ) linear dependence with time during the measured interval , and the two different slopes for millstone and altair stations due to their different locations . since range measurements are based on time - of - flight techniques , the validity of ( [ deltar ] ) means that the electromagnetic waves ( microwave ) of the dsn and ssn travel at different speeds . specifically , in the radar frame of reference , if the ssn waves travel at @xmath5 , then the dsn waves travel at @xmath5 plus the projection of the spacecraft velocity in the direction of the beam , in sharp contrast with the second postulate of special relativity theory ( srt ) . in view of the above result one may ask what is established , at present , about the relation of the speed of electromagnetic radiation ( light for short ) to the motion of the source . in order to elaborate this point the following questions are of relevance : 1 . are there _ simultaneous _ measurements of the speed of light from different moving macroscopic sources ( not moving images ) with different velocities ? ; 2 . since ballistic ( emission ) theories are ruled out ( see , for example , desitter @xcite@xcite , brecher @xcite and alvager et al @xcite ) , how else could the speed of light depend on the source movement ? how is it possible that there is a first order difference in @xmath7 in spacecraft range measurements , while at the same time there are many experiments on time dilation that are consistent with srt to second order in @xmath7 ( see , for example,@xcite ) ? ; 4 . if the velocity of light depend on the velocity of the source , why has this not been observed in other phenomena in the past ? ; in answer to the previous questions , so far as the author is aware , there is no known experimental work that simultaneously measures the speed of light from two different sources ( not images ) , or that simultaneously measures the speed of light and that of its source . for example , in the work by alvager et al,@xcite the speed of light is measured at a later time ( @xmath8 200 ns ) than the emission time , and there is no measurement of the speed of the source at the time of the _ detection _ of the light . note that measurements involving moving images produce different results from those produced by mobile sources . for example , under srt , a moving source is affected by time dilation while a moving image is not . therefore , to ensure the independence of the speed of light from its source movement , it is essential to have two sources with different movements . although controversial and beyond the scope of the this note , time dilatation phenomena may be of different physical origin from first order terms , as it may be inferred from the work of schrdinger @xcite . thus , measurements of time dilatation phenomena in accordance with srt , does not necessarily imply the independence of the speed of light with the movement of the source . the experiments mentioned above @xcite@xcite@xcite @xcite only rule out ballistic theories in which radiation maintains the speed of the source at the time of _ emission _ , but do not rule out other ideas , like faraday s 1846@xcite . in order to remove the ether , faraday introduced the concept of vibrating rays@xcite , in which an electric charge is conceived as a center of force with attached rays that extend to infinity . the rays move with their center , but without rotating . according to this view , the phenomenon of electromagnetic radiation corresponds to the vibration of these rays , that propagates at speed @xmath5 relative to the rays ( and the center ) . that is , the radiation remains linked to the source even after emitted . today we could describe the interaction as a kind of entanglement between the charge and the photon . a framework for the electromagnetic phenomena according to faraday s ideas was developed . it was called vibrating rays theory ( vrt ) @xcite in reference to faraday s vibrating rays . under faraday s idea , the velocity of radiation at a given epoch will be equal to _ c _ plus the velocity of the source at the _ same _ epoch , in contrast with ballistic theories in which the emitted light retains the speed of the source at the _ emission _ epoch . in this sense the radiation is always linked to the charge at every time after the emission . consequently , the measured doppler effect corresponds to the speed of the source at the time of _ reception _ , as well . further , a difference between active and passive reflection is expected , since the latter is still related to the original source according to vrt . the deep space network ( dsn ) works with the so called active reflection ( the spacecraft re - emits in real time a signal in phase with the received signal from earth ) , while the space surveillance network ( ssn ) works with passive radar reflection . in consequence , the downlink signal from the aproaching spacecraft will propagate faster that the reflected one . using the available orbital data@xcite we found that , under vrt , the theoretical time - of - flight difference between active and passive reflection gives exactly the same range disagreement as ( [ deltar ] ) , see part 6 of @xcite . the pioneer anomaly refers to the fact that the received doppler frequency differs from the modeled one by a blue shift that varies almost linearly with time , and whose derivative is @xmath9 where @xmath10 is the frequency difference between the measured and the modeled values . in the case of a source with variable speed , the main difference in doppler ( to first order ) between vrt and srt , is that srt relates to the speed of the source at the time of _ emission _ , while vrt relates to the speed of the source at the time of _ reception_. precisely , this difference seems to be present in the spacecraft anomalies . if vrt is valid , it automatically invalidates all calculations and data analysis of spacecraft tracking which are based on srt . so , it is not easy to make a direct comparison between the expected results from srt and vrt . however , to see whether or not the main features predicted by vrt are present in the measurements , we can evaluate the residual by simulating a measured doppler signal assuming that light propagates in accordance to vrt but analyzed according to srt . calling @xmath11 the emission time of the downlink signal from the spacecraft toward earth and @xmath12 the reception time at earth , the first order difference of the doppler shift between vrt and srt is ( see @xcite part 4 ) @xmath13 where @xmath14 and @xmath15 represent the velocities of the spacecraft at the corresponding epoch , @xmath16 is the unit vector from the spacecraft to the antenna , and @xmath17 the proper frequency of the signal . that is , the velocity used in the srt formula is that at the time of _ emission _ while according to vrt is that corresponding at the time of _ reception_. since the spacecraft slows down as it moves away , then @xmath18 , therefore the difference corresponds to a small blue shift mounted over the large red shift , as it has been observed in the pioneer anomaly . it should be noted that this difference appears because of the active reflection produced by the onboard transmitter . in case of a passive reflection ( for example , by means of a mirror ) the above difference vanishes . an estimate of the order of magnitude of [ df ] is obtained by using that the variation of the velocity of the spacecraft between the time of emission and reception is approximately @xmath19 where @xmath20 is a mean acceleration during the downlink interval . an estimate for the duration of the downlink is simply @xmath21 where @xmath22 is a mean position of the spacecraft between @xmath11 and @xmath12 , therefore @xmath23 since @xmath24 where @xmath25 is the gravitational constant , and @xmath26 the mass of the sun , then , the time derivative becomes @xmath27 if the difference ( [ ddfdtvrt ] ) is interpreted as an anomalous acceleration we get @xmath28 that is , the so - called anomalous acceleration is @xmath7 times the actual acceleration of the spacecraft . using data from horizons web - interface @xcite for the spacecraft ephemeris , some characteristic value for @xmath29 can be obtained . consider the anomalous acceleration detected at the shortest distance of the cassini spacecraft during solar conjunction in june , 2002 . the spacecraft was at a distance of @xmath30 au moving at a speed of @xmath31 km / s . the anomalous acceleration given by ( [ aa ] ) is @xmath32 m / s@xmath33 of the same order of the measured one ( @xmath34 m / s@xmath33 ) . also , the closest distance at which the pioneer anomaly has been detected was about @xmath35 au . the anomalous acceleration predicted by ( [ aa ] ) at that distance is @xmath36 m / s@xmath33 of the same order as the measured one . the anomaly given by ( [ aa ] ) decreases in time in a way that has not been observed . note , however , that according to markwardt @xcite the expected frequency at the receiver includes an additional doppler effect caused by small effective path length changes , given by @xmath37 where @xmath38 is the rate of change of effective photon trajectory path length along the line of sight . this is a first order effect that can partially hide the difference between srt and vrt . therefore , a more careful analysis should take into account the additional contribution of ( dfpath ) in ( [ aa ] ) . further , other first order effects may appear , for example , by a slight rotation of the orbital plane . since orbital parameters are obtained by periodically fitting the measurements ( mainly due to spacecraft maneuvers or random perturbations ) with theoretical orbits , there is no straightforward way to weight the importance of this effect in ( [ aa ] ) . in other words , data acquisition and analysis may hide part of the vrt signature . apart from the residual referred to in the preceding paragraph there is also an annual term . according to anderson et al @xcite the problem is due to modeling errors of the parameters that determine the spacecraft orientation with respect to the reference system . anyway , levy et al @xcite claim that errors such as errors in the earth ephemeris , the orientation of the earth spin axis or the stations coordinates are strongly constrained by other observational methods and it seems difficult to modify them sufficiently to explain the periodic anomaly . the advantage of studying the annual term over the main term , is that the former is less sensitive to the first order correction mentioned above , and , for the case of pioneer , also to the thermal propulsion correction@xcite . clearly , the earth orbital position does not modify those terms . as before , the annual term is explained by the difference between the velocity of the spacecraft at the time of emission and that at the moment of detection , which depends on whether the spacecraft is in opposition or in conjunction relative to the sun . when the spacecraft is in conjunction , light takes longer to get back to earth than in opposition . the time difference between emission and reception will be increased by the time the light takes in crossing the earth orbit . specifically , taking into account the delay due to the position of earth in its orbit , in opposition equation ( [ t3t2 ] ) should be written as @xmath39 while in conjunction it would be @xmath40 where @xmath41 is the mean orbital radius of earth . therefore , an estimate of the magnitude of the amplitude of the annual term is @xmath42 for the case of pioneer 10 at 40 au we get @xmath43 and at 69 au @xmath44 in good agreement with the observed values . using data from horizons web - interface @xcite a more complete analysis of the time variation of @xmath10 has be performed . the residual ( that is , simulated doppler using vrt but interpreted under srt ) during 12 years time span is plotted in figure [ anual ] . also the dumped sine best fit of the 50 days average measured by turyshev et al @xcite is plotted showing an excellent agreement between measurements and vrt prediction . the negative peaks ( i.e. , maximum anomalous acceleration ) occur during conjunction when the earth is further apart from the spacecraft , and positive peaks during opposition . also , the amplitude is larger at the beginning of the plotted interval and decreases with time , as it was observed@xcite@xcite . like the pioneer anomaly , the earth flyby anomaly can be associated to a modeling problem , in the sense that relativistic doppler includes terms that are absent in the measured signals . the empirical equation of the flyby anomaly is given by anderson et al@xcite , which , notably , can be derived using vrt , as is done in part 6 of @xcite . consider the case of near tracked by 3 antennas located in usa , spain , and australia ( a full description of the tracking system is found in a series of monographs of the jet propulsion laboratory@xcite ) . the receiving antenna was chosen as that having a minimum angle between the spacecraft and the local zenith . using available orbital data , a simulated doppler signal has been calculated using vrt . thus , the simulated residual is obtained by subtracting the theoretical srt doppler , from the vrt calculation . we observed , however , that the term that contains the velocity of the antennas , that is @xmath45 is not enough to completely remove the first order ( in @xmath46 ) earth signature ( @xmath47 is the velocity of the antenna , 1 refers to the emission epoch and 3 to the reception epoch , as in @xcite part 4 ) . this is so because the velocity of the antennas is not uniform and the evaluation of the emission time is different for vrt and srt . then , a small , first order related term remains . anyway , since orbital parameters are obtained by periodically fitting the measurements to theoretical orbits , thus a similar procedure is needed for vrt . by doing so the first order term is removed . the only difference between orbits adjusted by srt and vrt is a slight rotation of the orbit plane , as mentioned above . note that , in the case of range disagreement , two different orbital ajustment are needed by dsn and ssn due to the different propagation speed . in consequence , it will be impossible to fit a simultaneous measurement , as it seems to happen with the range disagreement . the final result shows that each antenna produces a sinusoidal residual with a phase shift at the moment of maximum approach . therefore , if we fit the data with the pre - encounter sinusoid a post - encounter residual remains and vice versa . in figure [ flyby ] are simultaneously plotted the result of fitting the residual by pre - encounter data ( right half in red , corresponding to figure 2a of @xcite ) and by post - encounter data ( left half in blue , corresponding to figure 2b of @xcite ) . note that the simulated plots are remarkably similar to the reported ones , including the amplitude and phase ( i.e. , minima and maxima ) of the corresponding antenna . the fitting of post - encounter data ( blue ) can be improved by appropriately setting the exact switching times of the antennas ( which are unknown to the author ) . the flyby doppler residual exhibits a clean signature of the vrt theory . in this work i have presented observational evidence favoring a dependence of the speed of light on that of the source , in the manner implied in faraday s ideas of vibrating rays . it is remarkable and very suggestive that , as derived from faraday s thoughts , simply by relating the velocity of light and the corresponding doppler effect with the velocity of the source at the time of detection , is enough to quantitatively and qualitatively explain a variety of spacecraft anomalies . also , it is worth mentioning that a formulation of electromagnetism compatible with faraday s conception is possible , as shown in @xcite , which is also compatible with the known electromagnetic phenomena . this new formalism introduces a striking concept : that both instantaneous interaction ( static terms ) and retarded interaction ( radiative terms ) are simultaneously present . finally , under vrt the manifestation of the movement of the source in the speed of light is more subtle than the naive @xmath48 hypothesis ( @xmath49 is a constant , @xmath50 ) usually used to test their dependence@xcite . thus , it is also of fundamental importance the fact that , from the experimental point of view , it is very difficult to detect differences between vrt and srt , as discussed in @xcite , which is also manifest in the smallness of the measured anomalies , and in the non clear manifestation of the effect in usual experiments and observations . for example , it produces a negligible effect on satellite positioning systems , see part 7 of@xcite . i am aware of how counterintuitive these conceptions are to the modern scientist , but also believe that , given the above evidence , a conscientious experimental research is needed to settle the question of the dependence of the speed of light on that of its source as predicted by vrt , and that has been observed during the 1998 near flyby . as a closure , i recall fox s words regarding the possibility of conducting an experiment on the propagation of light relative to the motion of the source : `` _ nevertheless if one balances the overwhelming odds against such an experiment yielding anything new against the overwhelming importance of the point to be tested , he may conclude that the experiment should be performed . i am thankful to fernando minotti who read this paper and improved the manuscript significantly , although he may not agree with all of the interpretations provided in this paper . g. antreasian and j. r. guinn , aiaa paper no . 98 - 4287 presented at the aiaa / aas astrodynamics specialist conference and exhibit ( boston , august 10 - 12 , 1998 ) . v. guruprasad , _ epl _ * 110 * , 54001 ( 2015 ) . j. d. anderson , p. a. laing , e. l. lau , a. s. liu , m. m. nieto , and s. g. turyshev , _ phys . lett . _ * 81 * , 2858 - 2861 ( 1998 ) . j. d. anderson , j. k. campbell , j. e. ekelund , j. ellis , and j. f. jordan , _ phys . lett . _ * 100 * , 091102 ( 2008 ) . s. g. turyshev , v. t. toth , g. kinsella , s .- c . lee , s. m. lok , and j. ellis , _ phys . rev . lett_. * 108 * , 241101 ( 2012 ) w. desitter , _ z. phys . _ * 14 * , 429 ( 1913 ) . w. desitter , _ z. phys . _ * 14 * , 1267 ( 1913 ) . k. brecher , _ phys . * 39 * , 1051 ( 1977 ) . t. alvager , f. j. m. farley , j. kjellman , and i. wallin , _ phys . * 12 * , 260 ( 1964 ) . b. botermann et al , _ phys . lett . _ * 113 * , 120405 ( 2014 ) . e. schrodinger ; _ ann . der physik _ * 77 * , 325 - 336 ( 1925 ) . m. faraday , _ phil . mag . _ * 28 * , 345 ( 1846 ) . l. bilbao , l. bernal , f. minotti ; `` vibrating rays theory , '' arxiv:1407.5001 [ physics.class-ph ] ( 2014 ) . http://ssd.jpl.nasa.gov/. c. b. markwardt , `` independent confirmation of the pioneer 10 anomalous acceleration , '' arxiv : gr - qc/0208046 v1 ( 2002 ) . j. d. anderson , p. a. laing , e. l. lau , a. s. liu , m. m. nieto , and s. g. turyshev , _ phys . d _ * 65 * , 082004 ( 2002 ) . a. levy , b. christophe , s. reynaud , j - m . courty , p. brio , and g. mtris , `` pioneer 10 data analysis : investigation on periodic anomalies , '' in journes scientifiques de la sf2a , paris , france pp.133 - 136 , 2008 . hal-00417743 g. turyshev et al , arxiv : gr - qc/9903024 v2 ( 1999 ) . descansoteam , jet propulsion laboratory , california institute of technology . ( accessed july 2014 ) . j. g. fox , _ am . _ * 30 * , 297 ( 1962 ) . ) is plotted ( full lines , millstone in blue and altair in red ) . for millstone , the error bars refer to the uncertainties in the extraction of the data from figure 10 of @xcite , rather than to its tracking error ( 5 m ) , while for altair , the accuracy is 25 m ]

data from spacecrafts tracking exhibit many anomalies that suggest the dependence of the speed of electromagnetic radiation with the motion of its source . this dependence is different from that predicted from emission theories that long ago have been demonstrated to be wrong . by relating the velocity of light and the corresponding doppler effect with the velocity of the source at the time of detection , instead of the time of emission , it is possible to explain quantitatively and qualitatively the spacecraft anomalies . also , a formulation of electromagnetism compatible with this conception is possible ( and also compatible with the known electromagnetic phenomena ) . under this theory the influence of the velocity of the source in the speed of light is somewhat subtle in many practical situations and probably went unnoticed in other phenomena .

You are an expert at summarizing long articles. Proceed to summarize the following text: many workers argue that a thermodynamic anomaly underlies the onset of glassy dynamics @xcite . evidence for this view is the rough , though not quantitative @xcite , correlation between dynamic fragility and excess heat capacity discontinuity @xcite . this thermodynamic view contrasts with the picture we have advocated @xcite , attributing glassy behavior to dynamic heterogeneity @xcite , with growing length scales appearing in space - time , but not space alone @xcite . indeed , direct observations of diffusive motion in colloidal glasses reveal excitations that are local and sparse @xcite . these findings seem consistent with excitations being local and uncorrelated , as assumed in the two - state model for the low temperature behavior of structural glass heat capacities @xcite . spatial correlations under such conditions can arise through constraints on particle motions that are relieved only when adjacent regions have exhibited some degree of mobility @xcite . trajectories are then correlated throughout space and time , with varying degrees of hierarchical structure determining the extent to which the system is fragile or strong @xcite . from this perspective , a non - thermodynamic explanation emerges for correlation between heat capacity and fragility @xcite : the concentration of excitations required to support fragile hierarchical dynamics is higher than that required for strong non - hierarchical dynamics . the juxtaposition of heat capacity discontinuities is therefore understood as a consequence of different excitation concentrations . biroli , bouchaud and tarjus ( bb&t ) @xcite have taken issue with this explanation . they show that the simplest treatment of defect models can not simultaneously fit the size and temperature dependence of the experimental heat capacity while simultaneously fitting dynamic properties . in general and in more current context , however , the terminology `` defect models '' refers to a broad class of kinetically constrained models @xcite . as a result of this generality , their question `` are defect models consistent with the entropy and specific heat of glass - formers ? '' has a non - trivial answer ( see also the analysis in @xcite ) . in particular , this class of models has an assortment of possible dynamical behaviors , so that relaxation data can be fit with many different functions of excitation concentration . in addition , kinetically constrained models are coarse grained caricatures of fluids , so that many degrees of freedom remain unspecified . here , we focus on this latter feature , the possible consequences of unspecified degrees of freedom . the starting point is the assumption that small length - scale and small time - scale features of a fluid can be integrated out leaving only simple stochastic rules for the dynamics of discrete variables on a lattice . these rules contain constraints imagined to be the consequence of the actual intermolecular interactions , interactions that limit the space or metric for molecular motions @xcite . the excitations or defects that survive coarse graining distinguish microscopic regions of space - time that exhibit molecular mobility from those where molecules are jammed or immobile . in particular , an excitation or defect is a microscopic region of space for which particle mobility emerges . this characterization is related to coarse graining in time , because it takes time to discern whether or not mobility is exhibited . bear in mind , `` defect '' does not distinguish disordered arrangements of atoms from those that are ordered because there are generally many disordered yet jammed configurations . similarly , a lack of `` excitation '' does not necessarily imply low energy because some immobile regions may have the same energy as mobile regions . uncertainty concerning this terminology may be the origin of criticisms of facilitated models leveled by bb&t @xcite and also by lubchenko and wolynes @xcite . as a simple illustration , consider dividing space into cells , with grid spacing larger than the equilibrium correlation length of the material . at a given time frame , the micro states of different cells are then uncorrelated . further , suppose there are two energy levels , @xmath0 and @xmath1 , for the states of a given cell , where the states with energy @xmath1 include those that will exhibit particle mobility after a coarse graining time @xmath2 . there is a fraction , @xmath3 , of states with energy @xmath1 that exhibit mobility in this way ; the others remain jammed . in this case the average excitation concentration is @xmath3 times the concentration of cells with energy @xmath1 , and thus the number of molecules contributing to energy or enthalpy fluctuations is @xmath4 times larger than that contributing to mobility fluctuations . in other words , when considering the thermodynamic implications of a kinetically constrained defect model , a factor @xmath4 is required to account for energetic states or degrees of freedom absent from explicit consideration . bb&t ignore this factor , which leads them to the difficulties they describe . in the next section we discuss this point in greater detail . we show that facilitated or kinetically constrained models such as the one we presented in ref . @xcite admit a variety of possibilities for partitioning micro states . this demonstration justifies the heat capacity formulas we used in ref . @xcite and therefore supports our argument about the juxtaposition of heat capacities for strong and fragile glass formers . the dynamics of facilitated models we consider are governed by master equations for distribution functions of the mobility field on a lattice . consistent with our earlier papers , we use the symbol @xmath5 to denote the value of the mobility field at lattice site @xmath6 during time period @xmath7 . it is a binary field in that @xmath8 takes on one of two values , @xmath0 ( corresponding to a cell that is unexcited , jammed ) or @xmath9 ( corresponding to excited , mobile ) . each cell is imagined to contain several molecules , so we expect that even after some coarse graining , there are still many ( not just two ) microstates possible for each cell at a given time @xmath7 . we further imagine these many microstates , specified with @xmath10 , evolve according to their own master equation . the master equation for the distribution of the mobility field @xmath11 is a contraction of that for the micro - state field @xmath10 . in this section , we show how this contraction works . we begin by considering the thermodynamics and statistics of the fields @xmath11 and @xmath10 , and then consider the master equations for the distributions of these fields . imagine we partition the liquid into a lattice with grid spacing @xmath12 , and denote the state in cell @xmath6 by @xmath10 . imagine further that the set of available cell levels @xmath13 can be split into two groups : one subset of @xmath13 is `` unexcited '' with respect to mobility , @xmath14 , and the complementary subset is `` excited '' , @xmath15 . in practice , @xmath11 is most conveniently determined by observing system for a coarse graining time @xmath2 . this splitting , or projection , of many micro states into two different mobility states ( or several , as in the generalization of @xcite ) is the the first central assumption of facilitated models @xcite . while the state @xmath16 of a cell will correspond to one of the two mobility states , @xmath17 , in general its energy , @xmath18 , will be distributed . ( for notational simplicity , when clarity is not diminished , we drop the subscripts on @xmath8 and @xmath19 . ) let @xmath20 and @xmath21 represent the normalized probability densities for the energy levels in states @xmath22 and @xmath23 , respectively . the cell partition function then reads ( we set @xmath24 ) : @xmath25 where @xmath26 and @xmath27 gives the total number of levels in each of the two mobility states . the excitation concentration is given by : @xmath28 the average energy per cell is : @xmath29 where @xmath30 the specific heat per cell is : @xmath31 where @xmath32 the quantity @xmath33 determines the concentration @xmath34 of excitations ; @xmath35 gives the average cell energy ; and @xmath36 sets the specific heat at small @xmath34 . except in the case where @xmath37 are delta functions , these scales are in principle all different . as a simple example , which will be useful below , consider the following model with just two energy levels . the unexcited states @xmath22 occupy both energy levels , which we denote @xmath38 and @xmath39 , with energies @xmath40 and @xmath41 , respectively : @xmath42 where @xmath43 sets the relative degeneracy between the two levels . the excited states @xmath23 have a single energy level , which we denote @xmath44 , with energy @xmath45 : @xmath46 in this case , the cell partition function , average energy and specific heat become : @xmath47 where @xmath48 denotes the relative occupation of the states @xmath39 with respect to states @xmath38 , @xmath49 $ ] . the total concentration of levels with energy @xmath1 is @xmath50 where we have defined @xmath3 as the ratio of mobile cells to the total number of cells with energy @xmath1 . the specific heat then reads @xmath51 we now show how an appropriate projection of the dynamics of the full states @xmath19 of individual cells into their mobility states @xmath52 gives a two - state facilitated model such as the fa or east models , and the slightly more general class of models introduced in ref.@xcite . the approach we take has been used to derive approximate real - space renormalization group equations @xcite . here , however , the approach requires no approximation . the master equation for the dynamics can be written @xcite : @xmath53 where @xmath54 is the state vector for the probability density of the system at time @xmath7 , and @xmath55 is the liouvillian operator for the dynamics . in the direct product representation it reads @xcite : @xmath56 where @xmath57 is the state vector for level @xmath19 in site @xmath6 . the second assumption in facilitated models is that transitions in a cell @xmath6 are only possible if neighbouring cells @xmath58 are mobile , @xmath59 @xcite . in a system with facilitated dynamics @xmath55 reads @xcite : @xmath60 where @xmath61 is the kinetic constraint operator imposing facilitation , and @xmath62 is the unconstrained dynamic operator at site @xmath6 . for example , in the east facilitated model @xmath63 , and in the direct product representation : @xmath64 the off diagonal elements of @xmath62 are given by the transition rates between levels : @xmath65 , while the diagonal elements are such that the sum of columns in @xmath62 vanish . a choice satisfying detailed balance is @xmath66 , where @xmath67 is the equilibrium distribution . notice that we have assumed that transitions between any of the levels in the cell are only possible if facilitated by a neighbour . in order to project the dynamics of the full set of levels @xmath68 into that of only the excitation occupancies @xmath69 we need projection , @xmath70 , and embedding , @xmath71 , operators @xcite : @xmath72 where the @xmath73 and @xmath74 s are constants . the projection is performed independently in each cell , so these are site diagonal operators . they must also satisfy @xmath75 . the dynamics of the projected state @xmath76 is then given by the projected liouvillian @xmath77 where the elements of the projection and embedding matrices ( [ t1t2 ] ) are given by : @xmath78 the choice of rates in @xmath55 , which depend only on the final level , @xmath79 , ensures that the projected dynamics is markovian @xcite . we now show explicitly how the projection procedure works in the simple the three - level example of the previous subsection . in this case @xmath80 , and the onsite operators can be represented by @xmath81 matrices : @xmath82 of the three levels @xmath68 , only @xmath83 corresponds to the excited state , and the projection matrix is : @xmath84 the corresponding embedding matrix is : @xmath85 the projected operators then read : @xmath86 these are precisely the operators for a two state facilitated model @xcite . using the facilitated models of ref . @xcite and compared to the formula used by biroli , bouchaud and tarjus ( indicated as bb&t ) @xcite . * top panel * : excess specific heat , @xmath87 , of supercooled otp ( left hand scale ) and 3bp ( right hand scale ) . symbols indicate experimental data @xcite . full curves are fits using the formulas for the heat capacity per molecule discussed in the text ( where @xmath88 ) : @xmath89 , where @xmath90 and @xmath91 . here g \exp \left ( -j / t + j / t_{\mathrm{ref } } \right)$ ] , where the first ( approximate ) equality neglects terms of order @xmath93 , and the second introduces the notation used in our ref . @xcite : @xmath38 is a degeneracy that we imagine measures the possible directions of mobility , and @xmath94 bounds from above the temperatures we consider . the values of @xmath95 and @xmath96 for otp and 3bp were determined in @xcite from kinetic data . on physical grounds we expect @xmath97 , and @xmath3 to increase with temperature . the fitting of @xmath87 gives @xmath98 and @xmath99 for otp and 3bp , respectively . @xmath100 at @xmath101 as argued in @xcite . clearly , it is possible to describe the size and modest change with @xmath102 of the specific heat in the vicinity of @xmath101 , in contrast to the observation of bb&t . * bottom panel * : excess entropy , @xmath103 , of otp ( left hand scale ) and 3bp ( right hand scale ) . symbols correspond to experimental data @xcite . full curves are fits using the formula for the entropy per molecule which follows from eq . ( [ c1 ] ) : @xmath104 } - ( \phi^{-1 } c ) \ln{(\phi^{-1 } c ) } - ( 1- \phi^{-1 } c ) \ln{(1-\phi^{-1 } c ) } + ( \phi^{-1 } c ) \ln{[(\omega_0 \alpha + \omega_1)/(1-\alpha ) \omega_0 ] } \}$ ] . the shown fits correspond to the same parameters as before plus @xmath105 } , s^{-1 } \ln{[(\omega_0 \alpha + \omega_1)/(1-\alpha ) \omega_0]}=2.8 , 1.2 $ ] and @xmath106 for otp and 3bp , respectively.,title="fig:",width=325 ] using the facilitated models of ref . @xcite and compared to the formula used by biroli , bouchaud and tarjus ( indicated as bb&t ) @xcite . * top panel * : excess specific heat , @xmath87 , of supercooled otp ( left hand scale ) and 3bp ( right hand scale ) . symbols indicate experimental data @xcite . full curves are fits using the formulas for the heat capacity per molecule discussed in the text ( where @xmath88 ) : @xmath89 , where @xmath90 and @xmath91 . here g \exp \left ( -j / t + j / t_{\mathrm{ref } } \right)$ ] , where the first ( approximate ) equality neglects terms of order @xmath93 , and the second introduces the notation used in our ref . @xcite : @xmath38 is a degeneracy that we imagine measures the possible directions of mobility , and @xmath94 bounds from above the temperatures we consider . the values of @xmath95 and @xmath96 for otp and 3bp were determined in @xcite from kinetic data . on physical grounds we expect @xmath97 , and @xmath3 to increase with temperature . the fitting of @xmath87 gives @xmath98 and @xmath99 for otp and 3bp , respectively . @xmath100 at @xmath101 as argued in @xcite . clearly , it is possible to describe the size and modest change with @xmath102 of the specific heat in the vicinity of @xmath101 , in contrast to the observation of bb&t . * bottom panel * : excess entropy , @xmath103 , of otp ( left hand scale ) and 3bp ( right hand scale ) . symbols correspond to experimental data @xcite . full curves are fits using the formula for the entropy per molecule which follows from eq . ( [ c1 ] ) : @xmath104 } - ( \phi^{-1 } c ) \ln{(\phi^{-1 } c ) } - ( 1- \phi^{-1 } c ) \ln{(1-\phi^{-1 } c ) } + ( \phi^{-1 } c ) \ln{[(\omega_0 \alpha + \omega_1)/(1-\alpha ) \omega_0 ] } \}$ ] . the shown fits correspond to the same parameters as before plus @xmath105 } , s^{-1 } \ln{[(\omega_0 \alpha + \omega_1)/(1-\alpha ) \omega_0]}=2.8 , 1.2 $ ] and @xmath106 for otp and 3bp , respectively.,title="fig:",width=302 ] subsection [ thermo ] tells us that if the mobility field is the result of coarse - graining and projecting the many mobile and immobile micro states , then different energy ( or enthalpy ) scales enter into the concentration of mobile cells @xmath34 , the average energy ( or enthalpy ) , and the specific heat . how these scales relate to each other depends on the distribution of energy levels . at equilibrium , the average excitation concentration , @xmath34 , is a function of thermodynamic state . its value should be indicative of the concentration of regions with excess energy or enthalpy , but to say more requires more assumptions or detail than needed to predict dynamical trends . in order to illustrate how facilitated models can be used to fit thermodynamic data the simple example given above of two immobile cell levels @xmath38 and @xmath39 and a single mobile level @xmath44 will suffice . from eqs . ( [ z3state])([cv3state ] ) we see that while @xmath34 gives the concentration of cells displaying mobility , both mobile and immobile cells can contribute to energy ( or enthalpy ) fluctuations . the relevant quantity to determine the specific heat is not @xmath34 but @xmath107 . from eqs . ( [ defphi ] ) and ( [ cv3state ] ) we have that the specific heat _ per cell _ is : @xmath108 the asymptotic equality of eq . ( [ c1 ] ) is , in effect , the expression we refer to in the penultimate paragraph of ref.@xcite . the factor @xmath109 accounts for the number of molecules contributing to energy or enthalpy fluctuations , this number being larger than that for molecules contributing to mobility fluctuations . this fact is implicit in the coarse graining procedure , as shown above . in ref . @xcite , @xmath4 is included in the quantity we called @xmath110 . in the few lines devoted to this topic in ref . @xcite , the meaning of @xmath110 was obscure and its description misleading . indeed , where we wrote `` @xmath110 is the number of molecules that contribute to enthalpy fluctuations per mobile cell '' , we should have written `` @xmath110 accounts for the number of molecules that contribute to enthalpy fluctuations beyond those associated with mobile cells . '' to be perfectly explicit , if @xmath111 is the average number of molecules in a cell , then @xmath112 . with this expression , the relative values of @xmath113 for several super - cooled liquids at @xmath101 are consistent with the relative values of their heat capacity jumps at the glass transition . this consistency is independent of the value of @xmath111 and @xmath110 , but the value of @xmath110 does play a role in the actual value of the heat capacity discontinuity . specifically , @xmath114 is consistent with the value of the discontinuities of the heat capacity per molecule at @xmath115 when @xmath116 . bb&t @xcite identify @xmath111 with @xmath110 , and further neglect the factor @xmath117 , and in so doing disagree with this consistency . bb&t s second criticism is about the temperature variation of @xmath113 . they consider the relationship @xmath118 this formula is consistent with the asymptotic equality of eq.([c1 ] ) , which requires @xmath119 to be small . while it is small at the glass transition , @xmath120 may not be small at the higher temperatures considered . the right - hand side of bb&t s formula is missing the requisite factor of @xmath121 present in eq . ( [ c1 ] ) . it is also missing a possible temperature dependence of @xmath3 . this temperature dependence would reflect that of transport in a liquid . it is typically more pronounced for isobaric temperature variation than for isochoric temperature variation , but weak and sub - arrhenius in either case @xcite . thus , we write @xmath122 , where the parameters @xmath123 and @xmath124 are temperature independent . in ref . @xcite , we chose to not consider @xmath113 above @xmath101 because doing so requires the additional parameters described in the previous paragraph . but since , bb&t raise this point , we show in fig.1 that the effects of these extra factors on the temperature dependence of @xmath113 is significant . this figure can be contrasted with that in bb&t s @xcite . while the asymptotic formula predicts an exponential variation with temperature , the full formula predicts modest variation with temperature that is not inconsistent with experiment . we also show fits to the corresponding excess entropy , @xmath125 . better agreement between theory and experiment for these properties would require modification of the models and perhaps further parameterization . in the us , this work was supported initially by the nsf , and more recently by doe grant no . de - fe - fg03 - 87er13793 . in the uk , it was supported by epsrc grants no . gr / r83712/01 and gr / s54074/01 , and university of nottingham grant no . fef 3024 . see , for example , d. kivelson , s.a . kivelson , x .- zhao , z. nussinov , and g. tarjus , physica a * 219 * , 27 ( 1995 ) ; m. mzard and g. parisi , phys . 82 * , 747 ( 1999 ) ; x. xia and p.g . wolynes , proc . usa * 97 * , 2990 ( 2000 ) ; s. franz and g. parisi , j. phys . c * 12 * , 6335 ( 2000 ) . for reviews see : h. sillescu , j. non - cryst . solids * 243 * , 81 ( 1999 ) ; m.d . ediger , annu . phys . chem . * 51 * , 99 ( 2000 ) ; s.c . glotzer , j. non - cryst . solids , * 274 * , 342 ( 2000 ) ; r. richert , j. phys . condens . matter * 14 * , r703 ( 2002 ) . l. berthier and j.p . garrahan , phys . e * 68 * , 041201 ( 2003 ) ; s. whitelam , l. berthier and j.p . garrahan , phys . 92 * , 185705 ( 2004 ) ; y.j . jung , j.p . garrahan and d. chandler , phys . e * 69 * , 061205 ( 2004 ) ; l. berthier , d. chandler and j.p . garrahan , europhys . lett . * 69 * , 230 ( 2005 ) .

in recent papers , we have argued that kinetically constrained coarse grained models can be applied to understand dynamic properties of glass forming materials , and we have used this approach in various applications that appear to validate this view . in one such paper [ j.p . garrahan and d. chandler , proc . nat . acad . sci . usa * 100 * , 9710 ( 2003 ) ] , among other things we argued that this approach also explains why the heat capacity discontinuity at the glass transition is generally larger for fragile materials than for strong materials . in the preceding article , biroli , bouchaud and tarjus ( bb&t ) have objected to our explanation on this point , arguing that the class of models we apply is inconsistent with both the absolute size and temperature dependence of the experimental specific heat . their argument , however , neglects parameters associated with the coarse graining . accounting for these parameters , we show here that our treatment of dynamics is not inconsistent with heat capacity discontinuities .

You are an expert at summarizing long articles. Proceed to summarize the following text: almost a decade elapsed since the first detection of the anisotropy of the cosmic microwave background at large angular scales ( @xmath2 ) @xcite , @xcite . today the cmb anisotropy ( cmba ) has been detected also at intermediate ( @xmath3 ) and small angular scales ( @xmath4 ) , so the cmba angular spectrum is now reasonably known down to the region of the first and second doppler peaks @xcite , @xcite , @xcite . its shape gives information e.g. on the spectrum of the primordial cosmological perturbations or can be used to test the inflation theory but rises new questions to which cmba can not answers . it is however possible to get answers looking at the cmb polarization ( cmbp ) . for instance one can use cmbp to disentangle the effects of fundamental cosmological parameters like density of matter , density of dark energy etc . this is among the goals of space and ground based experiments like @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and is the main goal of sport a polarization dedicated asi / esa space mission on the international space station @xcite . the relevance of the cmb polarization was remarked for the first time by m. rees @xcite . since his paper many models of the expected features of the cmbp have been published ( see for instance @xcite , @xcite , @xcite ) . they immediately stimulated the search for cmbp , but the first detection has been obtained only a few months ago ( @xcite and @xcite ) . because of its importance this discovery must be confirmed by new observations made with different systems and using different methods of extraction of the cmbp from the sky signal . the detection of the cmbp is in fact extremely difficult because the signal is at least an order of magnitude below the cmba level . moreover polarized foregrounds of galactic origin may cover the cmbp and/or mimic cmbp spots by their inhomogeneities . the signal to noise ( cmbp @xmath5 polarized foreground ) ratio is therefore @xmath6 . in this paper we discuss a method which improves this ratio and allows to disentangle cmbp and polarized foregrounds . in the microwave range the galactic foregrounds include : * synchrotron radiation ( strong polarization ) , * free - free emission ( null or negligible polarization ) , * dust radiation ( polarization possible ) beccause here we are interested in polarization , in the following we will neglect free - free emission.the effects of dust , if present , ( e.g. @xcite , @xcite ) , will be added to the synchrotron effects . in fact , as it will appears in the following , what matters in our analysis are the statistical properties of the spatial distribution of the foregrounds and , by good fortune , the spatial distribution of the dust polarized emission is similar to that of the synchrotron emission . behind both types of radiation there is in fact the same driving force , the galactic magnetic field which alignes dust grains and guides the radiating electrons ( @xcite and references therein ) . for the anisotropy the separation of foregrounds and cmb was successfully solved by the dmr / cobe team when they discovered cmba @xcite . the separation of cmbp and foregrounds is more demanding and recognizing true cmb spots among foreground inhomogneities severe . approaches commonly used ( e.g. @xcite , @xcite , @xcite , @xcite , @xcite , + @xcite ) , are based on the differences between the frequency spectra of foregrounds and cmb , therefore require multifrequency observations . in this paper we suggest a different method . it takes advantage of the fact that , as we will show in the following , at small angular scales the values of the parameters used to describes the polarization of the diffuse radiation at a given frequency in different directions fluctuate . we propose of analyzing the angular distribution of the polarized radiation on single frequency maps of the diffuse radiation and disentangling the main components , polarized synchrotron and cmbp , looking to their different statistical properties . this method was proposed and briefly discussed in @xcite . here we present a more complete analysis . convenient quantities commonly used to describe the polarization status of radiation are the stokes parameters ( see for instance + @xcite , @xcite , + @xcite ) . let s assume a monochromatic plane wave of intensity i and amplitude @xmath7 . in the _ observer plane _ , orthogonal to the direction of propagation of the electromagnetic wave , we can choose a pair of orthogonal axes @xmath8 and @xmath9 . on that plane the amplitude vector of an unpolarized wave moves in a random way . on the contrary it describes a figure , the _ polarization ellipse _ , when the wave is polarized . projecting the wave amplitude on @xmath8 and @xmath9 we get two orthogonal , linearly polarized , waves of intensity @xmath10 and @xmath11,(@xmath12 ) whose amplitudes are @xmath13 and @xmath14 respectively . if the original wave of intensity @xmath15 is polarized , @xmath10 and @xmath11 are correlated : lets call @xmath16 and @xmath17 their correlation products . by definition the stokes parameters are : @xmath18 , @xmath19 , @xmath20 , and @xmath21 . @xmath22 and @xmath23 describe the linear polarization , @xmath24 the circular polarization and @xmath15 the total intensity . the ratio @xmath25 gives the angle @xmath26 between the vector @xmath27 and the main axis of the polarization ellipse ( @xmath28 ) . rotation by an angle @xmath29 of the @xmath30 coordinate system give a new coordinate system @xmath31 in which the stokes parameters become @xmath32 so when @xmath33 the axes of the polarization ellipse coincide with the reference axes @xmath34 and @xmath35 ( see fig.[f2 ] ) . to analyze the properties of the cmb polarization it is sometimes convenient to use rotationally invariant quantities , like the radiation intensity @xmath15 and two combinations of @xmath23 and @xmath22 : @xmath36 and @xmath37 . the intensity @xmath15 can be decomposed into usual ( scalar ) spherical harmonics @xmath38 . @xmath39 the quantities @xmath40 can be decomposed into @xmath41 spin harmonics + @xcite , @xcite , @xcite + @xmath42 : @xmath43 the @xmath41 spin harmonics form a complete orthonormal system ( see , for instance , @xcite,@xcite , @xcite , @xcite ) and can be written @xcite , @xcite : @xmath44 where @xmath45 is a generalized jacobi polynomial , @xmath46 and : @xmath47 is a normalization factor . the harmonics amplitudes @xmath48 correspond to the fourier spectrum of the angular decomposition of rotationally invariant combinations of stokes parameters . because spin @xmath41 spherical functions form a complete orthonormal system : @xmath49 we can write @xmath50 following @xcite and the very nice introduction made more recently by @xcite we now introduce the so called _ @xmath51 ( electric ) _ and _ @xmath52 ( magnetic ) modes _ of these harmonic quantities : @xmath53 they have different parities . in fact when we transform the coordinate system @xmath54 into a new coordinate system @xmath55 , such that @xmath56 the e and b modes transform in a similar way : @xmath57 @xmath22 remains identical in both reference systems and @xmath23 changes sign . it is important to remark that @xmath58 and @xmath59 are uncorrelated . in terms of @xmath22 and @xmath23 we can write : @xmath60 therefore : @xmath61 here @xmath62 and @xmath63 designate values of delta correlated 2d stochastic fields @xmath22 and @xmath23 . omitting mathematical details , the correlation equations for @xmath22 and @xmath23 are : @xmath64 where @xmath65 is the dirac delta - function on the sphere . ( in the following we will sometimes omit indexes @xmath66 and @xmath67 ) . synchrotron radiation results from the helical motion of extremely relativistic electrons around the field lines of the galactic magnetic field ( see , for instance @xcite , @xcite , @xcite ) . the electron angular velocity @xmath68 is determined by the ratio between @xmath69 , the component of the magnetic field orthogonal to the particle velocity , and @xmath70 , the electron energy . as it moves around the magnetic field lines the electron radiates . a)_single electron _ until the electron velocity is small ( @xmath71 ) we speak of cyclotron radiation : the electron behaves as a rigid dipole which rotates with gyrofrequency ( [ gyr1 ] ) in a plane orthogonal to the magnetic field direction and emits a single line . the radiation has a dumbell spatial distribution ( see fig.[f3 ] ) : @xmath72 is circularly polarized along the dumbell axis ( @xmath73 ) and linearly polarized in directions orthogonal to it ( @xmath74 ) . when the electron velocity increases the radiation field changes until at @xmath75 , ( @xmath70 @xmath76 ) it assumes the peculiar charachters of synchrotron radiation : i)radiated power proportional to @xmath70 @xmath77 and @xmath78 , ii)continous frequency spectrum , peaked around : @xmath79 ( @xmath80 is the angle between the velocity vector @xmath81 and the wave vector @xmath82 , see fig.[f4 ] ) . the peak is so narrow that in a given direction the emission is practically monochromatic , iii)radiation almost entirely emitted in a narrow cone ) of the cyclotron dumbbell beam , seen by a fast moving observer ( @xmath75 ) becomes a cone folded around the direction of movement ] of aperture ( see fig.[f5 ] ) @xmath83 around the forward direction of the electron motion . inside the cone ( @xmath84 ) the frequency is maximum and equal to @xmath85 in the opposite direction ( @xmath86 ) intensity and frequency are sharply reduced . iv)radiation 100 @xmath87 linearly polarized at the surface of the cone.inside the cone the linear polarization is still dominant but a small fraction of circular polarization exists , ( @xmath88 and @xmath89 ) . outside the cone the very small fraction of radiation produced is elliptically polarized and becomes circularly polarized when seen along the @xmath69 direction . so the stokes parameters depend on @xmath90 , the angle @xmath91 between @xmath90 and the line of sight , and the dimensionless frequency @xmath92 , where @xmath93 is the so called _ critical frequency _ , @xcite,@xcite , itself function of @xmath90 ( see eq.[gyr1 ] ) . b)_cloud of monoenergetic electrons _ when the effects of many monoenergetic electrons with uniform distribution of pitch angles are combined , @xmath94 and @xmath23 are reinforced ( the stokes parameters are additive ) while @xmath24 is erased . in fact @xmath95 where @xmath96 and @xmath97 are constants , @xmath98 and @xmath99 almost monochromatic functions of @xmath100 , and @xmath26 the angle between the projection of the magnetic vector on the observer plane and an axis on that plane ( the projection of the magnetic vector on the observer plane is the minor axis of the polarization ellipse ) . if inside the emitting cloud the magnetic field varies , also @xmath26 varies , therefore @xmath101 and @xmath102 must be averaged along the line of sight across the cloud . in conclusion the degree of polarization @xmath103 varies between a maximum value , ( uniform magnetic field ) and zero ( magnetic field randomly distributed ) . c)_electrons with power law energy spectrum _ in the interstellar medium the radiating particles are the cosmic ray electrons whose energy spectrum : @xmath104 is a power law ( see for instance @xcite , @xcite ) and references therein ) with spectral index @xmath105 and space density proportional to @xmath106 . because the emission of a single electron is practically monochromatic the resulting radiation spectrum is a power law : @xmath107 with intensity spectral index @xmath108 , temperature spectral index @xmath109 and @xmath110 a slow function of @xmath111 @xcite . ( if the magnetic field is not uniform we use @xmath112 instead of @xmath69 and a slightly different function @xmath113 ) . the stokes parameters are products of the intensity @xmath114 , the degree of polarization @xmath103 and @xmath101 or @xmath102 , therefore we can write : @xmath115 where @xmath116 is a constant whose value depends on @xmath111 and the distribution of the magnetic field along the integration path . when the medium inside the synchrotron source , or in the medium where radiation propagates , is permeated by thermal electrons faraday rotation modifies the polarization charachteristics of the radiation . in fact by faraday effect , when the radiation crosses a region of thickness @xmath117 permeated by magnetic fields and thermal electrons with density @xmath118 , the angle of polarization of the radiation rotates by an angle ( see for instance @xcite ) @xmath119 inside the source this brings depolarization , because radiation produced at different points along @xmath117 suffers different rotations . additional depolarization inside the source comes about when the magnetic field along @xmath117 is not uniform because in eqs.([spec2])we have to use @xmath120 and @xmath121 instead of @xmath102 and @xmath101 . so the degree of polarization at the source @xmath103 varies between 0 ( random magnetic field distribution ) and : @xmath122 ( uniform magnetic field and faraday rotation absent ) . outside the source the angle of polarization rotates by farday effect so the stokes parameter of the synchrotron radiation @xmath123 and @xmath124 measaured by an earth observer are different from the stokes parameters at the source . a regular trend of the magnetic field inside and outside the source is insufficient to guarantee a regular trend of the spatial distribution of @xmath123 and @xmath124 . in fact as the line of sight moves among adjacent points on the sky @xmath118 , @xmath125 and @xmath117 fluctuate . so also the angle of faraday rotation @xmath126 fluctuates by a quantity @xmath127 ^ 2 ~+~ [ < n_e > \delta(l)]^2\}^{1/2 } ~\simeq~cost~\lambda^2~l~\delta(n_e ) \end{array}\ ] ] proportional to @xmath117 . because l is at least tens or hundred of pc @xmath128 can be very large . besides this effect produced by faraday rotation , when we look in different directions through the interstellar medium @xmath123 and @xmath124 may change also because @xmath90 , @xmath129 , @xmath111 , @xmath130 and @xmath106 vary ( see for instance @xcite , @xcite ) . is a nonzero mean , nonzero variance variable ; @xmath131 and @xmath132 are zero mean , nonzero variance variables ] we conclude that angle and degree of polarization of the synchrotron radiation fluctuates when the line of sight moves among adjacent regions on the sky , a conclusion supported by recent observations which show that the spatial distribution of the polarized component of the synchrotron radiation at intermediate and small angular scales is highly structured . on maps we see in fact lines ( _ channels _ ) along which @xmath103 goes to zero and sudden rotations of the plane of polarization when the line of sight goes from a side to the other of a channel @xcite . we expect therefore that at small and intermediate angular scales @xmath123 and @xmath124 are stochastic functions of the direction of observations so we can write : @xmath133 exceptions to this behaviour can be expected when along the line of sight there are peculiar regions characterized by special field configurations and/or by large densities of thermal electrons , like in the well known loops and spurs or in hii regions , where systematic effects overcome random effects . nature and origin of these features , clearly visible on brouw and spoelstra maps of the polarized component of the galactic background @xcite , are discussed for instance by @xcite . usually these regions are close to the observer and have large angular dimensions , so their effects can be removed if one evaluates the angular power spectrum of the radiation distribution and limits his analysis to angular scales below few degrees ( multipole order @xmath134 ) . being aware that this is a very important issue we tested its validity analyzing the distribution of the measured values of @xmath22 and @xmath23 on maps extracted from the effelsberg survey ( @xcite and @xcite ) of the diffuse radio emission . we found ( see appendix 2 ) that up to angular scales of 5 degrees ( l close to 36 ) the difference @xmath135 is fully consistent with zero . because of eq . ( [ main ] ) if @xmath136 also @xmath137 , so the synchrotron radiation has both electric and magnetic modes : @xmath138 and ( see eqs . ( [ syn1 ] ) and ( [ main ] ) ) : @xmath139 in a homogeneous and isotropic universe only temperature and intensity @xmath140 change as the universe expands : both decrease adiabatically . because this is true for @xmath10 and @xmath11 separately , we do not expect anisotropy nor polarization therefore @xmath141 and @xmath142 are natural consequences . on the contrary , inhomogeneities and perturbations of matter density or of gravitational field , induce anisotropy and polarization of the cmb . at the recombination epoch linear polarization appears as a by product of the thomson scattering of the cmb on the free electrons of the primordial plasma . the polarizarion tensor it gives can be calculated solving the boltzman transfer equations of the radiation in a nonstationary plasma permeated by a variable and inhomogeneous gravitational field @xcite , @xcite , @xcite , @xcite . the gravitational field is made of a background field , with homogeneous and isotropic frw metric , and an inhomogeneous and variable mix of waves : density fluctuations , velocity fluctuations , and gravitational waves . because of their transformation laws these waves are also said scalar , vector and tensor perturbations , respectively . scalar ( density ) perturbations affect the gravitational field , the density of matter and its velocity distribution . they were discovered studying the matter distribution in our universe on scales from @xmath143 mpc to @xmath144 mpc . it is firmly believed they are the seeds of the large scale structure of the universe and are reflected by the large scale cmb anisotropy detected for the first time at the beginning of the 90s @xcite , @xcite . their existence is predicted by the great majority of models of the early universe . observation shows that the effects of these perturbations are small , so we can treat them as small variations @xmath145 , @xmath146 , @xmath147 of @xmath10 , @xmath11 and @xmath148 . introducing the auxiliary functions @xmath149 and @xmath130 : @xmath150 ( @xmath91 is the angle between the line of sight and the wave vector ) for plane waves the boltzman equations ( see for instance @xcite , @xcite , @xcite and reference therein ) become : @xmath151 where @xmath98 is the gravitational force which drives both anisotropy and polarization , @xmath152 is the thomson cross - section , @xmath118 is the density of free electrons , and @xmath153 the scale factor . these equations give : @xmath154 where @xmath155 is the optical depth of the region where the phenomenon occurs . rotating the coordinate system we can generate a new pair of stokes parameters ( @xmath156 ) : no matter which is the system of reference we choose these parameters satisfy the symmetry parity condition . because there is always a system in which @xmath157 and @xmath158 , we may conclude that , in a system dominated by the primordial density perturbations ( see , for instance @xcite ) magnetic modes of the cmb polarization vanish and only electric modes exist @xcite . therefore : @xmath159 where index @xmath160 stays for _ density perturbation_. vector perturbations , associated to rotational effects , perturb only velocity and gravitational field . they are not predicted by the inflation theory and it is common believe that they do not contribute to the anisotropy and polarization of the cmb . gravitational waves ( tensor modes ) and gravitational lensing of large scale structures of the universe induce b modes in the distribution of cmbp(see for instance @xcite ) ) . gravitational lensing gives a power spectrum at least an order of magnitude below the power spectrum of the e mode polarized signal produced by scalar perturbations , with a maximum at @xmath161 . the power spectrum of the b mode polarization produced by gravity waves is maximun at @xmath162 , is definitely below the power spectrum produced by gravitational lensing for @xmath163 and only at very large angular scales ( @xmath164 ) it may be comparable to the scalar e modes . if one excludes very large angular scales we may conclude that cmbp is dominated by e - modes . b - modes are just a contamination by b - modes at levels of @xmath165 or less . the cmb radiation we receive is mixed with foregrounds of local origin . when the anisotropy of the cmb was detected , to remove the foregrounds from maps of the diffuse radiation , data were reorganized in the following way : @xmath166 where @xmath167 , is a two dimension vector ( map ) which gives the total signal measured at different points @xmath168 on the sky , @xmath169 the noise vector , @xmath170 the matrix which combines the components @xmath171 of the signal . at each point @xmath168 we can in fact write : @xmath172 where @xmath173 are weights , given by @xmath174 , @xmath175 is the synchrotron component , @xmath176 the free - free emission component , @xmath177 the dust contribution and so on . using just one map the signal components can not be disentangled . if however one has maps of the same region of sky made at different frequencies it is possible to write a system of equations . provided the number of maps and equations is sufficient , the system can be solved and the components of @xmath178 separated , breaking the degeneracy . we end up with a map of @xmath179 which can be used to estimate the cmb anisotropy . when we look for polarization at each point on the sky we measure tensors instead of scalar quantities , therefore to disentangle the polarized components of the cmb we need a greater number of equations . here we will concentrate on the separation of the two dominant components of the polarized diffuse radiation : galactic synchrotron ( plus dust ) foreground and cmbp . instead of observing the same region of sky at many frequencies , we suggest a different approach . it takes advantage of the differences between the statistical properties of the two most important components of the polarized diffuse radiation ( cmb ( background ) and synchrotron ( foreground ) radiation ) and does not require multifrequency maps . we define the estimator : @xmath181 where @xmath182 and @xmath183 ( here and in the following indexes @xmath184 or @xmath185 stay for _ synchrotron _ and cmb , respectively ) . because * @xmath51 and @xmath52 modes of synchrotron do not correlate each other neither correlate with the cmb modes + @xmath186 , + @xmath187 , * @xmath188 * @xmath189 where indexes @xmath160 and @xmath190 stay for @xmath191 ( or @xmath192 ) and @xmath193 perturbation , respectively . @xmath180 gives an estimate of the e - mode excess in maps of the polarized diffuse radiation . if tensor perturbations are negligible this excess is the cmbp signal . if tensor perturbations are important the excess is a lower limit with a systematic difference from the true value which in the worst condition ( maximum contribution to cmbp from gravitational waves and gravitational lensing ) reaches a maximum value of @xmath165 . let s now consider the angular power spectrum of @xmath180 . for multipole @xmath66 we can write : @xmath194 where @xmath195 and @xmath196 are random variables with gaussian distribution @xmath197 ( see eqs.([gauss1 ] ) and ( [ gauss2 ] ) in appendix a ) . according the ergodic theorem ( in the limit of infinite maps , the average over 2d space is equivalent to the average over realisations ) the average value of @xmath198 is equal to the difference of the average values of @xmath195 and @xmath196 summed over @xmath67 . taking into account equation ( [ gauss2 ] ) we can therefore write : @xmath199 where : ) is an explicit form of the average of the stochastic variables @xmath200 and @xmath201 over a probability density @xmath197 , the short form being triangle brackets ] @xmath202 comparing eq.([coco ] ) with the ordinary definition of multipole coefficients : @xmath203 we can write : @xmath204 for synchrotron radiation @xmath205 should be zero , non zero for density perturbations , but on real maps it is always different from zero . in fact a map is just a realization of a stochastic process and the amplitudes of @xmath195 and @xmath196 , averaged over 2d sphere , have uncertainties which add quadratically , so @xmath206 even in the case of synchrotron polarization . this effect , very similar to the well known _ cosmic variance _ of anisotropy @xcite , @xcite ( the real universe is just a realisation of a stochastic process , therefore there will be always a difference between the realization we measure and the expectation value ) does not vanish if observations are repeated . the variance of @xmath198 is @xmath207 the quantities @xmath58 and @xmath59 , being sums over @xmath67 of @xmath208 stochastic values with gaussian distribution , have a @xmath209 distribution with @xmath210 degrees of freedom , so their variance is @xmath211 more explicitly @xmath212 when the synchrotron foreground is dominant ( @xmath213 ) @xmath214 in agreement with ( [ var1 ] ) . the synchrotron foreground is a sort of _ system noise _ which hampers the detection of the _ signal _ , the cmb polarization . at frequencies sufficiently high ( above @xmath215 ghz , see next section ) the noise is small compared to the signal therefore direct detection of cmbp is possible . at low frequencies on the contrary the cmbp signal is buried in the noise created by synchrotron and dust emission . in this case to recognize the presence of the cmb polarization we can use our estimator @xmath180 . at angular scale @xmath66 , to be detectable the cmbp must satisfy the condition @xmath216 where @xmath217 are the coefficients of the multipole expansion of the @xmath51 modes and @xmath218 is the confindence level of the signal detectability . in a similar way we can write for our estimator : @xmath219 where @xmath220 if one neglects tensor perturbations ( @xmath221 ) the criterium for the cmbp detection becomes @xmath222 so by @xmath180 we get the cmbp level with an uncertainty @xmath223 which decreases as @xmath66 and the angular resolution increase . tensor perturbations , if present , add to this uncertainty a systematic uncertainty @xmath224 . the angular power spectra of the polarized component of the synchrotron radiation have been studied by @xcite , @xcite , @xcite , @xcite , @xcite and @xcite using the few partial maps of the polarized diffuse radiation available in literature . it appears that the power spectra of the degree of polarization @xmath103 and of electric and magnetic modes @xmath51 and @xmath52 follow power laws of @xmath66 up to @xmath225 . for the degree of polarization the spectral index is @xmath226 . for the @xmath51 and @xmath52 modes different authors get different values of the spectral index . according the authors of papers @xcite and @xcite , parkes data give : @xmath227 with @xmath228 and dependence of @xmath149 on the region of sky and the frequency . in paper @xcite , using effelsberg and parkes data , the authors get : @xmath229 with @xmath230 , @xmath231 and @xmath232 ( here we adjusted the original expression given in @xcite writing it in adimensional form ) . in paper @xcite the authors , using a completely different set of observational data ( @xcite and @xcite ) conclude that in the multipole range @xmath233 the spectral indexes of the @xmath51 and @xmath52 modes are @xmath234 and @xmath235 respectively , while for the polarized intensity the spectral index is @xmath236 . inside the multipole range @xmath237 the values of @xmath238 obtained by the three groups are marginally consistent and reasonably close to @xmath239 . at lower value of @xmath66 definite differences exist between the results obtained by @xcite and by @xcite : these differences must be understood , but they are not important here because our statistical method can not be used for small values of @xmath66 when the number of samples becomes insufficient to carry on statistical analyses . for high values of @xmath66 , if one excludes extrapolations and models ( eg . @xcite ) , there are no data in literature , but this is not a limitation because for large values of @xmath66 the results of our method are practically independent from the spectral index . all the above authors got their results analyzing low frequency data ( 1.4 , 2.4 and 2.7 ghz ) , therefore the extension of their spectra to tens of ghz , the region where cmb observations are usually made , depend on the accuracy of @xmath130 , the temperature spectral index of the galactic synchrotron radiation . a common choice is @xmath240 but in literature there are values of @xmath130 ranging between @xmath241 and @xmath242 . moreover @xmath130 depends on the frequency and the region of sky where measurements are made ( see @xcite , @xcite , @xcite , @xcite ) . last but not least many of the values of @xmath130 in literature have been obtained measuring the total ( polarized plus unpolarized ) galactic emission . in absence of faraday effect @xmath243 and @xmath244 , the spectral indexes of the total , polarized and unpolarized components of the galactic emission , should be identical ( see eqs . ( [ spec1]),([spec2 ] ) ) . however when faraday effect with its @xmath245 frequency dependence is present , we expect that , as frequency increases , the measured value of the degree of polarization ( se eq . ( [ degreep ] ) ) increases , up to @xmath246 therefore we should observe @xmath247 . the expected differences are however well inside the error bars of the data in literature so at present we can neglect them and assume @xmath248 . instead of extrapolating low frequency results it would be better to look for direct observations of the galactic emission and its polarized component at higher frequencies . unfortunately above 5 ghz observations of the galactic synchrotron spectrum and its distribution are rare and incomplete . at 33 ghz observations by @xcite give at some patches on the sky a galactic temperature of about @xmath249 from which follows that at the same frequency we can expect polarized foreground signals up to several @xmath250 . at 14.5 ghz observations made at ovro @xcite give synchrotron signals of 175 @xmath250 , equivalent to about 15 @xmath250 at 33 ghz of which up to 10 @xmath250 can be polarized . in conclusion there are large uncertainties on the frequency above which observations of the cmbp are practically unaffected by the polarized component of the galactic diffuse emission . to be on the safe side we can set it at 50 ghz ( see for instance @xcite , @xcite , @xcite , @xcite and @xcite ) . above 50 ghz cmbp definitely overcomes the polarized synchrotron foreground . below 50 ghz contamination by the galactic emission can be important but its evaluation , usually made by multifrequency observations , is dubious . this conclusion is supported by figure [ f9 ] where we plotted , versus the multipole order @xmath66 , the power spectra of cmbp and galactic synchrotron at 43 ghz . the cmb spectrum has been calculated by cmbfast @xcite assuming standard cosmological conditions ( cmb power spectrum normalized to the cobe data at low @xmath66 , @xmath251 , @xmath252 , @xmath253 , @xmath254 , @xmath255 km / sec / mpc , @xmath256 , @xmath257 , standard recombination ) . the synchrotron spectrum has been calculated assuming for @xmath258 and @xmath259 the scaling law ( [ synebb ] ) with @xmath260 ( most pessimistic case ) , @xmath261 and @xmath262 respectively . it appears that at 43 ghz the cmb power is comparable to the synchrotron power only at very small angular scales ( @xmath263 ) . similar calculations at other frequencies confirm that only above @xmath0 ghz and at small angular scales ( large values of @xmath66 ) the cmbp power spectrum overcomes the synchrotron spectrum . below @xmath0 ghz direct detection of the cmbp is almost impossible even at small angular scale . to overcome this limit we can use our estimator . to see how it improves the cmbp detectability we calculated the angular power spectrum of @xmath264 , the estimator we expect when the diffuse radiation is dominated by synchrotron radiation . from eqs.([multipole ] ) , ( [ dl21 ] ) and ( [ synebb ] ) we get : @xmath265 a quantity which can be directly compared with the power spectrum of cmbp . in fact eqs.([aed ] ) , ( [ d1 ] ) , ( [ equal ] ) and ( [ dl20 ] ) show that the power spectrum @xmath266 of the estimator evaluated when the sky is dominated by cmbp density perturbations coincides with the power spectrum of cmbp produced by density perturbations . figure [ fd ] shows at 37 ghz the power spectra at 37 ghz of : i)@xmath264 , the estimator for a sky dominated by galactic synchrotron ( solid line , calculated using eq.([synebb ] ) ) , ii)@xmath266 , the estimator for a sky dominated by cmbp . it coincides with the power spectrum of cmbp ( dotted line , calculated as in figure [ f9 ] with cmbfast using the same standard cosmological conditions ) , iii)the power spectrum of the polarized component of the galactic synchrotron radiation ( dashed line , calculated using eq.([synebb ] ) ) . as expected at 37 ghz the cmbp power is well below the synchrotron power , therefore direct observations of cmbp are impossible ( the maximum value of the cmbp power is about 2.5 times below the synchrotron power at the same @xmath66 ) . however above @xmath267 the power of the synchrotron estimator is definitely below the power of the cmbp estimator : at @xmath268 the ratio cmbp / d is maximum and close to @xmath269 . this confirms that the use of @xmath180 allows to recognize the cmbp also at frequencies well below 50 ghz . to further test the capability of our estimator we studied the separation of cmb and galactic synchrotron using measured instead of expected values of @xmath198.the measurements were simulated by random numbers , with gaussian distribution @xmath270 , zero mean and unity variance ( see eqs ( [ gauss1 ] ) - ( [ gauss2 ] ) ) . two series of @xmath208 random numbers gave representations of @xmath271 and @xmath272 respectively and from them we obtained @xmath198 ( see eq.([dest ] ) ) . then , to take into account that real data are collected with a finite angular resolution ( e.g. boomerang data come from regions whose angular extension is equivalent to @xmath273 @xcite , @xcite , @xcite ) we averaged the above values of @xmath198 on intervals @xmath274 : @xmath275 figure [ f6 ] , figure [ f7 ] and figure [ f8 ] show @xmath276 ( the sign of @xmath198 is arbitrary ) versus @xmath66 , for @xmath277 , @xmath278 and @xmath279 respectively : the very large fluctuations of the estimator are drastically reduced as soon as @xmath280 increases . figure [ f10 ] , figure [ f11 ] and figure [ f12 ] are similar to figure [ fd ] . here , instead of the expectation value , we plot simulated measurements of @xmath264 at 37 ghz , for @xmath277 ( figure [ f10 ] ) , @xmath281 ( figure [ f11 ] ) and @xmath279 ( figure [ f12 ] ) , respectively . once again the cmbp power spectrum comes from cmbfast assuming the same cosmological conditions we assumed above . the synchrotron power spectrum is obtained from eq.([synebb ] ) with @xmath282 ( most pessimistic condition ) . figure [ f13 ] shows at 17 ghz the same quantities we plotted in figure [ f12 ] . for a better appreciation of the differences among the three curves , on the vertical axis here we use a logarithmic scale . the power spectrum of the estimator now almost touches the two highest peaks of the cmbp spectrum . probably 17 ghz is the lowest frequency at which , in the most favorable conditions , one can use @xmath198 . in the most pessimistic case ( @xmath260 and @xmath283 ) the corresponding frequency is 25 ghz . we analyzed the impact of the real world experimental conditions on our method for disentangling cmbp and foregrounds in maps of the polarized diffuse radiation . a polarimeter is a two channel system which ( by hardware and/or software methods ) splits the sky signal in two polarized components , send them to separate channels where they are amplified and then looks for correlations between the two components . the system outputs are proportional to a pair of stokes parameters ( e.g. @xmath23 and @xmath22 ) , or to the electric and magnetic modes @xmath58 and @xmath59 or to a combination of them , e.g. our estimator @xmath180 . unfortunately as the signals propagate through the system , noise is added to them so at the system output the signal is mixed to noise . moreover if the channels are asymmetric or there are cross talks between them , the outputs contain additional signals which simulate spurious polarization , usually an offset of the system output from the level one should expect with an ideal system when polarization is absent ( for a discussion see for instance @xcite ) . so in the real world the signal we are looking for , @xmath284 , is accompanied by uncorrelated noise and offset . what one measures is @xmath285 where @xmath286 , @xmath287 is the cross talk coefficient , @xmath288 the noise signal and @xmath289 the noise standard deviation . in commercial systems @xmath290 while in dedicated cmbp experiments values of @xmath291 have been obtained @xcite . by accurate choice of the system components we can therefore minimize the offset and further reduce it by phase modulation techniques ( see e.g. @xcite ) . at this point we have noisy maps we can use to evaluate @xmath180 and its power spectrum instead of @xmath23 and @xmath22 or @xmath58 and @xmath59 . as shown by the above simulation the signal / noise ratio for @xmath180 is definitely better than the signal / noise ratio for the stokes parameters or @xmath58 or @xmath292 and brings it to values similar to the ones we find when studying weak radiosources or , in the worst situation , the cmb anisotropy . from here we can therefore go on and extract @xmath180 from the remaining noise using the well known methods of time integration , commonly used in radioastronomy . observations of the cmb polarization are hampered by the presence of a foreground , the polarized component of the galactic synchrotron radiation . only above @xmath293 the cosmic signal definitely overcomes the galactic synchrotron and direct measurements of the cmbp are possible . between @xmath294 ghz and @xmath215 ghz background and foreground are comparable . below @xmath295 ghz the polarized sky is dominated by the galactic signal . so when measurements are made at ground observatories where atmospheric absorption prevent observations above @xmath296 ghz , all programs for cmbp measurements must include observing and analysis strategies for disentangling the galactic synchrotron signals from the cmbp signals . a common approach is fitting models of the intensity , frequency dependence and spatial distribution of the cosmic and galactic signals to multifrequency maps of the polarized diffuse radiation . or , when observation are made at one frequency , looking for additional data in literature , but here the accuracy of the available data is insufficient to get firm evaluations of cmbp . in this paper we presented a different approach which at small angular scales ( @xmath297 ( @xmath298 ) ) and down to frequencies as low as @xmath299 ( @xmath300 in the most favorable conditions ) allows to extract the cmbp signal from single frequency maps of the polarized diffuse radiation . it takes advantage of the different statistical properties of the spatial distributions of cmbp and polarized galactic synchrotron . by our estimator @xmath180 , which evaluates the difference between e- and b - modes , we get the polarized component of cmb with a maximum systematic ( underestimate ) uncertainty of @xmath165 . this uncertainty is set by the contamination by the tensor perturbations which add b - modes to a cmbp map dominated by the e - modes generated by scalar(density ) perturbations . improving our knowledge of the tensor perturbations we will reduce the above uncertainty and improve the accuracy of our method of measuring cmbp . the accuracy we can get with our method is the maximun one can obtain at ground observatories with today ( 2nd generation ) systems for measuring the cmb polarization . these 2nd generation experiments are just arrived on the verge of detecting the polarized signals produced by density perturbations ( see for instance @xcite and @xcite).direct observations of the signals associated to tensor perturbations requires new , 3rd generation , intrinsically able to reject the foreground signals which , at present , are still in preparation ( see for instance @xcite ) . we are indebted to s.cortiglioni , e.caretti , e.vinjakin , j.kaplan and j. delabrouille for helpful discussion . mvs acknowledges the osservatorio astronomico di capodimonte , inaf , for hospitality during preparation of this paper . here and overall in paper we suppose that @xmath301 are complex random variables which satisfy the probability distribution law : @xmath302 with variance @xmath303 and @xmath304 . they have all the propertiers of gaussian variables ( below we omitt indexes @xmath51 and @xmath52 in first and second equations ) : @xmath305 @xmath306 setting @xmath307 with current index @xmath51 or @xmath52 , it immediately follows : @xmath308 in the our paper we assert that for the galactic synchrotron emission the measured values of the stokes parameters @xmath22 and @xmath23 behave as stochastic variables and random fields with gaussian distribution . this statement certainly holds at high galactic latitudes . at low galactic latitudes , where we observe large scale galactic structures , regular magnetic fields @xcite and quasi - periodic structures with typical sizes of about 250 pc , ( the amplitudes of regular and irregular components are approximately equal ) this assumption has to be checked . to do it we analyzed the distribution of the measured values of @xmath22 and @xmath23 on regions of different extension extracted from the effelsberg maps of the polarized diffuse radiation @xcite . the six fields were chosen within @xmath309 from the galactic plane , at galactic longitudes between @xmath310 and @xmath311 . they had dimensions of @xmath312 , @xmath313 , @xmath314 , @xmath315 , @xmath316 , @xmath317 , respectively . characteristics of the observed distributions of @xmath22 and @xmath23 measured at low galactic latitudes in areas of various extensions ( the larger is the number of pixels the closer is the histogramm to a gaussian shape ) [ cols="<,^,^,^,^,^ " , ] for each region we examined the shapes of the distributions , and calculated average value and the variance of the measured values of @xmath22 and @xmath23 . the distribution of @xmath22 in a @xmath318 area is shown in figure [ fa1 ] . very similar results are obtained for @xmath23 and for smaller regions . when the region is very small , ( figure [ fa2 ] is for an @xmath312 area ) , the distribution becomes broad , but its shape is still gaussian . results of analyses carried on on all the areas are presented in table 1 : for sufficiently large samples both @xmath22 and @xmath23 are compatible with zero . where @xmath288 is the number of pixels in the field , @xmath320 is the measured variance of @xmath22 and @xmath23 and @xmath321 is our estimator . the quantity ( [ appeq1 ] ) is plotted in figure [ fa3 ] versus the number @xmath288 of pixels . in the same figure we plot also : i)the calculated values of the differences @xmath322 in each field and ii)the calculated values of @xmath323

the polarization of the cosmic microwave background ( cmb)is a powerful observational tool at hand for modern cosmology . it allows to break the degeneracy of fundamental cosmological parameters one can not obtain using only anisotropy data and provides new insight into conditions existing in the very early universe . many experiments are now in progress whose aim is detecting anisotropy and polarization of the cmb . measurements of the cmb polarization are however hampered by the presence of polarized foregrounds , above all the synchrotron emission of our galaxy , whose importance increases as frequency decreases and dominates the polarized diffuse radiation at frequencies below @xmath0 ghz . in the past the separation of cmb and synchrotron was made combining observations of the same area of sky at different frequencies . in this paper we show that the statistical properties of the polarized components of the synchrotron and dust foregrounds are different from the statistical properties of the polarized component of the cmb , therefore one can build a statistical estimator which allows to extract the polarized component of the cmb from single frequency data also when the polarized cmb signal is just a fraction of the total polarized signal . our estimator improves the signal / noise ratio for the polarized component of the cmb and reduces from @xmath150 ghz to @xmath120 ghz the frequency above which the polarized component of the cmb can be extracted from single frequency maps of the diffuse radiation . , , polarimetry , mathematical procedures , radio and microwave , observational cosmology : 95.75.hi , 95.75.pq , 95.85.bh , 90.80.es

You are an expert at summarizing long articles. Proceed to summarize the following text: shear flow can drive dense colloidal dispersions into states far from equilibrium . especially of interest is the possibility to shear melt colloidal solids , in particular metastable colloidal glasses and gels , and to investigate shear - melted ( yielding ) colloidal glasses . does a yield stress and/or yield strain exist ( petekidis _ et al . are shear - melted states necessarily heterogeneous ( e.g. shear banded ) ? does ageing prevent stationary states under steady shearing ? do hydrodynamic interactions dominate the properties in flow ? these are just some of the questions whose answers will provide insights into the still murky glassy state . recently the integration through transients ( itt ) approach has been used to generalize the mode coupling theory ( mct ) of the structural glass transition to the case of steady shearing ( fuchs cates 2002 , 2009 ) , and to arbitrary homogeneous flows ( brader _ et al . _ 2008 ) , albeit based on a number of approximations . first , homogeneous flow profiles are assumed from the outset , second , hydrodynamic interactions are neglected , and third , the mode coupling ( mean - field - like ) decoupling approximation , splitting a four point density fluctuation function into the product of two - point density correlators , has been applied . while simplified ( schematic ) models of mct - itt ( fuchs cates 2003 ) can well be fitted to the linear rheology ( frequency dependent storage and loss moduli ) and non - linear rheology ( flow curves , viz . stress as function of shear rate ) of model dispersions ( siebenbrger _ et al . _ 2009 ) , no fully quantitative comparison between the theory and data without additional approximations has been presented up to now . we provide this first quantitative comparison , solving the mct - itt equations for hard discs confined to move in a plane , and performing brownian dynamics simulations for a binary hard disc mixture , also in two dimensions . computational ( memory ) constraints force us to work in two dimensions , which is still of some experimental relevance , as glasses have been observed in quasi - two - dimensional dispersions ( bayer _ et al . use of a binary mixture suppresses the nucleation rate sufficiently for even our long simulation runs to remain free of crystal nuclei . as hydrodynamic interactions are absent also in the simulation , and as lees - edwards boundary conditions ( le ) in combination with the thermostat impose a homogeneous shear rate , the integration of the equations of motion in the simulation ensure a stringent test of the theory . results for the stationary shear and normal stresses , and for the shear - distorted microstructure are presented and discussed . the input quantities entering the theory , and transient density correlation functions , which ( crucially ) enter during intermediate mct - itt steps , are characterized also . our work bears some similarity to the study by miyazaki , reichman , and yamamoto , who , however , concentrated on fluid states under shear , and on their time dependent fluctuations ( miyazaki _ et al . we focus here on the yielding glass state , and its stationary , time - independent averages , which are not accessible to the theory in ref . ( miyazaki _ et al . _ 2004 ) . in order to compare quantitatively with theory , we aimed for better statistics than in comparable previous two - dimensional simulation studies of sheared glasses ( yamamoto onuki 1998 ; furukawa _ et al . the basic concept of the algorithm has been described in detail in three dimensions in ( scala _ et al . _ 2007 ) and can easily be adapted to two dimensions @xmath0 and @xmath1 . in order to prevent crystallisation at high densities we consider a binary mixture with a diameter ratio of @xmath2 at equal number concentrations ( @xmath3 ) . @xmath4 hard discs move in a simulation box of volume @xmath5 with periodic lees edwards boundary conditions at packing fraction @xmath6 . the mass of the particles is set equal to unity , thermal energy @xmath7 sets the energy scale , and @xmath8 is used as unit of length in the following . we choose our coordinate axes such that flow is in the @xmath0-direction and the shear gradient @xmath9 is in the @xmath1-direction . after putting the particles on their initial positions we provide gaussian distributed velocities which will be overlain by the linear shear flow . to propagate the system forwards in time we employ a semi - event - driven algorithm . for every particle at the time @xmath10 the algorithm determines the possible collision time @xmath11 with any other particle . this is easily achieved by solving the equation @xmath12 where @xmath13 and @xmath14 denote the diameters of the particles @xmath15 and @xmath16 , @xmath17 the relative vector between both particles , and @xmath18 the relative velocities . the smallest solution @xmath19 for all particle pairs determines the next event in the algorithm . all particles can then be propagated according to : @xmath20 . with the conservation of energy and momentum the binary collision laws impose new velocities @xmath21 for the particle @xmath16 and @xmath15 @xmath22 due to the boundary conditions any particle in the vicinity of the box - boundary can collide with an image particle coming from the other end of the box . the boxes are constructed in such a way that they are translated with the velocity @xmath23 , where @xmath24 denotes the size of the box . the boundary conditions thus have the consequence that the velocities tend to increase . therefore a thermostat has to be introduced . after a time @xmath25 a so - called brownian step sets in , which assures that the particles move diffusively for longer times . in the brownian step at the time @xmath26 , all particle velocities are freshly drawn from a gaussian distribution with @xmath27 . after that linear shear flow is imposed so that @xmath28 holds . as the system starts from a cubic lattice it is necessary to wait for the system to relax before meaningful stationary averages can be taken . the quantity of interest in the work presented here is the potential part of the stress : @xmath29 , with the relative force components ( in @xmath30-direction ) of particles @xmath15 and @xmath16 @xmath31 , and the particles relative distance components @xmath32 . the kinetic part will play no role , and thus has already been omitted . as we consider hard particles the forces must be calculated from the collision events . by observing the collisions within a certain time window @xmath33 forces may be extracted using the change of momentum which occurs during the observation time . this leads to the evaluation algorithm ( lange _ et al . _ 2009 ; foss brady 2000 ) @xmath34 where the summation is over all collisions of particles @xmath15 and @xmath16 at time @xmath35 within the time window @xmath36 . the procedure effectively sums the momentum changes @xmath37 in the @xmath38 direction multiplied by the relative distance of the particles @xmath39 in the @xmath40 direction . here and below the brackets @xmath41 denote the average over different simulation runs . + additionally the shear stress can be computed via the contact value @xmath42 : @xmath43 where @xmath44 denote the @xmath45-dependent partial contact values of the two components and @xmath46 the minimal distance between two particles . @xmath45 is the polar angle . + the green - kubo relation @xmath47 holds for the non - sheared system . thus the shear viscosity can be extracted from the simulation via ( alder _ et al . _ 1970 ) @xmath48 where the sum runs over all collisions . the second pivot in this analysis is the equal - time structure factor and its deviation from the quiescent system . exploiting that @xmath49 is a real quantity , we can extract it via @xmath50 where the double sum runs over all pairs of particles @xmath15 and @xmath16 ( @xmath51 ) . the pairs @xmath52 are determined by the le boundary conditions and the constraint of having the lowest distance among all possible image particles in the surrounding boxes . + for low peclet numbers ( pe@xmath53=@xmath54)in fluid states , @xmath42 can be expanded : @xmath55 . this result can be used to derive the relative distortion of the structure factor in the linear response regime ( strating 1999 ) . @xmath56}\ ] ] while @xmath57 can be obtained via @xmath58 the mct - itt approach generalizes the mct of the glass transition to colloidal dispersions under strong continuous shear . it considers @xmath59 spherical particles with arbitrary interaction potential which move by brownian motion relative to a given linear shear profile . an equation of motion for a _ transient _ density correlator @xmath60 encodes structural rearrangements , and approximated generalized green - kubo laws relate stress relaxation to the decay of density fluctuations . the transient density correlator is defined by @xmath61 , where the density fluctuation is as usual @xmath62 , and , in itt , the average can be performed over the equilibrium gibbs - boltzmann ensemble . thus the normalization of the initial value @xmath63 is given by the equilibrium structure factor @xmath64 . the time - dependent or _ shear - advected _ wavevector @xmath65 appearing in the definition eliminates the affine particle motion with the flow field , and gives @xmath66 in the absence of brownian motion . shear flow coupled to random motion causes @xmath60 to decay , as given by an ( exact ) equation of motion containing a retarded friction kernel which arises from the competition of particle caging and shear advection of fluctuations @xmath67 where the initial decay rate contains taylor dispersion as @xmath68 . the generalized friction kernel @xmath69 , which is an autocorrelation function of fluctuating stresses , is approximated , following mct precepts by an expression involving the structural rearrangements captured in the density correlators @xmath70 with abbreviation @xmath71 , @xmath72 the particle density , and a vertex function given by @xmath73 where @xmath74 is the ornstein - zernicke direct correlation function @xmath75 . an additional memory kernel is neglected ( fuchs cates 2009 ) . the equilibrium structure factor , @xmath76 , encodes the particle interactions and introduces the experimental control parameters like density and temperature . similarly , the potential part of the stress @xmath77 in the non - equilibrium stationary state ( neglecting the diagonal contribution that gives the pressure ) is approximated assuming that stress relaxations can be computed from integrating the transient density correlations @xmath78 flow also leads to the build up of shear - induced micro - structural changes , which , again integrating up the transient density correlators , can be found from @xmath79 a far smaller isotropic term in @xmath80 ( see ( fuchs cates 2009 ) ) is neglected here , as it is of importance for the plane perpendicular to the flow only . the set of coupled eqs . [ mct1],[mct2 ] and [ mct3 ] was solved self - consistently using modifications of standard algorithms and the ng - iteration - scheme ( ng 2009 ) . the functions were discretized on a 2d - fourier - grid consisting of 101 grid points in either coordinate direction using a cutoff of @xmath81 . more explicitly the fourier - grid was discretized as @xmath82 with @xmath83 being the particle diameter . in order to enhance the accuracy , the advected direct correlation function @xmath84 in eq . [ mct3 ] was calculated from the input structure factor once it has been advected beyond the fourier - grid cutoff up to @xmath85 . for the time - integration of the convolution integral in eq . [ mct1 ] an initial time step of @xmath86 was used . the time - integration in the calculation of derived quantities eqs . [ sigma_xy ] and [ distsq ] was performed by dividing the integration interval into two sub - intervals , the first one containing times shorter , the other one times longer than @xmath87 . the integration was then carried out on the intrinsic quasi - logarithmic grid of the mct equation solver for times @xmath88 and on an equally - spaced linear time - grid with spacing @xmath89 for times @xmath90 . the mct - itt approach uses structural equilibrium correlations as input , computes transient structural density correlators to encode the competition between flow - induced and brownian motion , and calculates all stationary properties from time - integrals over the transient fluctuations . in this section , the input and intermediate quantities of the theoretical calculations are presented and discussed . the equilibrium structure factor @xmath64 is the only input quantity to the mct - itt equations . it varies smoothly with density or temperature , but leads to the transition from a shear - thinning fluid to a yielding glassy state at a glass transition density @xmath91 . figure [ fig1 ] shows modified hyper - netted chain structure factors of monodisperse hard discs in @xmath92 from ( bayer _ et al . _ 2007 ) used in the mct - itt calculations . for comparison , the averaged structure factors obtained from the binary mixture simulation are shown also . in both cases , the value of the glass transition density @xmath91 is included in the curves in fig . [ fig1 ] , and only smooth changes in @xmath64 are noticeable . the short range order at the average particle distance , as measured in the primary peak of @xmath64 , increases with densification . differences in the structure between the mono- and the bidisperse system become appreciable beyond the primary peak , and especially beyond the second peak in @xmath64 . also the height of the primary peak in @xmath64 at @xmath91 differs , indicating that the averaged structure factor in the bidisperse system does not well characterize the local structure and caging in the simpler system , and that a comparison with a bidisperse mct - itt calculation should be performed . as this is outside the present numerical reach , we proceed by comparing results for the characteristic wavevectors denoted @xmath93 to @xmath94 in fig . the angular dependence of the anisotropic structure will be explored along the special directions indicated in the insets . the central quantities in mct - itt , which encode the competition between shear driven motion and random fluctuations , are the transient density correlators . structural rearrangements manifest themselves as a ( second ) slow relaxation process in the @xmath95 , whose relaxation time depends sensitively on the distance to the glass transition , measured by the ( relative ) separation parameter @xmath96 , and on the magnitude of the shear rate . figure [ fig2 ] shows representative curves in a fluid state ( left column ) and in a shear - melted glassy state ( right column ) for the wavevectors and directions defined in fig . [ fig1 ] , and for nine different shear rates . the denoted bare peclet numbers measure the shear rate compared to the brownian diffusion time estimated with the single particle diffusion coefficient @xmath97 at infinite dilution . characteristically , for the present strongly viscoelastic system , shear affects the structural relaxation already at extremely small bare peclet numbers pe@xmath98 . the short time motion , which corresponds to the local diffusion of the particles within their neighbour cages , however , is not much affected as long as pe@xmath99 holds . + + in the fluid ( @xmath100 ) , a linear response regime , where shear does not affect the decay of thermal fluctuations , is observed for small ( dressed ) peclet or weissenberg numbers pe@xmath101 , where pe@xmath102 measures the shear rate relative to the intrinsic relaxation time . the latter can be estimated from the relaxation of @xmath103 , viz . at the primary peak of @xmath64 , where the structure relaxes most slowly in the quiescent case . the final ( or @xmath30- ) relaxation time @xmath11 increases strongly when approaching the glass transition ; with @xmath104 predicted by mct ( bayer _ et al . _ 2007 ) . for the fluid state in fig . [ fig2 ] , only pe@xmath105 is small enough that pe@xmath106 holds and the final relaxation curves agree for different shear rates . in the glass ( @xmath107 ) , shear is the origin of the decay of the otherwise frozen - in density fluctuations , and all nine shear rates lead to different final decays . while for small and larger wavevectors the final decay is rather isotropic , around the main peak in @xmath64 some anisotropy is noticeable in the transient correlators . in the direction perpendicular to the flow , @xmath108 ( red ) , shear is not as efficient in decorrelating the density as in the other directions , which fall closer together . + + one of the central predictions of mct - itt concerns the existence of a scaling law describing the yielding of glassy states , which manifests itself by an approach to a master function @xmath109 for decreasing shear rate , @xmath110 and @xmath107 , where the rescaled time @xmath111 agrees with the accumulated strain . this scaling law , which is quite obvious in fig . [ fig2 ] , is tested quantitatively for some wavevectors in fig . [ figyield ] , and holds extremely well ( for @xmath93 ) , or with pre - asymptotic corrections ( for @xmath112 ) . the shapes of the yielding master functions can be very well fitted with compressed exponentials , @xmath113 , but the parameters , including the exponent @xmath114 , depend on wavevector and orientation ; see table [ table1 ] for representative values . .parameters of the compressed exponentials @xmath115 fitted to the yield master functions shown in fig . [ figyield ] . [ cols="^,^,^,^,^,^,^",options="header " , ] the quite good fit to compressed exponentials is at present only a numerical observation . yet , close to the glass transition asymptotic expansions are possible which analytically predict the initial part of the yield master functions ( fuchs cates 2002,2003,2009 ) : @xmath116 as the critical glass form factor @xmath117 and the critical amplitude @xmath118 are isotropic , this result suggests a rather isotropic yielding process right at the glass transition density . approximating @xmath119 by an exponential , it also provides an estimate for the final relaxation time under shear . except for @xmath120 all quantities in the above formula have been determined in quiescent mct ( bayer _ et al . _ 2007 ) , and our values ( e.g. @xmath121 ) differ only because of the coarser discretization in @xmath122 space that is necessary under shear . close to the glass transition , we estimate @xmath123 . as the bottom left panel in fig . [ figyield ] and fig . [ fig2 ] both show , this estimate for the final relaxation time is quite good close to the transition , but deeper in the glass the correlators become somewhat more anisotropic . specifically for @xmath108 ( red ) with wavevector magnitudes around @xmath124 , the correlator slows down relative to other orientations . more detailed analytical predictions , are possible around @xmath117 , where spatial and temporal dependences in the transient fluctuations decouple ( fuchs cates 2003 ) : @xmath125 @xmath126 is a universal function , which only depends on @xmath127 , and @xmath128 is another mct time scale that diverges at the glass transition ( bayer _ et al . this factorization property known from quiescent mct generalizes to steady shear , and is an essential feature of the localization transition that underlies glass formation in mct . in fig . [ figbeta ] , the quantity @xmath129 is plotted which should become wavevector and orientation independent if factorisation holds . this holds in the liquid , and in the glass , where however shear leads to strong ( anisotropic ) pre - asymptotic corrections already for @xmath130 . based on the approximated generalized green - kubo relations of eqs . ( [ sigma_xy ] ) and ( [ distsq ] ) , and the properties of the transient correlators discussed in the previous section , mct - itt makes a number of predictions for stationary stresses and structural correlations ( fuchs cates 2002 , 2009 ) . in the following , we will test these qualitatively , but also quantitatively , by comparing them to the two - dimensional simulation data . the quantity of most interest in nonlinear rheology is the shear stress @xmath131 . flow curves giving the shear stress as function of the shear rate can be obtained in the simulations and are shown in fig . [ figstress ] . the viscosity decreases below its newtonian value @xmath132 upon increasing the shear rate already for small pe@xmath53 values . shear thinning in which the stress increases less than linearly with @xmath133 , sets in at pe of the order of unity , and this crossover shifts to lower and lower pe@xmath53 for increasing density . at the density around @xmath134 , the crossover leaves the accessible shear rate window . we use this as estimate of an ideal glass transition density , where the final relaxation time and the quiescent newtonian viscosity ( @xmath135 ) diverge . moreover , at this density the flow curves change from a characteristic s shape in the fluid , to exhibiting a rather @xmath133-independent plateau at small shear rates . this change is the hallmark of the transition in mct - itt between a shear thinning fluid and a yielding glass ( siebenbrger _ et al . _ 2009 ) . + the numerical mct - itt solutions show the same transition scenario , but because of the approximations involved , exhibit a different critical density than the simulations . even if theory and simulation considered the same ( binary ) system , a difference in critical packing fraction would be expected as is well known also in three dimensions ( voigtmann _ et al . thus the following mct - itt calculations match the relative separation @xmath136 from @xmath91 in order to compare with the simulations . as mct - itt is aimed at describing the long time structural motion , errors need to be anticipated in its description of short time properties . this is obvious in real dispersions , where hydrodynamic interactions ( neglected in mct - itt ) affect the short time diffusion coefficient @xmath137 , but may arise in the following comparisons , also . a rescaling of the effective peclet number pe@xmath138 would correct for this change in @xmath137 . in order not to introduce additional fit parameters we refrain from doing so , but anticipate that future comparisons may require @xmath139 . the approach to a yield scaling law , where the final decay of the transient correlators depends on the accumulated strain only ( see figs . [ fig2 ] and [ figyield ] ) , predicts the existence of a ( dynamical ) yield stress @xmath140 , which characterizes the shear melted glass . in the bidisperse hard disc mixture , at the glass transition it takes the ( critical ) value @xmath141 ; below the glass transition , the yield stress jumps to zero , @xmath142 . its quantitative prediction is quite a challenge for theory , because eq . [ sigma_xy ] shows that it requires an accurate calculation of the shear driven relaxation process . mct - itt overestimates the critical yield stress @xmath143 by roughly a factor 10 because , presumably , the decay of the transient correlators is too slow . yet , of course the difference between the monodisperse system in the mct - itt calculation and the bidisperse simulated system contributes in unknown way to the error . it appears reasonable to assume that mixing two species reduces the stresses under flow , which would explain part of the deviation . + at larger shear rates , the flow curves from simulation appear to approach a second newtonian plateau which , presumably , strongly depends on the hard - core character of excluded volume interactions and is outside the reach of the present mct - itt . the latter , by using its sole input @xmath64 rather than pair potential , is not directly aware of hard - core constraints . we checked , however , that the states remain homogeneous and random up to the pe@xmath53 values shown . reassuringly , the same rescaling factor of 0.1 as for the shear stress brings theoretical and simulational normal stress differences , @xmath144 , into register also ; see fig . [ fignorm ] . the normal stress differences are positive ( indicating that the dispersion would swell after flowing through a nozzle ) , and show similar behavior to the stress , increasing like @xmath145 in the fluid for small shear rates , and leveling out onto a plateau in the yielding glass . the macroscopic stresses in the flowing dispersion are experimentally most important , but provide only an averaged description of the local effects of shear . spatially resolved information can be learned from the distorted structure factor @xmath146 , which in mct - itt is connected to the stress via @xmath147 as follows from eqs . ( [ sigma_xy ] ) and ( [ distsq ] ) . + + figure [ figsqcolor ] shows colour - coded structure factors @xmath148 as function of the two - dimensional wavevector @xmath122 , with @xmath149 in the direction of flow and @xmath150 along the gradient direction . in the left column , panels with simulation data are compared to panels in the right column obtained in mct - itt . scattering intensities are presented @xmath151 ( top row ) in the linear response regime in the fluid , effectively measuring the equilibrium structure factors already shown in fig . [ fig1 ] , @xmath152 ( middle row ) at high shear in the glass , where all densities are in the shear thinning region , and @xmath153 ( bottom row ) in the glass at low shear rate , where the yielding glassy state is tested . while the fluid @xmath148 is isotropic for small pe@xmath53 ( case @xmath151 ) , as required by linear response theory , increasing pe@xmath53 to values around unity , leads to an ellipsoidal scattering ring , which is elongated along the so - called compressional axis @xmath154 , and more narrow along the extensional axis @xmath155 ; this indicates that shear pushes particles together along the compressional and pulls particles apart along the extensional diagonals ( vermant solomon 2005 ) . theory and simulation data in this representation qualitatively agree except for that mct - itt somewhat overestimates the anisotropic distortion of the glass structure at low pe@xmath53 . + + + a more careful look onto the distorted microstructure is provided by @xmath156-dependent cuts through @xmath148 along the directions colour - coded in fig . important for the direction @xmath108 ( red ) is the need , present in both simulation and theory , to average @xmath148 over a small but finite angle , because exactly at @xmath108 the structure oscillates noisily around zero . + especially of interest are the intensities of case @xmath153 , where the stationary structure of the shear melted solid is studied . the density @xmath157 is not yet high enough to lie in the glass , but close enough to the glass transition so that the correlations at quite low rate , namely pe@xmath158 , closely resemble the ones of glassy states at very low shear rates . ( while we can get good statistics for stresses at pe@xmath159 for all densities , structure factors can not be sampled sufficiently there . ) at this density of @xmath157 also the equilibrium structure factor @xmath64 can be obtained in long simulation runs , and the ( relative ) difference @xmath160 can thus be determined . it is shown in fig . [ figsqrel ] for two states in order to investigate the distorted structure of the shear melted glass in detail . we base the following discussion on the hypothesis that pe@xmath161 at @xmath157 captures a glass - like state in the limit of low bare peclet number . figure [ figsqrel ] provides a sensitive test of the accuracy of the theoretical predictions . the lower left panel shows that the structure factor at vanishing shear rate @xmath162 jumps discontinuously at the glass transition ; while @xmath163 holds in the fluid , @xmath164 holds in the glass . relative deviations @xmath165 of 20% remain . simulation finds quite isotropic deviations which show a maximum on the low-@xmath156 side of the primary peak in @xmath64 . mct - itt predicts the absence of a linear response regime in @xmath148 as function of the shear rate in the glass , and derives it from the existence of the yield scaling law in the transient correlators . because @xmath133 sets the time scale for the final relaxation into the stationary state , the limit @xmath166 does not agree with the quiescent result @xmath167 . + quantitatively , mct - itt overestimates the distortion again by a factor up to 10 , and finds a noticeable anisotropy , as discussed in context with fig . [ figsqcolor ] . while the difference between the bidisperse and the monodisperse system may influence the comparison , we believe that the major origin of the error is that mct - itt underestimates the speeding up of structural rearrangements caused by shear . the too slow transient correlators thus become anisotropic because the accumulated strain @xmath168 becomes too big before structural correlations have decayed . qualitatively , aspects of the anisotropy predicted by mct - itt can be seen in the simulations at only slightly larger shear rates , like at pe@xmath169 shown in the middle left panel of fig . [ figsqrel ] . while along the two axis- and the extensional diagonal direction , the low-@xmath156 wing of the primary peak in @xmath148 becomes enhanced under shear , along the compressional axis ( magenta ) it gets suppressed , and the high-@xmath156 wing is pushed up . simulation also finds a suppression of the peak height along all directions , which mct - itt reproduces along the diagonal directions . overall the anisotropy and the magnitude of the distortions predicted by mct - itt remain too large , but the deviation decreases . the differences between the equilibrium structure factors @xmath64 in the simulated and in the calculated system should be taken into account in future work , and presently preclude comparisons at larger wavevectors . the first fully quantitative solutions of the mct - itt equations for the distorted microstructure and the stresses in steadily sheared two - dimensional shear - thinning fluids and yielding glasses of brownian hard discs exhibit all the universal features discussed within schematic mct - itt models ( fuchs cates 2003 ) . they compare qualitatively well , but quantitatively with appreciable errors , with brownian dynamics simulations of a bidisperse mixture without hydrodynamic interactions in a linear shear profile , which for all states considered remains in an homogeneous and disordered state . the non - analytic behavior of the stationary properties , and the lack of a linear response regime throughout the ( shear - melted ) glass state , predicted by theory , can be found in the simulation . work funded in part by dfg - irtg 667 , and epsrc grants ep/045316 and ep/030173 . mec holds a royal society research professorship [ lastpage ]

brownian dynamics simulations of bidisperse hard discs moving in two dimensions in a given steady and homogeneous shear flow are presented close to and above the glass transition density . the stationary structure functions and stresses of shear melted glass are compared quantitatively to parameter free numerical calculations for monodisperse hard discs using mode coupling theory ( mct ) within the integration through transients ( itt ) framework . theory qualitatively explains the properties of the yielding glass but quantitatively overestimates the shear driven stresses and structural anisotropies . [ firstpage ]

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Proceed to summarize the following text: as a part of the third phase of the sloan digital sky survey ( sdss - iii ; * ? ? ? * ) , the marvels ( _ _ m__ulti - object _ _ a__po _ _ r__adial _ _ v__elocity _ _ e__xoplanet _ _ l__arge - area _ _ s__urvey ) project is searching for substellar companions by monitoring the radial velocities ( rvs ) of 3330 fgk stars @xcite . this sample size is large enough for the project to find relatively rare objects , such as brown dwarf ( bd ) companions to solar - type stars . the paucity of observed bd companions to solar - type stars with separations of @xmath85 au is typically referred to as the bd desert @xcite . since the size of the marvels sample allows us to begin to quantify how arid the bd desert may be , any marvels discovery of a bd in the desert ( or lack thereof ) is a step toward increasing our understanding of bd formation . in addition to its large homogeneous target sample , marvels differs from other surveys for substellar companions in two key ways . first , the project employs a dispersed fixed - delay interferometer ( dfdi ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? second , it uses a multi - object spectrograph to observe 60 stars simultaneously @xcite . the dfdi prototype instrument was used to discover the first extrasolar planet around hd 102195 in 2006 with this new rv method @xcite . the marvels dfdi technique combines an interferometer with a medium resolution spectrograph ( @xmath912,000 ) in order to obtain a precision of @xmath5100 m s@xmath7 . given its rv precision and survey design to monitor each target with at least 24 rv measurements over at least 1 yr , marvels is sensitive to bd and low - mass stellar companions with periods ranging from a few days to hundreds of days . nonetheless , certain specific types of astrophysical false positives can mimic substellar companions unless additional vetting is performed . this paper describes just such a case , tyc-3010 - 1494 - 1 ( hereafter tyc 3010 ) , a stellar binary that initially appeared as a single star with a substellar companion and that , through a confluence of orbital parameters , continued to masquerade as such despite a disconcertingly extensive amount of observation and analysis . when we began analysis of tyc 3010 , marvels and its pilot project had already detected two bd candidates orbiting late f stars in the bd desert ( ( * ? ? ? * ; * ? ? ? * at present , we have three more candidates in the desert : ) @xcite ) . the marvels discovery data indicated that tyc 3010 possessed a substellar companion with a minimum mass of @xmath10 and that it was on a @xmath5238-day moderately eccentric orbit with an rv amplitude of @xmath51.5 km s@xmath7 ( see the top panel of figure [ fig : mc10rv_curve_bd ] ) . however , given the cadence of marvels and the period of the orbit , there were significant gaps in the phase coverage and additional observations with a different spectrograph were required to constrain the rv solution . initially , the follow - up data remained fully consistent with the bd companion scenario . however , during the course of the program , we found two rv points that were shifted by @xmath520 km s@xmath7with respect to most of our data ; while investigating the source of these anomalous points , we realized that a few similar points had been rejected from our marvels discovery data by the team s outlier rejection procedures ( see bottom panel of figure [ fig : mc10rv_curve_bd ] ) . examining the cross - correlation function ( ccf ) of the anomalous rv points ( in both the discovery and subsequent data ) revealed evidence that there were two components in the ccf , which suggested that the companion to the primary was most likely a stellar - mass secondary . finally , including the initially flagged outlier measurements and disentangling the rv measurements of the two components , the system was found to be a nearly equal - mass stellar binary ( @xmath11 ) on a highly eccentric orbit ( @xmath12 ) . evidently , for a system like tyc 3010 , it is possible to clip just a few measurements and obtain an apparently reasonable solution that is convincing but completely incorrect . as large scale rv and transit surveys for exoplanets become more common , it is increasingly inevitable that any and all forms of astrophysical false positives , despite their rarity , will be found . indeed , the first bd candidate discovered by the marvels project , , appeared to exhibit evidence for an additional planet - mass companion , but turned out instead to likely be a quadruple system , comprising four stars with no detected bd or planetary - mass companion @xcite . akin to tyc 3010 , is a double - lined spectroscopic binary ; the stars have relative rvs which are sufficiently low that they are always blended , even at the resolution of the hobby - eberly telescope ( het ; @xmath13 mode ) . thus , with both and tyc 3010 , we actually measure a flux - weighted mean of two sets of stellar spectral lines . this flux - weighted mean exhibits a suppressed velocity shift that mimics a single - lined binary with a bd secondary . both systems possess geometries that allow them to masquerade as less massive systems : is nearly face - on , which leads to low projected velocities , while tyc 3010 is on a highly elliptical orbit with a semi - major axis oriented nearly perpendicular to our line of sight . similarly , @xcite describe what at first appeared to be a transiting bd companion to an f star from the tres transit survey , but turned out instead to be an f star blended with a g+m stellar eclipsing binary . the system that we describe here follows these unfortunate examples , and is similarly pernicious . in the following sections , we present our analysis as a kind of cautionary tale for other rv surveys to avoid similar false positives . in section [ sec : data ] , we describe the spectroscopic and photometric data obtained for tyc 3010 . in section [ sec : results ] , we discuss in detail the nature of the evidence that led us to conclude that tyc 3010 was an eccentric stellar binary instead of a bd companion to a solar - type star . we also present the properties we derived for both components of the spectroscopic binary . in section [ sec : discussion ] , we discuss the circumstances that allowed this false positive to masquerade for so long and through several vetting steps as a compelling detection of a substellar companion , and we describe methods that the marvels team and other rv surveys can use to recognize this kind of astrophysical false positive in the future . finally , in section [ sec : summary ] , we conclude with a summary of the main results . we obtained a total of 65 rv measurements from the sloan 2.5 m , the apo 3.5 m , and the het 9.2 m telescopes . we will briefly summarize the characteristics of the data from all three telescopes . for more details of the analysis , please see @xcite , @xcite , and @xcite . a total of 28 spectra ( see table [ tbl : sing_rvs ] ) of tyc 3010 were obtained with the sloan 2.5 m telescope @xcite at apache point observatory ( apo ) . the multi - fiber marvels spectrograph @xcite can simultaneously measure the rvs of 60 stars during each telescope pointing . both beams of the interferometer are imaged onto the detector , so each 50-minute observation results in two fringed spectra in the wavelength range of @xmath5500570 nm with a resolving power of @xmath14 . the marvels interferometer delay calibrations are described in @xcite . for more details on how the data were reduced and analyzed to yield rvs , see @xcite . cccc 2454927.82470 & m & 62.681 & 0.148 + 2454928.85061 & m & 62.564 & 0.139 + 2454964.76792 & m & 61.479 & 0.108 + 2454965.77714 & m & 61.374 & 0.113 + 2454994.69536 & m & 59.933 & 0.115 + 2455193.91250 & m & 62.102 & 0.165 + 2455197.96727 & m & 61.753 & 0.134 + 2455198.94828 & m & 61.714 & 0.095 + 2455199.96552 & m & 61.664 & 0.139 + 2455200.98947 & m & 61.585 & 0.097 + 2455201.97760 & m & 61.587 & 0.116 + 2455202.99063 & m & 61.528 & 0.149 + 2455258.88272 & m & 39.192 & 0.091 + 2455259.83118 & m & 41.327 & 0.092 + 2455260.82412 & m & 45.097 & 0.145 + 2455261.82050 & m & 48.416 & 0.096 + 2455280.77587 & m & 61.103 & 0.105 + 2455280.76844 & m & 61.174 & 0.117 + 2455283.81484 & m & 61.411 & 0.154 + 2455284.75054 & m & 61.461 & 0.112 + 2455311.68421 & m & 62.493 & 0.209 + 2455313.62591 & m & 62.402 & 0.174 + 2455369.64423 & m & 62.531 & 0.333 + 2455551.99403 & m & 62.788 & 0.120 + 2455552.98222 & m & 62.856 & 0.104 + 2455553.98561 & m & 62.795 & 0.121 + 2455556.97163 & m & 62.821 & 0.123 + 2455557.97465 & m & 62.801 & 0.104 + 2455471.98302 & a & 60.138 & 0.116 + 2455519.95995 & a & 61.359 & 0.052 + 2455519.98157 & a & 61.371 & 0.051 + 2455637.88366 & a & 62.452 & 0.055 + 2455637.92209 & a & 62.278 & 0.048 + 2455654.83350 & a & 62.390 & 0.059 + 2455665.65219 & a & 62.323 & 0.065 + 2455665.69165 & a & 61.664 & 0.075 + 2455669.60113 & a & 61.827 & 0.052 + 2455686.82409 & a & 60.946 & 0.076 + 2455695.66512 & a & 60.949 & 0.039 + 2455695.70529 & a & 60.931 & 0.053 + 2455703.61994 & a & 60.942 & 0.116 + 2455709.77767 & a & 59.749 & 0.098 + 2455903.90846 & h & 62.448 & 0.051 + 2455917.87269 & h & 62.237 & 0.060 + 2455928.84083 & h & 61.759 & 0.046 + 2455940.80855 & h & 61.122 & 0.058 + 2455946.80490 & h & 60.285 & 0.055 + 2455950.80134 & h & 59.539 & 0.045 + 2455953.82447 & a & 58.385 & 0.049 + 2455954.00566 & h & 58.467 & 0.050 + cccccc 2455725.68377 & a & 46.012 & 0.167 & 75.222 & 0.257 + 2455735.62781 & a & 43.197 & 0.251 & 81.056 & 0.175 + 2455956.76037 & h & 53.788 & 0.030 & 69.104 & 0.063 + 2455959.78075 & h & 51.409 & 0.025 & 71.807 & 0.055 + 2455964.75592 & h & 44.163 & 0.026 & 80.066 & 0.055 + 2455964.83117 & a & 44.884 & 0.071 & 79.700 & 0.372 + 2455967.75334 & h & 36.755 & 0.030 & 88.389 & 0.062 + 2455967.82824 & a & 37.684 & 0.076 & 88.651 & 0.471 + 2455968.74640 & h & 34.456 & 0.025 & 90.966 & 0.054 + 2455971.73989 & h & 36.359 & 0.029 & 88.685 & 0.060 + 2455972.97350 & h & 40.368 & 0.025 & 84.354 & 0.054 + 2455976.73787 & h & 50.072 & 0.024 & 73.253 & 0.051 + 2455977.71541 & h & 51.788 & 0.028 & 71.580 & 0.058 + 2455978.71767 & h & 52.990 & 0.026 & 69.840 & 0.056 + 2455979.71599 & h & 54.160 & 0.029 & 68.450 & 0.062 + as described below , it proved essential to examine the ccfs of the individual spectra . however , performing a cross - correlation on a dfdi spectrum requires a few steps beyond what one performs for a typical slit or cross - dispersed echelle spectrograph . in both cases the images are reduced using standard techniques ( bias subtraction , trace correction , flat fielding etc . ) once a fully processed two - dimensional spectrum has been extracted , there is a divergence in the techniques . in the case of a normal spectrum , one merely sums the flux in the slit ( channel ) direction to produce a one - dimensional spectrum . this approach is not possible in the dfdi technique because the fringing pattern will introduce false fluctuations in total flux if one just sums in the slit direction . these fluctuations will be a function of the phase of the fringe pattern in each pixel channel . to correct for this effect , a sinusoidal function of the form @xmath15 is fit to each pixel column . for the purposes of cross - correlation the only term of interest is @xmath16 , or the mean flux in each channel . a one dimensional spectrum is then constructed using the @xmath16 term in each channel . from this point forward the ccf is determined using standard techniques . a total of 19 rv observations were taken with the apo 3.5 m telescope using the arc echelle spectrograph ( arces ; * ? ? ? this spectrograph operates in the optical regime from @xmath53,60010,000 with a resolving power of @xmath17 . the first set of observations were taken from 2010 october to 2011 june . the second set of observations , which were undertaken with the goal of increasing phase coverage of periastron , were obtained during 2012 january february . as shown in tables [ tbl : sing_rvs ] and [ tbl : doub_rvs ] , there were 15 arces points observed outside of periastron , and 4 points during periastron ( the first two of these periastron points are where we initially resolved both the primary and secondary spectral lines see bottom panel of figures [ fig : mc10rv_curve_bd ] , [ fig : spectra ] , and [ fig : ccfs]and began to suspect that the system might be a double - lined spectroscopic binary ) . to achieve high - accuracy rv measurements with the echelle spectrograph , we obtained a thorium - argon ( thar ) exposure after every science exposure . in order to place tyc 3010 on an absolute rv scale , we also frequently bracketed our observations of tyc 3010 with observations of the rv standard hd 102158 , which has an absolute rv of 28.122 km s@xmath7 @xcite . from the standard deviation of the 13 rv measurements we obtained for hd 102158 ( see table [ tbl : rvstd ] ) , we were able to determine that the arces spectrograph possesses an rv stability of @xmath50.5 km s@xmath7 . two of the arces spectra were taken with longer exposure times in order to achieve a high signal - to - noise ratio ( s / n ) for deriving the fundamental stellar parameters ( see section [ sec : starpars ] ) . these two spectra were taken with an exposure time of 200 s and with the default slit setting described in @xcite . the data were reduced with iraf , and after barycentric corrections and continuum normalization , the two spectra were combined to produce a final spectrum with an s / n of @xmath5170 per resolution element at @xmath56500 . however , once we realized that tyc 3010 was a double - lined spectroscopic binary , we re - derived the spectroscopic parameters with a double - lined spectrum obtained near periastron , as described in section [ sec : starpars ] . upon realizing the eccentric binary - star nature of the object from the apo 3.5 m data , observations where initiated with the 9.2 m het @xcite and the higharcsec resolution spectrograph ( hrs ; * ) at a resolving power of using a 2 arcsec optical fiber . a total of 18 observations were obtained to completely cover periastron , and thereby fully constrain the orbit . the queue - scheduled observing mode of the het @xcite is extremely well suited for investigating objects that require monitoring over a long timespan , as well as targeted observations near periastron passage . for wavelength calibration , thar images were obtained immediately before and after the science exposure to aid in calibrating any possible instrument drift . the data were reduced and wavelength calibrated using custom optimal extraction scripts written in idl . rvs were measured using two different techniques , which we describe below . the het observations clearly resolve the orbit for tyc 3010 , and constrain the eccentricity to a value of @xmath2 ( see section [ sec : sb2_soluxn ] ) . ccc 2455654.81577 & 28.734 & 0.052 + 2455665.67180 & 28.050 & 0.034 + 2455665.71479 & 28.169 & 0.043 + 2455669.58461 & 27.736 & 0.045 + 2455686.80825 & 27.470 & 0.054 + 2455695.68591 & 28.030 & 0.038 + 2455695.72570 & 28.090 & 0.034 + 2455703.60341 & 28.244 & 0.039 + 2455709.76151 & 27.476 & 0.030 + 2455725.66705 & 26.692 & 0.123 + 2455735.61237 & 28.126 & 0.109 + 2455964.81467 & 28.093 & 0.058 + 2455967.81369 & 27.728 & 0.050 + rvs were measured using a cross - correlation mask derived from national solar observatory fourier transform spectroscopic solar data @xcite , and a technique similar to that described by @xcite . the resultant ccf encodes information from the @xmath5400600 nm region , and we elected not to use redder wavelengths due to issues with telluric contamination . figure [ fig : ccfs ] shows the resulting ccf for an epoch during periastron and one outside of periastron ; as is the case for the arces data , during periastron the primary and secondary peaks are clearly visible in the het ccfs , but outside of periastron only a single peak is resolved . the centroid of the ccf peak is determined by fitting a gaussian . this technique has been used successfully for isolated stars to derive precise rvs by the teams using fiber - fed high resolution spectrographs ( e.g. , harps , sophie , elodie , coralie ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , since psf stability is an important component of deriving precise rvs with this technique . any mismatch between the ccf and the simple gaussian model is absorbed as a zero - point offset in the derived rvs as long as the psf is stable ( resulting in a stable ccf shape ) . the het / hrs spectrograph is also fiber - fed , enabling this technique to also be applied to binary stars . this method is computationally efficient , and also does not require that the spectra be normalized , resulting in a quick turn around in determining rvs once the data are in hand . the rvs derived enabled us to plan and obtain observations as soon as the peaks began to separate on the approach to peri - passage . table [ tbl : sing_rvs ] shows the het rvs obtained with this technique for those epochs where the ccf appears as a single peak . while the ccf mask technique described above works quite well , it does not yield the best rvs possible for spectra with two ccf peaks since only one mask ( g2 spectral type ) was used in determining peak positions . once all the data were in hand , we were able to apply the two - dimensional cross - correlation algorithm , todcor @xcite . todcor can simultaneously cross - correlate two stellar templates against a blended target stellar spectrum to disentangle the stellar rvs of the components as well as derive a flux ratio . we used todcor along with hrs observations of hd161237 ( g5v ) and hd 198596 ( k0v ) as templates to measure the rvs of tyc 3010 . the hrs spectrum was divided into different bandpasses , and each bandpass was solved independently following @xcite and the resulting cross - correlation surface combined with a maximum likelihood analysis . further details on our implementation of the todcor algorithm , as well as details of our custom hrs spectral extraction pipeline , can be found in @xcite . table [ tbl : doub_rvs ] shows the rvs of the primary and secondary determined using this algorithm at those epochs where the ccf is double peaked . we add 0.05 km s@xmath7 in quadrature to the todcor formal errors to account for additional noise effects like wavelength calibration , small tracking induced psf changes , etc . while the het observed the target on 18 epochs , the secondary rvs are only reliably measured for 11 epochs . these are the epochs where the primary and secondary peaks are sufficiently separated to determine an independent rv for each . while rvs can be determined for the other 7 epochs , they are rvs of blended spectra , and the associated systematic error is not only larger , but also more difficult to quantify . since both peaks are unambiguously detected in todcor at these epochs , we are also able to measure the secondary to primary flux ratio , @xmath18 , which we determine to be @xmath19 by averaging the flux ratio of the templates ( g5v and k0v ) over four bandpasses spanning 4663 - 5863 . finally , the mass ratio derived from these 11 epochs is @xmath11 . the marvels team obtained lucky imaging for tyc 3010 in order to detect any spatially resolvable companions . in 2011 april , using the fastcam @xcite instrument on the 1.5 m tcs telescope at observatorio del teide in spain , we obtained 47,000 frames in the @xmath20-band with a 70 ms exposure time for each frame . data processing was accomplished with a custom - made idl pipeline . as described in @xcite , the best frames are selected via the brightest pixel ( bp ) method . the frames with the brightest @xmath21% of bps are combined to generate a final image , where @xmath22 \{@xmath23 } for tyc 3010 . figure [ fig : lucky_exp ] shows the resulting final images for each particular percentage of the best frames . no companions are detected , but we can place constraints on the upper limit of the masses of resolvable companions . using the spectroscopic @xmath24 for tyc 3010 ( see section [ sec : starpars ] ) , and the relations from @xcite , we determine the bolometric magnitude . combining the bolometric magnitude with mass luminosity relations @xcite , we convert the detection limit for the @xmath20-band magnitude into a lower limit for the masses of detectable companions at different separations . at the @xmath25 level , where @xmath26 is defined in @xcite as the rms of the counts within concentric annuli centered on tyc 3010 , and using 8 pixel boxes , we can rule out the presence of detectable companions above a mass of @xmath27 outside of 50 au ( see figure [ fig : ao_mass_limit ] ) . in addition to the lucky imaging , we were also able to obtain adaptive optics ( ao ) images of tyc 3010 on 2012 october 21 ut using the nirc2 imager at keck ( instrument pi : keith matthews ; * ? ? ? * ) . observations consist of a sequence of nine dithered frames in the @xmath28 filter ( central @xmath29 m ) using the narrow camera ( plate scale = 10 mas pix@xmath7 ) setting . each frame consisted of 20 coadds with 0.1814 s of integration time per coadd , totaling 32.65 s of on - source exposure time . images were processed using standard techniques to remove hot pixels , subtract the sky - background , and align and coadd the cleaned frames . no candidate companions were identified in either raw or processed images . figure [ fig : ao_mass_limit ] shows our sensitivity to off - axis sources as a function of angular separation . our diffraction - limited observations rule out the presence of companions 6.5 magnitudes fainter than the primary star for separations beyond 0.5@xmath30(@xmath25 ) . using theoretical isochrones from @xcite , we convert this magnitude limit to a mass upper limit , as shown in figure [ fig : ao_mass_limit ] ; we can exclude companions with a mass above outside of 100 au . in this section we present the orbit solution of the tyc 3010 system . first we show how the data initially suggested a spurious solution in which tyc 3010 is a single star with a bd companion . next we present the correct solution , in which tyc 3010 is shown to be a double - lined spectroscopic stellar binary ( sb2 ) with two solar - type stars , and we provide a full characterization of the system properties . of the 28 rv measurements collected with the marvels instrument , 24 passed the data quality checks and were therefore included in the automated orbit solution fitting procedures . for the arces data , the first 14 consecutive rv points obtained during the initial set of observations were fully consistent with our working solution , that tyc 3010 was a candidate bd ( see figure [ fig : mc10rv_curve_bd ] and table [ tbl : orb_params ] ) . these rv points are well fit by a solution consistent with a substellar object ( @xmath31 ) orbiting in the bd desert around a solar - type star . a robust fit to the low amplitude ( @xmath512 km s@xmath7 ) variations was found with the exofast program @xcite , which uses a set of markov chain monte carlo trials to find the best fit . this solution , shown in figure [ fig : mc10rv_curve_bd ] ( top panel ) , is a very convincing fit to the 38 originally included marvels ( red points ) and apo ( blue points ) measurements . this fit yielded a @xmath32 of 34.63 after scaling the error bars to force @xmath32/dof@xmath51 . these scalings were not unreasonable compared to other marvels candidates . as noted previously , four of the original marvels rv measurements were initially rejected as outliers . the outlier rejection procedure included a 40@xmath26 statistical clipping to avoid phase wrapping , and rejection of consecutive points deviating by a large systematic offset from the bulk of the measurements . the latter rejection step was specifically implemented in an attempt to account for cases of fiber mis - pluggings , which are known to happen on occasion , in which the wrong star is observed for a few observations in a row and those few measurements appear at a very different systemic velocity relative to the majority of the measurements . the four rejected marvels measurements are also shown in figure [ fig : mc10rv_curve_bd ] ( bottom panel , red points ) near hjd 2455250 . the final ( correct ) orbit solution is also shown ( see details below ) , but it must be noted that this final orbit solution is only a good fit after properly disentangling the rvs from epochs where just a single set of spectral lines is resolved ; it is not a good fit to the _ directly observed _ single - lined rv measurements , since these are in fact a flux - weighted average of the true primary and secondary rvs . the six outlier " measurements from this first set of observations ( four marvels points and two arces points ) appear systematically displaced by 1520 km s@xmath7 relative to the other 38 measurements , which are well fit by the spurious orbit ( solid curve ) but not by the correct orbit ( dashed curve ) . ) on the mass of companions as a function of angular separation . with this upper limit , we can rule out the presence of companions above a mass of @xmath27 outside of @xmath550 au , and above a mass of @xmath33 outside of @xmath5100 au . , scaledwidth=38.0% ] lcrcr @xmath34 & & @xmath35 & & @xmath36 + @xmath37 & & @xmath38 & & @xmath39 + @xmath40 & & @xmath41 & & @xmath42 + @xmath43 & & @xmath44 & & @xmath45 + @xmath46 ( km s@xmath7 ) & & @xmath47 & & @xmath48 + @xmath49 ( km s@xmath7 ) & & ... & & @xmath50 + @xmath51 ( km s@xmath7 ) & & @xmath52 & & @xmath53 + @xmath54 & & ... & & @xmath55 + cccc @xmath18 ( 2000 ) & 11 00 11.45 & & ( 1 ) + @xmath56 ( 2000 ) & + 39 43 24.74 & & ( 1 ) + pmra [ mas yr@xmath7 ] & @xmath58 & 1.7 & ( 1 ) + pmde [ mas yr@xmath7 ] & @xmath59 & 1.6 & ( 1 ) + b@xmath60 & @xmath61 & @xmath62 & ( 1 ) + v@xmath60 & @xmath63 & @xmath64 & ( 1 ) + @xmath65 & @xmath66 & @xmath67 & ( 2 ) + @xmath68 & @xmath69 & @xmath70 & ( 2 ) + @xmath71 & @xmath72 & @xmath73 & ( 2 ) + @xmath74 & @xmath75 & @xmath76 & ( 2 ) + @xmath77 & @xmath78 & @xmath79 & ( 2 ) + @xmath80 & @xmath81 & @xmath82 & ( 2 ) + @xmath83 & 9.977 & 0.021 & ( 3 ) + @xmath84 & 9.554 & 0.016 & ( 3 ) + @xmath85 & 9.488 & 0.019 & ( 3 ) + wise1 ( 3.4 @xmath86 m ) & 9.407 & 0.006 & ( 4 ) + wise2 ( 4.6 @xmath86 m ) & 9.482 & 0.006 & ( 4 ) + wise3 ( 12 @xmath86 m ) & 9.470 & 0.038 & ( 4 ) + k , @xmath87 , [ fe / h]@xmath88 , and @xmath89 , yielding a distance of @xmath90 pc . _ bottom : _ a second nextgen fit that uses two stellar components ( corresponding to the primary and secondary stars of tyc 3010 ) with one of the components constrained to the spectroscopically determined stellar parameters for the primary ( @xmath91 k , @xmath92 , [ fe / h]@xmath93 ) . this fit estimates the secondary stellar parameters to be @xmath94 k , @xmath95 , @xmath96 , and the distance to tyc 3010 to be @xmath97 pc , with an @xmath98 ( @xmath99dof@xmath100 ) . + , title="fig:",scaledwidth=35.0% ] k , @xmath87 , [ fe / h]@xmath88 , and @xmath89 , yielding a distance of @xmath90 pc . _ bottom : _ a second nextgen fit that uses two stellar components ( corresponding to the primary and secondary stars of tyc 3010 ) with one of the components constrained to the spectroscopically determined stellar parameters for the primary ( @xmath91 k , @xmath92 , [ fe / h]@xmath93 ) . this fit estimates the secondary stellar parameters to be @xmath94 k , @xmath95 , @xmath96 , and the distance to tyc 3010 to be @xmath97 pc , with an @xmath98 ( @xmath99dof@xmath100 ) . + , title="fig:",scaledwidth=35.0% ] in addition , as we have done with all marvels candidates , we performed a fit to the spectral energy distribution ( sed ) of the system to verify that it is consistent with a single stellar source and to provide a consistency check on the spectroscopically determined stellar properties ( see below ) . we constructed the sed using fluxes ( see table [ tbl : mc10ab_cat_props ] ) from the tycho catalogue @xcite , apass ( _ _ a__avso _ _ p__hotometric _ _ a__ll-__s__ky _ _ s__urvey ; data release 6 , see * ? ? ? * ) , two micron all sky survey @xcite , and _ wise _ @xcite . nextgen models ( hauschildt et al . 1999 ) are used to generate theoretical seds by holding @xmath101 , @xmath102 , and [ fe / h ] at the spectroscopically determined values ( see below ) , and the maximum extinction @xmath103 was limited to 0.05 mag based on the dust maps of @xcite . the best fit model can be seen in the top panel of figure [ fig : sed ] ; it corresponds to an @xmath103 of @xmath104 , and a distance of @xmath105 pc . this single - star sed fit to the available photometry spanning 0.212@xmath86 m is quite good , with the only hint of a discrepancy being a mild excess that appears in the _ galaxy evolution explore ( galex ) _ near - uv ( nuv ) passband , despite the lack of any strong emission in the observed ca hk lines . however , this by itself was not deemed to be a compelling reason to suspect the high quality orbit solution . thus , at this point in our analysis , fully 38 rv measurements from two separate instruments were well fit by the same orbit solution of a single , solar - type star with a @xmath106 companion on a modestly eccentric orbit . the sed of tyc 3010 was furthermore consistent with being a single solar - type star , and the lack of any companions in the high - resolution imaging ruled out a blend scenario in which the rv variations might be caused by a binary beyond 0.5@xmath30 of the line of sight . only four of the discovery rv measurements appeared to be discrepant , and these were rejected for what appeared to be good reasons , behaving not unlike fiber mis - pluggings that the marvels team had observed in other stars before . however , the last two rv measurements from the first set of arces observations appeared as strong outliers ( see figure [ fig : mc10rv_curve_bd ] , blue points near hjd 2455730 ) . as they were observed with a standard echelle spectrograph , these could not be attributed to fiber mis - pluggings , and inspection of the ccfs revealed double lines ( see bottom panel of figure [ fig : ccfs ] ) , immediately nullifying the bd companion hypothesis . 238 days , an eccentricity of @xmath50.79 , with @xmath10715.38 km s@xmath7 and @xmath10817.50 km s@xmath7 . finally , for the rv points outside of periastron , it was necessary to de - blend the observed rvs with the method described in section [ sec : rv_fitting ] . + , scaledwidth=45.0% ] to further confirm that tyc 3010 was indeed a stellar binary , we closely observed the next peripassage with the hrs spectrograph on het . with het , we obtained complete coverage of periastron , permitting a complete double - lined orbit solution . in this section we present the correct orbit solution for tyc 3010 , including all the points from the discovery and subsequent data , which shows that tyc 3010 is an sb2 with a period of @xmath3 days , an eccentricity of @xmath109 , and a mass ratio of @xmath11 . with this eccentricity and orbital period , tyc 3010 lies near the upper bound of ( but within ) the distribution of orbital eccentricities of solar - type binaries with orbital periods of 100300 days ( see , e.g. , * ? ? ? * ; * ? ? ? the orbital parameters for the binary are summarized in table [ tbl : orb_params ] , the rv solution is shown in figure [ fig : mc10rv_curve_sb ] , and a schematic of the orbit is shown in figure [ fig : mc10_schematic ] . in this section we also describe our determination of the stellar parameters for the primary in tyc 3010 , and we estimate its mass and radius using the relations described in @xcite . since the secondary is comparable in mass to the primary , we had to take special care in accounting for the flux contamination from the secondary , both in our determination of the stellar parameters and with the rv values that we measured for the system outside of periastron . for the orbital solution of the binary , we used the rv fitting software described in @xcite . since we do not resolve two sets of spectral lines for the phases outside of periastron , most of the rv points correspond to a flux - weighted average of the primary and secondary rvs . in order to de - blend the flux - weighted rvs that we measured , and derive the corresponding primary rvs , we used the following prescription . we treat the blended velocities as a flux - weighted average of the primary and secondary velocities : @xmath110 where @xmath111 and @xmath112 are the primary and secondary velocities respectively , and @xmath113 and @xmath114 are the primary and secondary fluxes . we normalize the flux weights by setting the sum of the fluxes , @xmath115 , to unity . using the flux ratio , @xmath116 , from the todcor analysis ( which was only performed for the het / hrs epochs where it was possible to resolve two sets of spectral lines ) , we can solve for @xmath113 and @xmath114 in terms of @xmath18 : @xmath117 in addition , we can use the mass ratio , @xmath118 , from the rv solution to write @xmath112 in terms of @xmath111 , since @xmath119 . @xmath120 returning to ( [ eqn : vblend ] ) , we can now write @xmath121 with equation ( [ eqn : deblend ] ) , we can iteratively solve for a final set of de - blended rvs for the primary . for the first iteration , we provide an initial guess for @xmath122 by performing a joint fit to the primary rvs ( blended@xmath123unblended ) combined with the secondary rvs ( unblended ; only measured during periastron ) . inserting this initial guess for @xmath122 into equation ( [ eqn : deblend ] ) , we derive an initial set of de - blended primary rvs . then we perform another joint fit to the primary ( de - blended+unblended ) and secondary ( unblended ) rvs to refine our value for @xmath122 . we repeat the process until @xmath122 converges . the value we find for @xmath122 ( @xmath124 ) from this de - blending analysis is in excellent agreement with the value for @xmath122 ( @xmath50.88 ) that we found from the ratio of the primary and secondary rvs that were measured for the 11 het / hrs epochs where two peaks were resolved in the ccfs . thus , @xmath122 has been determined very precisely by the orbital solution ( better than 3% ) , and is more precise than the individual quoted errors on the masses . as a further consistency check on @xmath18 and @xmath122 , we also note that according to the relationship between mass and bolometric luminosity from @xcite , there should be a relationship between @xmath18 and @xmath122 . since @xmath18 is derived from a set of finite wavelength bands , it is not bolometric . however , since the stars have temperatures that are not too dissimilar , @xmath18 is likely to be approximately equal to the ratio of the bolometric luminosities . for stars with @xmath125 , a fit to the @xcite data yields @xmath126 . thus , @xmath127 , so @xmath128 , which is within 3@xmath26 of the value obtained from the rv analysis . the stellar parameters for the primary were determined with a double - lined spectrum obtained near periastron ( see section [ sec : arces_data ] ) . the spectroscopic analysis used to determine the atmospheric parameters is similar to the one described in @xcite , where we use two independent methods that require the conditions of excitation and ionization equilibria for and lines . these methods are referred to as the `` bpg '' ( brazilian participation group ) method and the `` iac '' ( instituto de astrofsica de canarias ) method . crr parameter & value & uncertainty + @xmath129 & 0.335 & 0.035 + @xmath54 & 0.878 & 0.016 + a@xmath130 & 0.03 & 0.02 + @xmath131 ( pc ) & 225 & 40 + & tyc 3010 a & tyc 3010 b + @xmath24 ( k ) & @xmath132 & @xmath133 + @xmath102 ( cgs ) & @xmath134 & @xmath135 + @xmath136fe / h@xmath137 $ ] & @xmath138 & ... + @xmath139 ( @xmath140 ) & @xmath141 & @xmath142 + @xmath143 ( @xmath144 ) & @xmath145 & @xmath146 the `` bpg '' analysis was done in local thermodynamic equilibrium ( lte ) using the 2002 version of moogchris / moog.html ] @xcite and one - dimensional plane - parallel model atmospheres interpolated from the odfnew grid of models @xcite . in previous marvels papers ( e.g. , * ? ? ? * and references therein ) , the equivalent widths ( ews ) of the fe lines were determined in an automated fashion . however , in this case , the ews were manually measured to carefully account for visible blends on the fe lines from the secondary s spectrum . we note that contaminations from very weak lines could have affected the ew measurements . in order to correct the ews measured for the primary for the veiling from the continuum flux of the secondary star , we followed a procedure similar to the one described in section 5.2.1 of @xcite . according to their prescription , we can relate the value of the true equivalent width ( ew@xmath147 ) of a given line to the observed equivalent width ( ew@xmath148 ) through the following relationship , @xmath149 where @xmath150 is the so - called veiling factor for the primary . the veiling factors for the two components are related by @xmath151 where @xmath152 and @xmath153 are the fluxes for the primary and secondary . furthermore , the veiling factors satisfy the equation @xmath154 to simplify our analysis , we treated the veiling factors and flux ratio as if they were wavelength independent . using the average flux ratio derived by todcor ( @xmath155 ; see section [ sec : todcor ] ) , and the added constraint from equation [ eqn : veiling_norm ] , we find the veiling factor for the primary to be @xmath156 . thus , after correcting the ews , we find the stellar parameters to be @xmath91 k , @xmath92 , and [ fe / h]@xmath93 ( see table [ tbl : mc10ab_sys_props ] ) . the uncertainties for these parameters are larger than the typical errors that we achieve with our spectroscopic analysis because of the flux contamination from the secondary star . the `` iac '' analysis extracted the stellar parameters of the primary and secondary stars by considering veiling factors that were wavelength - dependent . these veiling factors are estimated using low - resolution kurucz fluxes ( * ? ? ? * and references therein ) and the following equation : @xmath157 where @xmath158 and @xmath159 correspond to the surface brightness of the primary and the secondary respectively . to determine the ratio of the radii , we derived an empirical mass radius relationship from a sample of 55 stars from @xcite , with the masses restricted to 0.7 m@xmath160 1.4 m@xmath161 . we fit a function to the data of the form @xmath162\ ] ] where @xmath163 and @xmath164 . thus , the ratio of the radii for the components of tyc 3010 can be written as @xmath165 the mass ratio was determined from the todcor analysis to be @xmath166 , so we find that @xmath167 . as a first guess , we adopt the above values to estimate the stellar mass and radius of the primary @xcite , from solar - scaled theoretical isochrones @xcite . the mass ratio allows us to derive a first guess of the @xmath168 value for the secondary to be roughly 5100 k , assuming @xmath169 and the same metallicity as the primary . the stellar radii we get from the comparison with isochrones are 0.89 @xmath170 and 0.77 @xmath170 , and thus the ratio is @xmath171 , which is very similar to the value previously estimated ( @xmath172 ) . thus , the derived veiling factors lie in the range @xmath173 and @xmath174 in the spectral region 45007000 . we then measure automatically , using the code ares @xcite , the ews of the and lines @xcite for both stellar components and correct them using the wavelength - dependent veiling factors . we then use the code stepar @xcite to automatically derive the stellar parameters of each component and we get @xmath175 k , @xmath176 , [ fe / h]@xmath177 and @xmath178 from 162 and 18 lines . the uncertainties are unexpectedly large and may be due to the contamination of neighboring lines of other elements of the companion star . thus the results for the secondary are fairly tentative and the errors are even larger . we were only able to measure 64 and 3 lines to get @xmath179 k , @xmath180 , [ fe / h]@xmath181 and @xmath182 . compared to the `` bpg '' analysis , the lower @xmath183 of the primary may be related to the different methods used to derive the veiling factors . nevertheless , the `` iac '' stellar parameters for the primary star are very similar to those previously derived and are actually consistent within the large uncertainties so we decide to adopt the `` bpg '' values . , @xmath184 ) , and the error bars correspond to the 68.27% confidence intervals . the contours are lines of equal probability density which enclose 68% , 90% , and 95% of the cumulative probability relative to the maximum of the probability density . in the top and right panels , the probability distribution ( solid line ) and cumulative probability ( dashed line ) are shown for the mass and radius respectively.,scaledwidth=38.0% ] with the `` bpg '' stellar parameters for the tyc 3010 primary , we again performed a fit to the observed sed of the system as in section [ sec : spurious ] , but now also including the contribution of the secondary star . once again , nextgen models @xcite are used to generate theoretical seds by holding @xmath101 , @xmath102 , and [ fe / h ] at the spectroscopically determined values for the primary , while the @xmath101 for the secondary is found by the value that minimizes @xmath185 ( @xmath99dof@xmath100 ) . the best fit model can be seen in the bottom panel of figure [ fig : sed ] ; it corresponds to an @xmath103 of @xmath186 , and a distance of @xmath187 pc . compared to the sed fit performed in section [ sec : spurious ] , which assumed a single stellar contribution , this two - component sed fit no longer exhibits an excess in the _ galex _ nuv passband , and more generally is an excellent fit to all of the available photometry . finally , from this two - component fit to the sed , we also obtain a set of values for the stellar parameters of the secondary of tyc 3010 . we find that @xmath188 k , @xmath189 , and [ fe / h]@xmath190 . and [ fe / h]@xmath191 . ages ( in gyr ) of 1.0 , 5.0 , 8.0 , and 11.0 are represented by blue dots , and the 1@xmath26 deviations from the evolutionary track are shown in the shaded region.,scaledwidth=34.0% ] given the spectroscopic stellar parameters , we can derive the mass and radius of the tyc 3010 primary star using the empirical relationships described in @xcite . figure [ fig : mc10_mass_radius ] shows the result of a set of mcmc trials for the best estimate of the mass and radius . for the precise parameters of the primary ( @xmath192 k , @xmath193 , [ fe / h]=0.09 ) , the torres relations give 0.98 @xmath140 and 0.75 @xmath144 . once one includes the fairly large uncertainties in the stellar parameters , the median values for the mass and radius become @xmath194 and @xmath195 , respectively . the means are @xmath196 and @xmath197 , so the distributions are quite skewed as shown in figure [ fig : mc10_mass_radius ] . compared to a yonsei - yale evolutionary track ( see figure [ fig : y2_models ] ) , we do not have a strong constraint on the age , but tyc 3010 is unlikely to have evolved off the main sequence . we can also derive the mass and radius for the secondary given the stellar parameters determined from the two - component sed fit and the @xcite relations . we find that @xmath198 and @xmath199 . this value for the mass of the secondary agrees within 1@xmath26 of the value that can be derived using the primary mass we determined above and the mass ratio from the rv solution , i.e. , @xmath200 . the rv signal from tyc 3010 initially seemed to indicate that it was a bd orbiting a solar - type star in the bd desert . over 80% of the marvels discovery data agreed with this interpretation , and there seemed to be plausible reasons for excluding the outliers . however , once similar outliers were found in the subsequent observations , we began to suspect the validity of the bd interpretation . in this section , we discuss in detail why we initially favored the bd interpretation , as well as how this conclusion was abruptly overturned by a few surprising data points . in the discovery data , there were four outliers in total , each offset by @xmath520 km s@xmath7 from the rest of the data . the most anomalous of the outliers was extracted from a spectrum with a low s / n , so its rv value did not seem trustworthy . the remaining outliers ( considering that they corresponded to a @xmath520 km s@xmath7 offset in rv that was only captured once during the three orbits contained in the discovery data ) , also seemed likely to be spurious . the marvels spectrograph is a fiber - fed spectrograph that can observe 60 objects simultaneously . each fiber is plugged by hand to observe the correct target , and occasionally a mistake may occur . indeed , the marvels data vetting procedures were evolved to specifically include an outlier rejection step that sought to mitigate such errors , by searching for consecutive strings of measurements that were offset from the bulk of the data in a similar fashion to how these four measurements behave . remarkably , excluding these few apparent `` outliers''and in fact _ only _ by excluding them permits a convincing orbit solution . it is not intuitive that this should be the case , in particular because only @xmath515% of the measurements are excluded ( including both the discovery data and the initial follow - up data which appeared to corroborate the spurious solution ) and because the resulting solution is so dramatically different from the true solution . evidently , a system such as tyc 3010 ( with its extreme eccentricity , leading to punctuated large rv excursions , and its orbital orientation being nearly perpendicular to the line of sight , leading to very small rv variations for @xmath595% of the orbit ) is able to mimic a more circular orbit of a low - mass companion about a single star . moreover , the similarity of the two stars in tyc 3010 leads to a combined light sed that is only slightly different from that of a single star at a nearer distance . thus many lines of evidence supported the initial solution , considering that the bd interpretation appeared to be supported by two years of discovery rv data , six months of additional rv observations , lucky imaging , and a well - constrained sed . indeed , when the two follow - up rv measurements observed near periastron appeared , indicating a possible problem with the original orbit solution , we began to search for reasons to suspect the validity of these two anomalous points . at first , we thought the situation might be similar to the fiber mis - pluggings believed to have occurred with the discovery data , and we considered that the arces outliers were the result of pointing at the wrong star . but after investigating the data from those two nights , we confirmed that we had observed the correct target . next we learned of a recent change that had been made to the arces instrument : the thar lamp had recently been replaced . the thar lamp is used to perform the wavelength calibration , and it was plausible that the new lamp might have caused problems with the wavelength solution . therefore , the arces outliers may have merely been the result of an artificial doppler shift generated by an incorrect wavelength solution . in the end , we were only able to accept that the bd interpretation was incorrect after we inspected the ccf for each of the outliers . the ccfs for the outliers both showed two peaks instead of one , indicating the presence of a second stellar component . furthermore , the secondary peak was comparable in height to the primary peak ( see bottom panel of figure [ fig : ccfs ] ) , which led us to suspect that tyc 3010 was in fact a spectroscopic stellar binary . but how did most of the data that we had for tyc 3010 conspire to imply that it was a much less massive system ? the period , shape , and orientation of the orbit with respect to the line of sight ( see figure [ fig : mc10_schematic ] ) made it such that for most of the orbit the two stars possess relatively low rvs with respect to each other . in particular , the difference between the magnitude of their rvs is smaller than the typical ccf width for our instruments , resulting in their ccf peaks being blended into one . since the flux ratio is not too different from unity , and the mass ratio is also close to unity , for epochs where the spectral lines are blended , there is a near - cancellation ( or strong suppression ) of the true orbital velocities for the primary and secondary , which are nearly equal in magnitude but oppositely signed ( see equation [ eqn : deblend ] , and recall that @xmath201 is what we actually measure ) . thus , for @xmath595% of the orbit , the amplitude of the variations ( @xmath512 km s@xmath7 ) suggest a bd companion to a solar - type star ; furthermore , the eccentricity and the orbital period ensure that the stars spend a long time ( @xmath57 months ) away from periastron , which is precisely the moment when the rvs of the components are disparate enough for it to be fairly easy to resolve the two sets of spectral lines , and the large rv amplitude ( @xmath51520 km s@xmath7 ) is indicative of a stellar binary with two solar - type stars . moreover , the orientation makes it so that only a relatively small component of the orbital velocities is directed along our line of sight . finally , the cadence of the marvels survey made it unlikely to observe multiple epochs of periastron . for any given rv survey , the lower the resolution of the spectrograph , the more vigilant one must be for these kinds of false positives . for tyc 3010 in particular , a spectrograph with a resolution of @xmath202 is required to resolve the spectral lines throughout most of the orbit . but in general , as the resolution ( and cadence of observations ) decreases , the wider the range of eccentricities , arguments of periastron , and orbital periods by which stellar binaries could masquerade as substellar companions for significant fractions of their orbits . furthermore , longer period orbits ( @xmath203yr ) should be handled with special care , for in these cases the phase coverage is more likely to be incomplete . in order to survey @xmath53,000 stars over four years , marvels required a cadence that made it less likely to observe multiple epochs of periastron for a binary with the period of tyc 3010 . for marvels and similar rv surveys for substellar companions , it can be costly to use precious resources to examine false positives . therefore , in this section , we describe a method that the marvels team currently employs to identify binaries like tyc 3010 during the candidate - vetting process . for typical rv surveys today , a standard line bisector analysis can usually be performed to assess the presence of blended double - lined binaries . however , this was not possible for the marvels discovery data due to its limited spectral resolution . thus , following our experience with tyc 3010 , marvels has developed an internal pipeline for inspecting the widths of the ccf peaks for all of our candidates . this way , we can readily monitor the ccfs for signs that indicate that there may be more than one stellar component present ( e.g. , the large excursions in the width of the ccf peak that occur near periastron for tyc 3010 ; see figure [ fig : ccf_widths ] ) . there are two properties of the ccfs that we now monitor : ( 1 ) the average width of the ccf peak compared to other stars in the survey , and ( 2 ) any other significant changes in the shape of the ccf over time . for a typical solar - type star that is not rotating too rapidly ( i.e. , the kinds of stars that marvels targets ) , one would expect the width of the ccf peak to be @xmath510 km s@xmath7 , which is largely the result of thermal broadening and micro - turbulence . however , when binary systems like tyc 3010 are unresolved , the widths of the ccf peak are broader ( @xmath520 km s@xmath7 ) , indicating that there may be multiple stellar components contributing to the flux from the system ( see figure [ fig : ccf_widths ] ) . in fact , an atypically broad ccf peak could also be the result of a single star rotating atypically fast , so a broad peak is not in itself sufficient to identify the system as a binary . nevertheless , a broad peak should be taken as a sign to proceed with caution . furthermore , changes in the skewness of the ccf peak might provide an even more sensitive diagnostic for these kinds of systems . thus , by monitoring changes in the ccf peak , even if one misses the small fraction of the orbit where , depending on the resolution , the ccf peak either broadens dramatically or separates into distinct peaks ( or if one is suspicious of the relatively few epochs where the system happened to be caught near periastron ) , it is possible to flag systems like tyc 3010 , which may contain much more mass than most of the rv data suggests . the case of tyc 3010 is also a pertinent lesson on how important it is to handle outliers carefully , especially in this era of large surveys where thousands of objects must be screened for the most favorable candidates . we possessed plausible reasons for suspecting that the outliers in the discovery data might be spurious ( known issues with fiber mis - pluggings ; low s / n ; and the outliers were only detected during one of the three orbits observed ) . moreover , and perhaps ironically , the spurious orbit solution is actually a better fit to the discovery data ( excluding the outliers ) than the true orbit solution , because of the need to disentangle the primary and secondary rv components from the ( apparently ) single - lined rv measurements . however , even when faced with such a compelling initial solution and sensible reasons for considering the outliers to be invalid , it is imperative to investigate further and provide evidence that the reasons for rejecting the outliers are not only plausible but justified . furthermore , when the analysis is distributed among multiple team members like it is within marvels , it is necessary to make sure each step of the analysis is documented as clearly as possible . for marvels , the members who perform the candidate - vetting are usually different from those who perform the subsequent analysis for each candidate , so it is important for each team member to be able to readily discover if any outliers were rejected and why . marvels has now modified its internal analysis tracking system in order to make the entire analysis process more transparent . finally , if we had been monitoring the widths of the ccf peaks , we could have considered the evidence of the broad peak , as well as the changing peak width around periastron , though in truth neither the changing width nor the broad peak by themselves would have likely been sufficiently compelling to reject the initial orbit solution . in the end , the most important part of our analysis was to strategically focus our het / hrs observations on periastron , the phase where the outliers occurred and where it was easiest to resolve the spectral lines . this strategy would have been more difficult with a conventionally scheduled telescope , but was readily achieved with the queue - scheduled nature of the het . we have demonstrated , using high resolution spectroscopy , that tyc 3010 is an sb2 . we have shown how , with a spectrograph below a given resolution ( @xmath204 ) , the eccentricity and the orientation of the system with respect to our line - of - sight allowed a large fraction of the rv curve to appear remarkably similar to the kind of signal one would expect from a bd secondary as opposed to a stellar - mass secondary . furthermore , as a result of the cadence of the marvels survey and the orbital period of the system , we were more likely to miss periastron during a given orbit . thus , we were more susceptible to rejecting the periastron points we did obtain as outliers , even though these points are where the spectral lines are most widely separated , and thereby where it is easiest to determine that the system is an sb2 . finally , we concluded with a word of warning to rv surveys , since for a given resolution and cadence , there are a range of orbital parameters that can make a stellar - mass binary companion appear to be substellar . the lower the resolution or cadence , the greater the number of stellar binaries that can masquerade in a fashion similar to tyc 3010 . therefore , if other surveys can carefully monitor the widths of the ccf peaks for their targets ( or monitor their line bisectors if they have high enough resolution ) , and when possible , focus their resources on observations of peripassage , then we hope that they will be able to avoid similar astrophysical false positives . this research was partially supported by the vanderbilt initiative in data - intensive astrophysics ( vida ) and nsf career grant ast 0349075 ( cem , kgs , lh , jp ) , nsf aapf ast 08 - 02230 ( jpw ) , nsf career grant ast 0645416 ( ea ) , cnpq grant 476909/2006 - 6 ( gfpm ) , faperj grant apq1/26/170.687/2004 ( gfpm ) , nsf career grant ast-1056524 ( bsg , jde ) , and a papdrj capes / faperj fellowship ( lg ) . based on observations with the sdss 2.5-meter telescope . funding for the marvels multi - object doppler instrument was provided by the w.m . keck foundation and nsf grant ast-0705139 . the marvels survey was partially funded by the sdss - iii consortium , nsf grant ast-0705139 , nasa with grant nnx07ap14 g and the university of florida . the center for exoplanets and habitable worlds is supported by the pennsylvania state university , the eberly college of science , and the pennsylvania space grant consortium . data presented herein were obtained at the hobby - eberly telescope ( het ) , a joint project of the university of texas at austin , the pennsylvania state university , stanford university , ludwig - maximilians - universitt mnchen , and georg - august - universitt gttingen . the het is named in honor of its principal benefactors , william p. hobby and robert e. eberly . this research has made use of the simbad database , operated at cds , strasbourg , france , and the aavso photometric all - sky survey ( apass ) , funded by the robert martin ayers sciences fund . it also made use of the iraf software distributed by the national optical astronomy observatory , which is operated by the association of universities for research in astronomy ( aura ) under cooperative agreement with the national science foundation . this publication makes use of data products from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by the national aeronautics and space administration and the national science foundation . we also make use of data products from the wide - field infrared survey explorer , which is a joint project of the university of california , los angeles , and the jet propulsion laboratory / california institute of technology , funded by the national aeronautics and space administration . funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s . department of energy office of science . the sdss - iii web site is http://www.sdss3.org/. sdss - iii is managed by the astrophysical research consortium for the participating institutions of the sdss - iii collaboration including the university of arizona , the brazilian participation group , brookhaven national laboratory , university of cambridge , university of florida , the french participation group , the german participation group , the instituto de astrofisica de canarias , the michigan state / notre dame / jina participation group , johns hopkins university , lawrence berkeley national laboratory , max planck institute for astrophysics , new mexico state university , new york university , ohio state university , pennsylvania state university , university of portsmouth , princeton university , the spanish participation group , university of tokyo , university of utah , vanderbilt university , university of virginia , university of washington , and yale university . , j. , lee , b. , de lee , n. , et al . 2009 , in society of photo - 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we report the discovery of a highly eccentric , double - lined spectroscopic binary star system ( tyc 3010 - 1494 - 1 ) , comprising two solar - type stars that we had initially identified as a single star with a brown dwarf companion . at the moderate resolving power of the marvels spectrograph and the spectrographs used for subsequent radial - velocity ( rv ) measurements ( @xmath0 ) , this particular stellar binary mimics a single - lined binary with an rv signal that would be induced by a brown dwarf companion ( @xmath1 ) to a solar - type primary . at least three properties of this system allow it to masquerade as a single star with a very low - mass companion : its large eccentricity ( @xmath2 ) , its relatively long period ( @xmath3 days ) , and the approximately perpendicular orientation of the semi - major axis with respect to the line of sight ( @xmath4 ) . as a result of these properties , for @xmath595% of the orbit the two sets of stellar spectral lines are completely blended , and the rv measurements based on centroiding on the apparently single - lined spectrum is very well fit by an orbit solution indicative of a brown dwarf companion on a more circular orbit ( @xmath6 ) . only during the @xmath55% of the orbit near periastron passage does the true , double - lined nature and large rv amplitude of @xmath515 km s@xmath7 reveal itself . the discovery of this binary system is an important lesson for rv surveys searching for substellar companions ; at a given resolution and observing cadence , a survey will be susceptible to these kinds of astrophysical false positives for a range of orbital parameters . finally , for surveys like marvels that lack the resolution for a useful line bisector analysis , it is imperative to monitor the peak of the cross - correlation function for suspicious changes in width or shape , so that such false positives can be flagged during the candidate vetting process .

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Proceed to summarize the following text: anomaly - mediated supersymmetry breaking ( amsb ) models have received much attention in the literature due to their attractive properties@xcite : the soft supersymmetry ( susy ) breaking terms are completely calculable in terms of just one free parameter ( the gravitino mass , @xmath11 ) , the soft terms are real and flavor invariant , thus solving the susy flavor and @xmath12 problems , the soft terms are actually renormalization group invariant@xcite , and can be calculated at any convenient scale choice . in order to realize the amsb set - up , the hidden sector must be `` sequestered '' on a separate brane from the observable sector in an extra - dimensional universe , so that tree - level supergravity breaking terms do not dominate the soft term contributions . such a set - up can be realized in brane - worlds , where susy breaking takes place on one brane , with the visible sector residing on a separate brane . the soft susy breaking ( ssb ) terms arise from the rescaling anomaly . in spite of its attractive features , amsb models suffer from the well - known problem that slepton mass - squared parameters are found to be negative , giving rise to tachyonic states . the original solution to this problem is to suppose that scalars acquire as well a universal mass @xmath13 , which when added to the amsb ssb terms , renders them positive . thus , the parameter space of the `` minimal '' amsb model ( mamsb ) is given by m_0 , m_3/2 , , sign ( ) . an alternative set - up for amsb has been advocated in ref . @xcite , known as hypercharged anomaly - mediation ( hcamsb ) . it is a string motivated scenario which uses a similar setup as the one envisioned for amsb . in hcamsb , susy breaking is localized at the bottom of a strongly warped hidden region , geometrically separated from the visible region where the mssm resides . the warping suppresses contributions due to tree - level gravity mediation@xcite and the anomaly mediation@xcite can become the dominant source of susy breaking in the visible sector . possible exceptions to this sequestering mechanism are gaugino masses of @xmath14 gauge symmetries @xcite . thus , in the mssm , the mass of the bino the gaugino of @xmath15 can be the only soft susy breaking parameter not determined by anomaly mediation@xcite . depending on its size , the bino mass @xmath16 can lead to a small perturbation to the spectrum of anomaly mediation , or it can be the largest soft susy breaking parameter in the visible sector : as a result of rg evolution its effect on other soft susy breaking parameters can dominate the contribution from anomaly mediation . in extensions of the mssm , additional @xmath17s can also communicate susy breaking to the mssm sector @xcite . besides sharing the same theoretical setup , anomaly mediation and hypercharge mediation cure phenomenological shortcomings of each other . the minimal amsb model predicts a negative mass squared for the sleptons ( and features relatively heavy squarks ) . on the other hand , the pure hypercharge mediation suffers from negative squared masses for stops and sbottoms ( and features relatively heavy sleptons ) : see sec . [ sec : pspace ] . as a result , the combination of hypercharge and anomaly mediation leads to phenomenologically viable spectra for a sizable range of relative contributions @xcite . we parametrize the hcamsb ssb contribution @xmath18 using a dimensionless quantity @xmath2 such that @xmath19 so that @xmath2 governs the size of the hypercharge contribution to soft terms relative to the amsb contribution . then the parameter space of hcamsb models is given by , m_3/2 , , sign ( ) . in the hcamsb model , we assume as usual that electroweak symmetry is broken radiatively by the large top - quark yukawa coupling . then the ssb @xmath20 term and the superpotential @xmath21 term are given as usual by the scalar potential minimization conditions which emerge from requiring an appropriate breakdown of electroweak symmetry . in hcamsb , we take the ssb terms to be of the form : m_1 & = & _ 1+m_3/2,m_a & = & m_3/2 , a=2 , 3 m_i^2 & = & -14\ { _ g+_f}m_3/2 ^ 2 a_f & = & m_3/2 , where @xmath22 , @xmath23 is the beta function for the corresponding superpotential coupling , and @xmath24 with @xmath25 the wave function renormalization constant . the wino and gluino masses ( @xmath26 and @xmath27 ) receive a contribution from the bino mass at the two loop level . thus , in pure hypercharge mediation , they are one loop suppressed compared to the scalar masses . for convenience , we assume the above ssb mass parameters are input at the gut scale , and all weak scale ssb parameters are determined by renormalization group evolution . we have included the above hcamsb model into the isasugra subprogram of the event generator isajet v7.79@xcite . after input of the above parameter set , isasugra then implements an iterative procedure of solving the mssm rges for the 26 coupled renormalization group equations , taking the weak scale measured gauge couplings and third generation yukawa couplings as inputs , as well as the above - listed gut scale ssb terms . isasugra implements full 2-loop rg running in the @xmath28 scheme , and minimizes the rg - improved 1-loop effective potential at an optimized scale choice @xmath29@xcite to determine the magnitude of @xmath21 and @xmath30 . all physical sparticle masses are computed with complete 1-loop corrections , and 1-loop weak scale threshold corrections are implemented for the @xmath31 , @xmath32 and @xmath33 yukawa couplings@xcite . the off - set of the weak scale boundary conditions due to threshold corrections ( which depend on the entire superparticle mass spectrum ) , necessitates an iterative up - down rg running solution . the resulting superparticle mass spectrum is typically in close accord with other sparticle spectrum generators@xcite . once the weak scale sparticle mass spectrum is known , then sparticle production cross sections and branching fractions may be computed , and collider events may be generated . then , signatures for hcamsb at the cern lhc may be computed and compared against standard model ( sm ) backgrounds . our goal in this paper is to characterize the hcamsb parameter space and sparticle mass spectrum , and derive consequences for the cern lhc @xmath34 collider , which is expected to begin operation in fall , 2009 . some previous investigations of mamsb at lhc have been reported in ref . @xcite . the remainder of this paper is organized as follows . in sec . [ sec : pspace ] , we calculate the allowed parameter space of hcamsb models , imposing various experimental and theoretical constraints . we also show sample mass spectra from hcamsb models , and show their variation with @xmath2 and @xmath11 . we show typical values of @xmath35 and @xmath36 that result . in sec . [ sec : lhc ] , we explore consequences of the hcamsb model for lhc sparticle searches . typically , collider events are characterized by production of high @xmath37 @xmath32 and @xmath31 quarks , along with @xmath38 and observable tracks from late decaying charginos @xmath39 . for small @xmath2 , slepton pair production may be visible , while for large @xmath2 , direct @xmath40 and @xmath41 production may be visible . the lhc reach for 100 fb@xmath3 should extend up to @xmath42 ( 105 ) tev , corresponding to a reach in @xmath43 ( 2.2 ) tev , for small ( large ) values of @xmath2 . the hcamsb model should be easily distinguishable from the mamsb model at the lhc if @xmath6 is not too large , due to the presence of @xmath44 candidates in cascade decay events . the presence of these reflects the mass ordering @xmath0 in the hcamsb model , while @xmath45 in the mamsb model . in sec . [ sec : conclude ] , we present our conclusions and outlook for hcamsb models . we begin our discussion by plotting out in fig . [ fig : m10 ] the mass spectra of various sparticles versus _ a _ ) . @xmath46 in mamsb and _ b _ ) . @xmath2 in the hcamsb model , for @xmath11 fixed at 50 tev , while taking @xmath47 , @xmath48 and @xmath49 gev . for @xmath13 and @xmath50 , the yellow - shaded region yields the well - known tachyonic slepton mass - squared values , which could lead to electric charge non - conservation in the scalar potential . in mamsb , as @xmath13 increases , all the scalars increase in mass , while @xmath51 , @xmath52 and @xmath53 remain roughly constant , and the superpotential @xmath21 term decreases . the large @xmath13 limit of parameter space is reached around @xmath54 , where ewsb is no longer properly broken ( signaled by @xmath55 ) . we also see the well - known property of mamsb models that @xmath56 . in addition , an important distinction between the two models is the mass ordering which enters into the neutralino mass matrix : we find typically that @xmath0 in the hcamsb model , while @xmath1 in mamsb . thus , both models will have a wino - like @xmath57 state . however , in the hcamsb model , the @xmath58 are dominantly higgsino - like states , with @xmath59 being bino - like , while in the mamsb model , we expect @xmath60 to be bino - like with @xmath61 being higgsino - like . this mass ordering difference will give rise to a crucial distinction in lhc susy cascade decay events ( see sec . [ sec : lhc ] ) which may serve to distinguish the two models . in the hcamsb case , as @xmath2 increases , the gut scale gaugino mass @xmath16 increases . thus , the bino mass increases with @xmath2 , while the light charginos @xmath39 and neutralino @xmath57 remain wino - like with mass fixed near @xmath26 , and the gluino remains with mass fixed at nearly @xmath62 . many of the scalar masses also vary with @xmath2 . the reason is that as @xmath2 increases , so does the gut scale value of @xmath16 . the large value of @xmath16 feeds into the scalar masses via their renormalization group equations , causing many of them to increase with @xmath2 , with the largest increases occurring for the scalars with the largest weak hypercharge assignments @xmath63 . thus , we see strong increases in the @xmath64 , @xmath65 and especially the @xmath66 masses with increasing @xmath67 . the @xmath68 squark only receives a small increase in mass , since its hypercharge value is quite small : @xmath69 . from fig . [ fig : m10]_b _ ) . , we already see an important distinction between mamsb and hcamsb models : in the former case , the @xmath65 and @xmath66 states are nearly mass degenerate , while in the latter case these states are highly split , with @xmath70 . an exception to the mass increase with @xmath2 in fig . [ fig : m10]_b _ ) . occurs in the values of @xmath71 and @xmath72 . in these cases , the large increase in @xmath73 feeds into the rge @xmath74 term@xcite , and _ amplifies _ the top - quark yukawa coupling suppression of the @xmath75 term . since the doublet @xmath76 contains both the @xmath77 and @xmath78 states , both of these actually suffer a _ decrease _ in mass with increasing @xmath2 . thus , we expect in hcamsb models with moderate to large @xmath2 that the third generation squark states will be highly split . for large @xmath67 , we expect the light third generation squarks @xmath79 and @xmath80 to be quite light , with a dominantly left- squark composition . the heavier squarks @xmath81 and @xmath82 will be quite heavy , and dominantly right - squark states . in addition , we see from fig . [ fig : m10]_b _ ) . that the superpotential @xmath21 term _ decreases _ with increasing @xmath2 . at moderate - to - large @xmath6 , the @xmath21 term is from the tree - level scalar potential minimization conditions @xmath83 . the running of @xmath84 versus energy scale @xmath85 is shown in fig . [ fig : mhu ] for @xmath86 and 0.195 . we see that as @xmath2 increases , the value of @xmath87 actually decreases , leading to a small @xmath88 value . the relevant rge reads = ( - g_1 ^ 2m_1 ^ 2 -3g_2 ^ 2m_2 ^ 2 + g_1 ^ 2 s+3f_t^2 x_t ) . a large value of @xmath16 thus leads to an _ upwards _ push to @xmath84 in its early running from @xmath89 , which is only later compensated by the downward push of the yukawa - coupling term involving the top yukawa coupling @xmath90 . in the figure , for the case of @xmath91 , the weak scale value of @xmath84 is actually positive . upon adding the large 1-loop corrections to the effective potential ( due to the light top - squark ) , the rg - improved scalar potential yields a positive value of @xmath88 . thus , in the region of large @xmath2 , where @xmath21 becomes small and comparable to @xmath26 , we expect the neutralino @xmath57 to become a mixed wino - higgsino particle , and the corresponding @xmath92 mass gap to increase beyond the value @xmath93 mev which is expected in amsb models@xcite . parameter as a function of energy scale @xmath85 for @xmath94 , @xmath95 and @xmath96 for @xmath97 tev and @xmath47 , @xmath48 in the hcamsb model . , scaledwidth=50.0% ] an interesting coincidence related to the rg evolution of @xmath84 in the limit where hypercharge mediation dominates is that the electroweak symmetry breaking _ requires _ the electroweak scale to be @xmath98 orders of magnitude below the scale @xmath99 ( @xmath99 may be of order the gut scale or string scale ) at which the bino mass @xmath16 is generated . if the hierarchy between the electroweak scale and @xmath99 was smaller , then a susy breaking scenario in which hypercharge mediation dominates would not be capable of triggering ewsb ( the energy interval for rg evolution would not be large enough to drive the @xmath84 parameter to negative values ) . this is a very uncommon feature among susy breaking scenarios . for a more detailed comparison , we list in table [ tab : cases ] the sparticle mass spectrum for a mamsb point with @xmath100 gev , @xmath97 tev , @xmath47 and @xmath48 , and two hcamsb points with small and large @xmath2 values equal to @xmath101 and @xmath96 . while all three cases have a comparable gluino mass , we see that the rather small splitting amongst @xmath102 and also @xmath103 states in mamsb is turned to large left - right splitting in the hcamsb cases . we also see that the @xmath104 mev mass gap in amsb and hcamsb1 which leads to long - lived and possibly observable @xmath105 tracks in collider detectors opens up to a few gev in the hcamsb2 case . the latter mass gap is large enough to make the @xmath105 state less long lived , although still maintaining possibly measureable tracks in collider scattering events . the value of @xmath106 versus @xmath2 is shown in fig . [ fig : ctau ] , where we usually get @xmath107 mm for most @xmath2 values . the value drops to shorter lengths for large @xmath2 . the shorter travel time of the @xmath105 would distinguish the large @xmath2 hcamsb case with a mixed higgsino - wino @xmath57 state from the low @xmath2 hcamsb case where @xmath57 is instead nearly pure wino - like . .masses and parameters in gev units for three case study points amsb , hcamsb1 and hcamsb2 using isajet 7.79 with @xmath49 gev and @xmath48 . we also list the total tree level sparticle production cross section in fb at the lhc . [ cols="<,^,^,^",options="header " , ] in this paper , we have examined some phenomenological consequences of hypercharged anomaly - mediated susy breaking models at the lhc . we have computed the expected sparticle mass spectrum , and mapped out the relevant parameter space of the hcamsb model . we have computed sparticle branching fractions , production cross sections and expected lhc collider events , and compared against expectations for sm backgrounds . our main result was to compute the reach of the lhc for hcamsb models assuming 100 fb@xmath3 of integrated luminosity . we find an lhc reach to @xmath42 tev ( corresponding to @xmath43 tev ) for low values of @xmath2 , and a reach to @xmath108 tev ( corresponding to @xmath109 tev ) for large @xmath2 . we expect the reach for @xmath110 to be similar to the reach for @xmath48 , due to similarities in the spectra for the two cases ( see fig . [ fig : m10 ] . ) we also expect the reach for large @xmath6 to be similar to the reach for low @xmath6 in the @xmath111 and @xmath112 channels ( differences in the multi - lepton channels can occur due to enhanced -ino decays to taus and @xmath32s at large @xmath6 ) . the lhc reach for hcamsb models tends to be somewhat lower than the reach for mamsb models , where ref . @xcite finds a 100 fb@xmath3 reach of @xmath113 tev for low values of @xmath13 . this is due in part because , in mamsb , the various squark states are more clustered about a common mass scale @xmath13 , while in hcamsb the squark states are highly split , with @xmath114 . the hcamsb lhc event characteristics suffer similarities and differences with generic mamsb models . both hcamsb and mamsb give rise to multi - jet plus multi - lepton plus @xmath38 event topologies , and within these event classes , it is expected that occasional hits of length a few _ cm _ will be found , arising from production of the long - lived wino - like chargino states . some of the major differences between the models include the following . a severe left - right splitting of scalar masses is expected in hcamsb , while left - right scalar degeneracy tends to occur in mamsb . this may be testable if some of the slepton states are accessible to lhc searches . it is well known that in mamsb , @xmath56 , while in hcamsb , @xmath115 , since the @xmath66 state has a large weak hypercharge quantum number . in addition , the lightest stau state , @xmath116 , is expected to be mainly a left- state in hcamsb , while it is mixed , but mainly a right- state in mamsb . while it is conceivable that the left - right mixing might be determined at lhc ( using branching fractions or tau energy distributions ) , such measurements would be easily performed at a linear @xmath117 collider , especially using polarized beams@xcite . in hcamsb models , the light third generation squarks @xmath79 and @xmath80 are expected to be generically lighter than the gluino mass , and frequently much lighter . this leads to cascade decays which produce large multiplicities of @xmath32 and @xmath31 quarks in the final state . thus , in hcamsb models , a rather high multiplicity of @xmath32 jets is expected . in mamsb , a much lower mutiplicity of @xmath32-jets is expected , although this depends also on the value of @xmath6 which is chosen . in hcamsb models , the @xmath14 gaugino mass @xmath16 is expected to be the largest of the gaugino masses , with a mass hierarchy of @xmath0 . this usually implies that the @xmath59 neutralino is mainly bino - like , while @xmath60 and @xmath118 are higgsino - like , and @xmath57 is wino - like . in contrast , in the mamsb model , usually the ordering is that @xmath1 , so that while @xmath57 is again wino - like , the @xmath60 state is bino - like , and @xmath118 and @xmath59 are higgsino - like . the compositions of the @xmath119 for @xmath120 will not be easy to determine at lhc , but will be more easily determined at a linear @xmath117 collider . however , the mass ordering gives rise to os dilepton distributions with a prominent @xmath121 peak in hcamsb , while such a peak should be largely absent in mamsb models ( except at large @xmath6 where there is greater mixing in the neutralino sector ) . thus , cascade decay events containing hits along with a @xmath121 peak in the os dilepton invariant mass distribution may be a smoking gun signature for hcamsb models at the lhc , at least within the lower range of @xmath6 . this work was supported in part by the u.s . department of energy . 99 l. randall and r. sundrum , _ nucl . * b 557 * ( 1999 ) 79 ; 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we investigate the phenomenological consequences of string models wherein the mssm resides on a d - brane , and the hypercharge gaugino mass is generated in a geometrically separated hidden sector . this hypercharged anomaly - mediated susy breaking ( hcamsb ) model naturally solves the tachyonic slepton mass problem endemic to pure amsb scenarios . in hcamsb , one obtains a mass ordering @xmath0 with split left- and right- scalars , whereas in mamsb models , one obtains @xmath1 with nearly degenerate left- and right- scalars . we compute the allowed parameter space and expected superparticle mass spectrum in the hcamsb model . for low values of the hc and amsb mixing parameter @xmath2 , the spectra is characterized by light left - sleptons , while the spectra for large @xmath2 is characterized by light top- and bottom- squarks . we map out the approximate reach of lhc for hcamsb , and find that with 100 fb@xmath3 of integrated luminosity , a gravitino mass of @xmath4 ( 105 ) tev can be probed for low ( high ) values of @xmath2 , corresponding to a gluino mass reach of @xmath5 ( 2.2 ) tev . both cases contain as is typical in amsb models long lived charginos that should yield visible highly ionizing tracks in the lhc detector . also , in the lower @xmath6 range , hcamsb models give rise to reconstructable @xmath7 candidates in susy cascade decay events , while mamsb models should do so only rarely . * prospects for hypercharged anomaly mediated + susy breaking at the lhc * + howard baer@xmath8 , radovan derm ' iek@xmath9 , shibi rajagopalan@xmath10 , heaya summy@xmath10 + _ 1 . dept . of physics and astronomy , university of oklahoma , norman , ok 73019 , usa + 2 . dept . of physics , indiana university , bloomington in 47405 , usa + _

You are an expert at summarizing long articles. Proceed to summarize the following text: high - harmonic generation ( hhg ) is a highly nonlinear optical phenomenon @xcite of increasing interest because it can provide coherent xuv and soft x - ray radiation with attosecond ( @xmath1 s ) durations . this property offers the opportunity to investigate unexplored research areas in atoms and molecules with unprecedented time resolution @xcite . the hhg optical spectrum has a distinctive shape : a rapid decrease of the intensity for the low - order harmonics consistent with perturbation theory , followed by a broad plateau region where the harmonic intensity remains almost constant , and then an abrupt cutoff , beyond which almost no harmonics are observed . the hhg process can be understood by means of semi - classical pictures , such as the celebrated three - step model @xcite : ( i ) an electron escapes from the nuclei through tunnel ionization associated with the strong laser field , ( ii ) it is accelerated away by the laser field until the sign of the field changes , ( iii ) whereupon the electron is reaccelerated back to the nucleus , where it may emit a photon as it recombines to the ground state . a key quantity emerging from the model is the maximum energy the field can provide to the electron , @xmath2 , where @xmath3 is the ionization potential and @xmath4 is the ponderomotive energy @xcite . hhg has been studied for many years with theoretical methods solving the time - dependent schrdinger equation using a real - space representation of the wave function @xcite . these grid - based methods are taken as the numerical reference for this kind of calculations . indeed , these approaches have proven to be accurate enough to explain key features of atomic and molecular hhg spectra . however , grid calculations imply memory and cpu requirements that rapidly become prohibitive with increasing numbers of electrons . because of this limitation , multielectron systems are handled in practice via the use of effective potentials keeping a single - active electron . by contrast , quantum - chemistry methods such as time - dependent configuration interaction ( tdci ) @xcite , multiconfiguration time - dependent hartree - fock @xcite , or time - dependent density - functional theory @xcite using local basis functions can more easily handle multielectron systems such as molecules , including the treatment of electron correlation . the main problem of these methods lies in the difficulty to accurately represent the continuum part of the system eigenstate spectrum . addressing this issue can be done on one - electron systems , such as the h atom , since only one electron is promoted into the continuum during the hhg process . in this context , the tdci method with a gaussian - type orbital ( gto ) basis set and a heuristic lifetime model @xcite was recently applied to the calculation of the dipole form of the hhg spectrum for the h atom @xcite . the role of the rydberg and the continuum states was discussed in detail , and reasonable hhg spectra ( plateau / cutoff ) have been obtained , when compared with the prediction from the three - step model @xcite and grid - based calculations @xcite . however , the background region , beyond the harmonic cutoff , was higher than expected and spurious harmonics were present . a possible reason of this behavior is that the basis sets adopted in ref . describe rydberg states better than the continuum ones . indeed , while gto basis sets have been successfully applied for calculations of bound - state electronic properties ( even for non - linear optical properties such as second - order hyperpolarizabilities , see e.g. ref . ) , the inherent local nature of gto functions makes it difficult to properly describe continuum states extending over large distances ( see , e.g. , ref . ) . in refs . and , standard gto basis sets have been augmented with a large number of diffuse basis functions and/or basis functions centered away from the nucleus in order to cover the large spatial extension of the time - dependent wave function . however , this strategy has the serious drawback of only increasing the number of rydberg states while the number of continuum states is not substantially changed . this results in an unbalanced description of the rydberg and continuum states . few attempts have been reported in the literature to further improve gto basis sets for a better description of the continuum states . et al . _ @xcite proposed to fit gto basis functions to slater - type orbital basis functions having a single fixed exponent @xmath5 , supposed to be adequate for scattering calculations . nestmann and peyerimhoff @xcite proposed to fit a linear combination of gto basis functions to a set of spherical bessel functions , which are the spherically - adapted continuum eigenfunctions for zero potential . et al . _ @xcite extended the work of nestmann and peyerimhoff to the possibility of fitting a linear combination of gto basis functions to a set of coulomb continuum functions ( i.e. , the continuum eigenfunctions obtained in presence of the coulomb potential @xmath6 , with @xmath7 the nuclear charge ) . finally , some hybrid methods have also been proposed , combining gaussian functions with finite - element / discrete - variable representation techniques @xcite or with b - spline basis sets @xcite . note that an alternative approach to gaussian basis sets is given by the use of sturmian functions @xcite . + in this article , we study the merits of the gaussian continuum basis functions proposed by kaufmann _ et al . _ @xcite for calculating the hhg spectra in atomic hydrogen within the tdci framework . while the present results are focused on hhg , our work is relevant for the calculation of any property involving electronic transitions toward the continuum such as , e.g. , photoionization cross sections @xcite or above - threshold ionization rates @xcite . the paper is organized as follows . we first describe the theory and give computational details . we then present and discuss our results . in particular , we show velocity hhg spectra extracted from the dipole , velocity , and acceleration power spectra calculated for different laser intensities , and basis sets . we study in detail the effect of increasing the basis set cardinal number , the number of diffuse basis functions , and the number of gaussian continuum basis functions . we directly compare our results with data from grid calculations , for three values of the laser intensity and two values of the laser wavelength , and adjust the heuristic lifetime model . finally , we conclude with final comments and perspectives . unless otherwise noted , hartree atomic units , i.e. @xmath8 , are used throughout the paper . the time - dependent schrdinger equation for the h atom in an external time - dependent uniform electric field @xmath9 in the length gauge is @xmath10 where @xmath11 is the time - independent field - free hamiltonian and @xmath12 is the interaction potential between the atom and the field in the semiclassical dipole approximation . we consider the case of an electric field @xmath9 linearly polarized along the @xmath13-axis , representing a laser pulse , @xmath14 where @xmath15 is the maximum field strength , @xmath16 is the unit vector along the @xmath13 axis , @xmath17 is the carrier frequency , @xmath18 is the carrier - envelope phase , and @xmath19 is the envelope function chosen as @xmath20 where @xmath21 is the full width at half maximum of the field envelope . the target quantity to be computed is the power spectrum @xmath22 defined as @xmath23 where @xmath24 and @xmath25 are the initial and final propagation times . in eq . ( [ fft ] ) , the operator @xmath26 can be either equal to the position operator @xmath27 , or to the velocity operator @xmath28 $ ] , or to the acceleration operator @xmath29 $ ] ( where @xmath30 is the total time - dependent hamiltonian ) , defining three different forms of the power spectrum : the dipole @xmath31 , the velocity @xmath32 , and the acceleration @xmath33 forms . according to recent works @xcite , the velocity form @xmath32 best represents the hhg spectrum of a single atom or molecule . the three forms are related to each other by ( see appendix [ app : powerspectrum ] ) : @xmath34 in this work , we always show the same quantity , i.e. the velocity hhg spectrum , either extracted directly from the velocity power spectrum , or indirectly from the dipole or the acceleration power spectrum with the appropriate frequency factors following eq . ( [ eq : equi ] ) . the time - dependent schrdinger equation is solved using the tdci method ( see , e.g. , refs . ) applied to the special case of the h atom . the wave function @xmath35 is expanded in the discrete basis of the eigenstates @xmath36 of the field - free hamiltonian @xmath37 ( projected in the same basis ) , composed of the ground state ( @xmath38 ) and all the excited states ( @xmath39 ) @xmath40 where @xmath41 are time - dependent coefficients . inserting eq . ( [ tdcis_state ] ) into eq . ( [ tdse ] ) , and projecting on the eigenstates @xmath42 , gives the evolution equation @xmath43 where @xmath44 is the column matrix of the coefficients @xmath41 , @xmath45 is the diagonal matrix of elements @xmath46 ( where @xmath47 is the energy of the eigenstate @xmath48 ) , and @xmath49 is the non - diagonal matrix of elements @xmath50 . the initial wave function at @xmath51 is chosen to be the field - free ground state , i.e. @xmath52 . to solve eq . ( [ ciequation ] ) , time is discretized and the simple split - propagator approximation is used to separate the contributions of the field - free hamiltonian @xmath45 and the atom - field interaction @xmath49 @xmath53 where @xmath54 is a small time step . since the matrix @xmath45 is diagonal , @xmath55 is a diagonal matrix of elements @xmath56 . the exponential of the non - diagonal matrix @xmath49 is calculated as @xmath57 where @xmath58 is the unitary matrix describing the change of basis between the original eigenstates of @xmath37 and a basis in which the atom - field interaction @xmath59 is diagonal , i.e. @xmath60 where @xmath61 is the diagonal atom - field interaction matrix and @xmath62 is the diagonal representation matrix of the position operator . since the time dependence is simply factorized in a multiplicative function independent of @xmath63 , the unitary matrix @xmath58 is time - independent and can be calculated once and for all before the propagation . once the time - dependent coefficients are known , it is possible to calculate the time - dependent dipole , velocity , or acceleration as @xmath64 which , after taking the square of its fourier transform , leads to the corresponding power spectrum of eq . ( [ fft ] ) . the field - free states ( simply corresponding to the atomic orbitals for the h atom ) are expanded on a gaussian basis set , @xmath65 where @xmath66 are real - valued gto basis functions centered on the nucleus . in spherical coordinates @xmath67 , @xmath68 where @xmath69 is a normalization constant , @xmath70 are exponents , @xmath71 are real spherical harmonics . we built the gaussian basis set starting from the dunning basis sets @xcite , adding first diffuse gto functions to describe the rydberg states , and a special set of gto functions adjusted to represent low - lying continuum states . for the latter , we follow kaufmann _ et al . _ @xcite who proposed to fit gto basis functions to slater - type orbital basis functions having a single fixed exponent @xmath5 . for each angular momentum @xmath72 , kaufmann _ et al . _ found a sequence of optimized gto exponents which are well represented by the simple formula @xcite @xmath73 where @xmath74 is not associated to the quantum principal number but is just an index identifying a given value in the list of all exponents for a fixed @xmath72 , and the parameters @xmath75 and @xmath76 are given in table 2 of ref . . the gto basis functions obtained with these exponents ( collected in table [ tab : continuum ] ) will be in the following referred to as `` gaussian continuum functions '' or `` kaufmann ( k ) functions '' . .exponents @xmath77 [ see eq . ( [ kauf_sca ] ) ] of the gaussian functions for describing the continuum proposed by kaufmann _ _ @xcite and used in the present work for @xmath78 and @xmath79 . [ tab : continuum ] [ cols="^,^,^,^",options="header " , ] the gto basis set incompleteness is responsible for an incorrect description of the continuum eigenfunctions . they decay too fast for large @xmath80 , which prevents the description of the above - threshold ionization and leads to unphysical reflections of the wave function in the laser - driven dynamics . to compensate for this , we use the heuristic lifetime model of klinkusch _ et al . _ @xcite which consists in interpreting the approximate field - free eigenstates @xmath81 above the ionization threshold ( taken as the zero energy reference ) as non - stationary states and thus replacing , in the time propagation , the energies @xmath47 by complex energies @xmath82 , where @xmath83 is the inverse lifetime of state @xmath48 . for the special case of the h atom the @xmath83 are chosen as @xcite @xmath84 where @xmath85 is an empirical parameter representing the characteristic escape length that the electron in the state @xmath48 is allowed to travel during the lifetime @xmath86 . these complex energies are used in the propagation described by eq . ( [ coefficients ] ) , in the field - free hamiltonian matrix @xmath45 . the heuristic lifetime model is a simple alternative to using complex scaling @xcite , a complex - absorbing potential @xcite , or a wave - function absorber @xcite . in this work , we also introduce and test a modified version of the original heuristic lifetime model . in this version , two different values of the escape length , @xmath87 and @xmath88 , are used to increase the flexibility in the definition of the finite lifetimes , adapted to the present context of hhg . a large value of @xmath87 ( small value of @xmath83 ) is used for all the above - ionization - threshold states with positive energy below the energy cutoff of the three - step model @xmath89 , while a smaller @xmath88 ( larger @xmath83 ) is used for the continuum states with energies above @xmath89 , which are not expected to contribute to hhg . this allows us to better retain the contribution of low - energy continuum states for the recombination step of the hhg process . c c c c @xmath90 & 5@xmath9110@xmath92 w/@xmath93 & 10@xmath94 w/@xmath93 & 2@xmath9110@xmath94 w/@xmath93 + @xmath95 nm + @xmath96 & 1.51 & 1.06 & 0.76 + @xmath97 & 0.11 & 0.22 & 0.44 + @xmath89 & 0.85 & 1.20 & 1.89 + @xmath98 & 15 & 21 & 33 + @xmath99 & 23 & 33 & 46 + + @xmath100 nm + @xmath96 & 1.13 & 0.79 & 0.57 + @xmath97 & 0.19 & 0.40 & 0.78 + @xmath89 & 1.10 & 1.77 & 2.97 + @xmath98 & 26 & 41 & 69 + @xmath99 & 41 & 59 & 82 + the field - free calculations are performed using a development version of the molpro software package @xcite from which all the electronic energies , as well as the dipole , velocity , and acceleration matrix elements over the electronic states have been obtained . the external code light @xcite is used to perform the time - propagation using a time step @xmath54 = 2.42 as ( 0.1 a.u . ) and the fourier transformations with a hann window function . an escape length @xmath85=1.41 bohr is used for the original heuristic lifetime model , while @xmath101 bohr and @xmath102 bohr are chosen for the modified version of the heuristic lifetime model as explained in the section results and discussion . correlation - consistent @xmath103-aug - cc - pv@xmath104z @xcite basis sets are used , where @xmath104 is the cardinal number ( @xmath105 t , q , 5 ) connected to the maximum angular momentum ( @xmath106 for the h atom ) , and @xmath103 is the number of shells of diffuse functions for each angular momentum . we only employ @xmath107 or @xmath108 because @xmath107 can be considered as the minimum augmentation needed to reasonably describe hhg spectra for the h atom @xcite . in particular , the 6-aug - cc - pvtz basis set describes up to @xmath109-shell rydberg states , 6-aug - cc - pvqz up to @xmath110-shell rydberg states , and 6-aug - cc - pv5z up to @xmath111-shell rydberg states . furthermore , we investigate the effect of adding to the 6-aug - cc - pvtz basis set 3 , 5 , and 8 gaussian continuum functions ( or k functions ) for each angular momentum . the extra diffuse and continuum gaussian functions are uncontracted . for comparison , we also perform accurate grid calculations in the length gauge . the wave function is expanded on a set of spherical harmonics @xmath112 up to @xmath113 , and the resulting coupled equations are discretized on a radial grid with a step size of @xmath114 bohr ( see ref . ) . a box size of 256 bohr is used with a mask function @xcite at 200 bohr to absorb the part of the wave function accounting for ionized electrons that will not rescatter towards the nucleus . the mask function multiplying the wave function at each time step has been chosen to be cos(@xmath80)@xmath115 , which is effective in modeling the ionization @xcite . the time step used is @xmath54=0.65 as ( 0.027 a.u . ) . the grid - based calculations , being converged with respect to the parameters mentioned above , represent the numerical reference for the current gto results . we note that performing the grid - based calculations takes hours on a standard workstation , while the field - free and time - propagation calculations in the gto basis sets take only a few minutes . unless otherwise noted , the calculations are done with the carrier laser frequency @xmath116 = 1.550 ev ( @xmath117 nm ) , corresponding to a ti : sapphire laser . for the comparison with the grid calculations , we also use the laser frequency @xmath116 = 1.165 ev ( @xmath118 nm ) for which higher - energy regions are probed . the pulse duration is @xmath119 oc where 1 optical cycle ( oc ) is @xmath120 ( 110.23 a.u . ) . we use three peak laser intensities @xmath121 : @xmath122 w/@xmath93 , @xmath123 w/@xmath93 , and @xmath124 w/@xmath93 . we have thus chosen a range of intensities encompassing the over - barrier ionization threshold ( i.e. the critical intensity above which the electron can classically overstep the barrier ) of hydrogen , @xmath125 w/@xmath126 . we can therefore study the performance of our method in realistic conditions for which hhg progressively becomes less pronounced with increasing laser intensity . the physical parameters relevant to hhg are reported in table [ tab : lasers ] . we start by studying the performance of several gaussian basis sets for the calculation of hhg spectra of the h atom , continuing the previous work of luppi and head - gordon @xcite . the optimal basis set including gaussian continuum functions is then used for a direct comparison with reference hhg spectra from grid calculations . ( left ) , velocity @xmath127 ( middle ) , and acceleration @xmath128 ( right ) calculated with the 6-aug - cc - pvtz basis set for laser intensities @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . [ fig1],title="fig:",width=288 ] + ( left ) , velocity @xmath127 ( middle ) , and acceleration @xmath128 ( right ) calculated with the 6-aug - cc - pvtz basis set for laser intensities @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . [ fig1],title="fig:",width=288 ] + ( left ) , velocity @xmath127 ( middle ) , and acceleration @xmath128 ( right ) calculated with the 6-aug - cc - pvtz basis set for laser intensities @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . [ fig1],title="fig:",width=288 ] we have reported on figure [ fig1 ] the time evolution of the dipole @xmath131 , the velocity @xmath127 , and the acceleration @xmath128 with the 6-aug - cc - pvtz basis set for the three laser intensities . the evolution of @xmath131 , @xmath127 , and @xmath128 follows the shape of the laser field given in eq . ( [ enve ] ) , with the shape of their envelopes changing with the intensity of the pulse . note that @xmath127 is one order of magnitude smaller than @xmath131 and its oscillations have a finer structure . similarly , @xmath128 is one order of magnitude smaller than @xmath127 and has even more structured oscillations . even though some fast oscillations are still present after the laser is switched off due to the population of electronic excited states , the conditions @xmath132 and @xmath133 ( see appendix [ app : powerspectrum ] ) are approximately fulfilled , which will allow us to use eq . ( [ eq : equi ] ) . our results are in reasonable agreement with the results of bandrauk _ et al . _ @xcite and those of han and madsen @xcite who used grid - based methods . similar findings have been reported for the he atom in a low - field regime using time - dependent hartree - fock and time - dependent kohn - sham with gaussian basis sets @xcite . ( i.e. @xmath134 ) , the velocity power spectrum @xmath135 ( i.e. @xmath136 ) , and the acceleration power spectrum @xmath137 ( i.e. @xmath138 ) calculated with the 6-aug - cc - pvtz basis set and laser intensities @xmath12910@xmath92 w/@xmath126 , @xmath123 w/@xmath126 , and @xmath13910@xmath94 w/@xmath126 . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig2],width=336 ] in figure [ fig2 ] the velocity hhg spectrum , extracted from the dipole , velocity , and acceleration power spectra according to eq . ( [ eq : equi ] ) , calculated with the 6-aug - cc - pvtz basis set and the three laser intensities are shown . the typical form of the hhg spectrum ( plateau / cutoff / background ) is obtained . we note that the harmonic peaks that we obtained are sharper than those calculated by bandrauk _ et al . _ @xcite based on a direct propagation of the time - dependent schrdinger equation on a grid . the dipole , velocity , and acceleration formulations of the velocity hhg spectrum give similar spectra in the plateau region , but different backgrounds beyond the harmonic cutoff . in particular , the hhg spectrum calculated from the dipole power spectrum presents a higher background than the hhg spectra calculated from the velocity and acceleration power spectra , in agreement with the calculations of bandrauk _ et al . _ @xcite . these differences reflect the sensitivity to the basis set . indeed , the expectation value of the dipole operator probes the time - dependent wave function in spatial regions further away from the nucleus than the expectation values of the velocity and acceleration operators do . in the following , since the dipole is the most difficult to converge with our basis set we will focus on the basis set convergence of the ( velocity ) hhg spectrum computed from the dipole power spectrum . we first analyze the effect of the basis - set cardinal number @xmath104 , before examining the effect of adding gaussian continuum basis functions in sec . [ conti ] . we use the following series of basis sets : 6-aug - cc - pvtz ( s , p , and d shells ) , 6-aug - cc - pvqz ( s , p , d , and f shells ) , and 6-aug - cc - pv5z ( s , p , d , f , and g shells ) . the number of total , bound ( i.e. , energy below 0 ) , and continuum ( i.e. , energy above 0 ) states , and the maximum energy obtained with these basis sets are reported in the upper half of table [ angmom ] . going from 6-aug - cc - pvtz to 6-aug - cc - pv5z the total number of states increases considerably , from 68 to 205 . the percentage of continuum states also tends to increase with the cardinal number . however , these added continuum states are not necessarily in the energy range relevant to the hhg spectrum . indeed , the maximum energies obtained are 3.45 hartree for 6-aug - cc - pvtz , 7.74 hartree for 6-aug - cc - pvqz , and 15.94 hartree for 6-aug - cc - pv5z , while the maximal kinetic energy that can be transmitted to the electron ( @xmath141 ) in the three - step model are between 0.35 and 2.47 hartree for the parameters considered ( see table [ tab : lasers ] ) . l > + m10 mm > + m05 mm > + m05 mm > + m05 mm > + m05 mm > + m10 mm & total & & & @xmath142 + 6-aug - cc - pvtz & 68 & 42 & ( 62% ) & 26 & ( 38% ) & 3.45 + 6-aug - cc - pvqz & 126 & 63 & ( 50% ) & 63 & ( 50% ) & 7.74 + 6-aug - cc - pv5z & 205 & 90 & ( 44% ) & 115 & ( 56% ) & 15.94 + 6-aug - cc - pvtz+3k & 95 & 42 & ( 44% ) & 53 & ( 56% ) & 6.31 + 6-aug - cc - pvtz+5k & 113 & 46 & ( 41% ) & 67 & ( 59% ) & 6.68 + 6-aug - cc - pvtz+8k & 140 & 51 & ( 36% ) & 89 & ( 64% ) & 6.93 + in figure [ fig3 ] , we compare the velocity hhg spectrum extracted from the dipole power spectrum for the 6-aug - cc - pvtz , 6-aug - cc - pvqz , and 6-aug - cc - pv5z basis sets for the laser intensity @xmath123 w/@xmath126 . the three basis sets give very similar results , in the plateau as well as beyond the harmonic cutoff . we thus conclude that the hhg spectrum is not strongly affected by the cardinal number @xmath104 of the basis set and therefore , in the following , we will use a triple - zeta ( @xmath105 t ) basis set . in figure [ fig3 ] , we also compare the spectra calculated using the @xmath103-aug - cc - pv@xmath104z basis sets with @xmath107 and @xmath108 . the results show that the convergence in terms of diffuse basis functions is achieved with 6 diffuse shells . calculated with the 6-aug - cc - pv@xmath104z and 9-aug - cc - pv@xmath104z basis sets with @xmath104= t ( left ) , q ( middle ) and 5 ( right ) . the laser intensity is @xmath123 w/@xmath126 . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig3],title="fig:",width=336 ] + the sensitivity of the hhg spectrum to the cardinal number and to the number of diffuse functions led us to select the 6-aug - cc - pvtz basis set as the reference basis set to include the gaussian continuum functions of kaufmann _ et al . _ we have added 3 , 5 , and 8 gaussian continuum functions ( denoted by k ) for each angular momentum in the 6-aug - cc - pvtz basis set . in the lower half of table [ angmom ] , the number of total , bound , and continuum states and the maximum energy obtained with these 6-aug - cc - pvtz+3k , 6-aug - cc - pvtz+5k , and 6-aug - cc - pvtz+8k basis sets is reported . it is noteworthy that increasing the number of k functions hardly affects the number of bound states , in favor of positive energy states , thus focusing the improvement on the description of the continuum . more precisely , as the maximum energy obtained with these three basis sets is nearly unchanged ( 6.313 , 6.681 , and 6.927 hartree , respectively ) , the k functions increase the density of states in the energetically important region of the continuum . we show in figure [ fig4 ] the distribution of the state energies for the different basis sets . increasing the number of k functions essentially does not change the energy spectrum below the ionization threshold , while an almost continuum distribution builds up in the low - energy region above the ionization threshold . when compared with the 6-aug - cc - pvtz basis set , the distribution of the continuum states becomes more dense ( closer to a `` real '' continuum ) and the gaps between ( near-)degenerate sets of states become smaller . in particular , the density of states is improved in the region from the ionization threshold to around 1 hartree , which is also the most relevant energy region for hhg for the laser intensity range studied here , according to the three - step model . by contrast , luppi and head - gordon @xcite showed that adding diffuse functions increases the density of rydberg states , leaving the density of continuum states mostly unchanged . , width=288 ] the upper panel of figure [ fig6 ] compares the radial wave function @xmath143 of a s continuum state at the energy @xmath144 hartree obtained with the 6-aug - cc - pvtz+8k basis set with the analytical solution of the time - independent schrdinger equation @xcite . for completeness , the radial wave function from the grid calculation is also shown and is perfectly superimposed with the analytical solution . the radial wave function obtained with the 6-aug - cc - pvtz+8k basis set is a reasonable approximation to the exact solution , the continuum gaussian functions correctly reproducing the oscillations of the function up to a radial distance as large as 30 bohr . this radial distance is consistent with the maximum distance @xmath99 ( see table [ tab : lasers ] ) traveled by the electron predicted by the three - step model with the laser parameters used here . for comparison , the lower panel of figure [ fig6 ] shows the radial wave function @xmath143 obtained with the 6-aug - cc - pvtz basis set for a similar s continuum state at the closest energy obtained with this basis set , @xmath145 hartree . clearly , the basis set without the continuum gaussian functions is only able to describe the short - range part of function @xmath143 but not the long - range oscillating part . @xcite and the radial wave function obtained using the 6-aug - cc - pvtz+8k basis set for a s continuum state at the energy @xmath144 hartree ( upper panel ) . in the lower panel , the same comparison is done for a similar state of close energy @xmath145 hartree but with the 6-aug - cc - pvtz basis set , i.e. without the kaufmann basis functions . the radial wave functions obtained in the grid calculations are also shown . since continuum wave functions can not be normalized in the standard way , the curves have been scaled in order to approximately have the same value at the first minimum.,title="fig:",width=288 ] + @xcite and the radial wave function obtained using the 6-aug - cc - pvtz+8k basis set for a s continuum state at the energy @xmath144 hartree ( upper panel ) . in the lower panel , the same comparison is done for a similar state of close energy @xmath145 hartree but with the 6-aug - cc - pvtz basis set , i.e. without the kaufmann basis functions . the radial wave functions obtained in the grid calculations are also shown . since continuum wave functions can not be normalized in the standard way , the curves have been scaled in order to approximately have the same value at the first minimum.,title="fig:",width=288 ] calculated with the 6-aug - cc - pvtz basis set plus 3 ( left ) , 5 ( middle ) , and 8 ( right ) gaussian continuum ( k ) functions . the laser intensity is @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig5],title="fig:",width=288 ] + calculated with the 6-aug - cc - pvtz basis set plus 3 ( left ) , 5 ( middle ) , and 8 ( right ) gaussian continuum ( k ) functions . the laser intensity is @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig5],title="fig:",width=288 ] + calculated with the 6-aug - cc - pvtz basis set plus 3 ( left ) , 5 ( middle ) , and 8 ( right ) gaussian continuum ( k ) functions . the laser intensity is @xmath12910@xmath92 w/@xmath126 ( top ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( bottom ) . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig5],title="fig:",width=288 ] in figure [ fig5 ] the velocity hhg spectrum extracted from the dipole power spectrum is shown for the 6-aug - cc - pvtz+3k , 6-aug - cc - pvtz+5k , and 6-aug - cc - pvtz+8k basis sets and for the three laser intensities . we focus our attention to the post - cutoff background region of the spectrum since diminishing the background in this region is an important goal of the present work . considering the laser intensity @xmath146 w/@xmath93 and analyzing the spectra between the 20th and 40th harmonics , we observe that the hhg spectrum with the 6-aug - cc - pvtz+3k basis set resembles the one obtained with the original 6-aug - cc - pvtz basis set , with no obvious improvement . when adding 5 or 8 k functions the background is strongly diminished , while the harmonics before the cutoff are not substantially changed . the same trend is also observed for laser intensities @xmath147 w/@xmath93 and @xmath148 w/@xmath93 , even if the lowering of the background is not as strong . as demonstrated in ref . , the rydberg bound states strongly contribute to the background of the hhg spectrum . the addition of gaussian continuum functions to the basis set allows one to appropriately describe the low - lying continuum states , leading to a more balanced basis set yielding a lower background and therefore a much clearer identification of the cutoff region . of course , such an improvement depends on the intensity of the laser pulse , since larger intensities require to describe continuum states of higher energy and therefore require more gaussian continuum functions . calculated with the 6-aug - cc - pvtz+8k basis set with two lifetime models and with grid calculations , for the two laser wavelength @xmath95 nm ( upper panel ) and 1064 nm ( lower panel ) , the laser intensities @xmath14910@xmath92 , 10@xmath94 , and 2@xmath9110@xmath94 w/@xmath93 . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig8],title="fig:",width=288 ] calculated with the 6-aug - cc - pvtz+8k basis set with two lifetime models and with grid calculations , for the two laser wavelength @xmath95 nm ( upper panel ) and 1064 nm ( lower panel ) , the laser intensities @xmath14910@xmath92 , 10@xmath94 , and 2@xmath9110@xmath94 w/@xmath93 . the ionization threshold ( @xmath140 , vertical dashed line ) and the harmonic cutoff in the three - step model @xmath98 ( vertical dot - dashed line ) are also shown . [ fig8],title="fig:",width=288 ] for the laser wavelength @xmath95 and the intensities @xmath12910@xmath92 w/@xmath126 ( left ) , @xmath123 w/@xmath126 ( middle ) , and @xmath13010@xmath94 w/@xmath126 ( right ) obtained with the 6-aug - cc - pvtz+8k basis set with the original ( one parameter ) and modified ( two parameters ) lifetime models , compared with the results from the grid calculations.,width=288 ] we now investigate in more detail the performance of the 6-aug - cc - pvtz+8k basis set by comparison with grid calculations . in figure [ fig8 ] we compare the velocity hhg spectrum extracted from the acceleration power spectrum , obtained with the 6-aug - cc - pvtz+8k basis set with two lifetime models , and with grid calculations for the same laser intensities as before and for @xmath95 and 1064 nm . the acceleration power spectrum , rather than the dipole power spectrum , was chosen here because the grid calculation is easier to converge for the acceleration power spectrum . for the intensity @xmath14910@xmath92 w/@xmath93 and for the two wavelengths , the spectra obtained with the gaussian basis set and the original lifetime model ( with @xmath150 bohr ) are in good agreement with the ones from the grid calculations . in particular , the cutoff appears at almost the same energy . however , for the larger intensities , the intensity of the higher harmonics in the plateau obtained with the gaussian basis set decrease too rapidly . this can be attributed to a limitation of the original lifetime model which assigns too large lifetimes to high - energy continuum states . for this reason we introduce a modified version of the lifetime model , with two different values of the parameter @xmath85 : a large value , @xmath101 bohr , for continuum states with positive energies below the energy cutoff of the three - step model @xmath141 , and a small value , @xmath102 bohr , for continuum states with energies above @xmath141 . the ionization rates @xmath83 are thus smaller than in the original model for low - lying continuum states , and larger for high - lying continuum states . this choice allows us to get a more accurate description of the harmonics in the plateau and close to the cutoff . the values of @xmath87 and @xmath88 have been chosen comparing with the corresponding grid hhg spectra . not surprisingly , the value of @xmath87 is of the order of magnitude of the electron excursion distance @xmath99 ( see table [ tab : lasers ] ) . we test our modified lifetime model by calculating the ionization probability ( for both the grid and gaussian - basis - set calculations ) @xmath151 where the sum runs over all the bound states . figure [ fig7 ] reports @xmath152 obtained with the original and the modified lifetime models and from the grid calculations for the three laser intensities . the original lifetime model leads to largely overestimated ionization probabilities in comparison to the results obtained from the grid calculations . our modified lifetime model reduces the ionization probability and is in better agreement with the grid calculations , especially for the intensities @xmath147 w/@xmath93 and @xmath148 w/@xmath93 . coming back to figure [ fig8 ] , it is seen that the combined use of the 6-aug - cc - pvtz+8k basis set and of the modified lifetime model results in a hhg spectrum that is in good agreement with the one obtained with the grid calculation at wavelength @xmath95 nm and the intensity @xmath123 w/@xmath93 . the general shape of the spectrum and the position of the harmonic cutoff are well reproduced with the gaussian basis set , the only remaining differences being larger peaks and a larger background after the cutoff in comparison to the grid results . for the same wavelength and the largest intensity @xmath15310@xmath94 w/@xmath93 , the agreement is also fairly good even though the position of the harmonic cutoff predicted with the gaussian basis set is slightly too low . the longer laser wavelength @xmath100 nm represents a more stringent test for our method since higher - energy regions are probed ( see table [ tab : lasers ] ) . the agreement between the hhg spectra obtained with the gaussian basis set and from the grid calculations is still pretty good for the intensity @xmath14910@xmath92 w/@xmath93 , while the position of the cutoff is slightly underestimated for the intensity @xmath123 w/@xmath93 and significantly underestimated for the largest intensity @xmath154 w/@xmath93 . this likely comes from a too poor description of the continuum states above 1 hartree with the 6-aug - cc - pvtz+8k basis set , which can be populated for these wavelengths and intensities . a larger number of continuum gaussian functions is needed in order to improve the high - energy part of the hhg spectrum for the largest intensities . we note , however , that increasing the number of continuum gaussian functions can lead to near - linear dependencies in the basis set ( seen with the presence of very small eigenvalues of the overlap matrix of the basis functions ) and thus numerical instability issues in self - consistent - field calculations . in this work , we have explored the calculation of the velocity hhg spectrum of the h atom extracted from the dipole , velocity , and acceleration power spectra with gaussian basis sets for different laser intensities and wavelengths . while all the three power spectra give reasonable velocity hhg spectra with similar harmonic peaks before the cutoff , they tend to differ in the background region beyond the cutoff . the hhg spectrum extracted from the dipole power spectrum is the most sensitive to the basis set . with the 6-aug - cc - pvtz basis set it leads to a high background which blurs out the location of the plateau cutoff . increasing the cardinal number of the basis set ( from @xmath155 to @xmath156 ) or the number of diffuse basis functions ( from @xmath107 to @xmath108 ) does not improve the hhg spectrum . by contrast , adding 5 or 8 gaussian continuum functions , as proposed by kaufmann _ et al . _ @xcite , leads to an improvement of the velocity hhg spectrum extracted from the dipole power spectrum at least for laser intensities up to @xmath157 w/@xmath93 by decreasing the background , which thus allows one to better identify the cutoff region . the combined use of gaussian continuum functions and a heuristic lifetime model with two parameters for modeling ionization results is in a fairly good agreement with the reference hhg spectra from grid calculations , in terms of the general shape of the spectrum , the number and intensity of peaks , and the position of the cutoff . the agreement is less satisfactory for the largest intensities because the high - energy continuum states are poorly reproduced by the gaussian basis set calculations . improving the accuracy for the largest intensities would require a larger number of gaussian continuum functions . gaussian continuum functions thus appear as a promising way of constructing gaussian basis sets for studying electron dynamics in strong laser fields , allowing one to define a balanced basis set to properly describe both bound and continuum eigenstates . the present work therefore opens the way to the systematic application of well established quantum chemistry methods with gaussian basis sets to the study of highly nonlinear phenomena ( such as hhg , photoionization cross sections , above - threshold ionization rates , ... ) in atoms and molecules . this work was supported by the labex michem and calsimlab part of french state funds managed by the anr within the investissements davenir programme under references anr-11-idex-0004 - 02 and anr-11-labx-0037 - 01 . we thank a. savin for having pointed out to us ref . and v. vniard for useful comments . in this appendix , we review the relationship between the dipole , velocity , and acceleration forms of the power spectrum @xcite . if we define @xmath158 , where @xmath159 stands for position @xmath13 , velocity @xmath160 , or acceleration @xmath161 , and its fourier transform @xmath162 the three forms of the power spectrum are commonly expressed as @xmath163 and the relationship between the three forms is the relationship between the three @xmath164 . applying eq . ( [ xiomega ] ) for @xmath165 , performing an integration by parts over @xmath166 , and using @xmath167 , gives @xmath168 which , if we have the condition @xmath169 , can be simplified as @xmath170 the relation between @xmath171 and @xmath172 is then @xmath173\bigl ) , \ ; \nonumber\\ \label{vel}\end{aligned}\ ] ] which , in the case where we can make the approximation @xmath174 , simplifies as @xmath175 similarly , applying now eq . ( [ xiomega ] ) for @xmath176 , using @xmath177 , and integrating by parts gives @xmath178 where we have used the condition @xmath179 . this leads to the relation between @xmath172 and @xmath180 @xmath181\bigl ) , \nonumber\\\end{aligned}\ ] ] which , if we can make the approximation @xmath182 , gives the final approximate relations between the three forms of the spectrum @xmath183 g. sansone , f. kelkensberg , j. f. prez - torres , f. morales , m. f. kling , w. siu , o. ghafur , p. johnsson , m. swoboda , e. benedetti , f. ferrari , f. lpine , j. l. sanz - vicario , s. zherebtsov , i. znakovskaya , a. lhuillier , m. y. ivanov , m. nisoli , f. martin , and m. j. j. vrakking , nature * 465 * , 763 ( 2010 ) .

we explore the computation of high - harmonic generation spectra by means of gaussian basis sets in approaches propagating the time - dependent schrdinger equation . we investigate the efficiency of gaussian functions specifically designed for the description of the continuum proposed by kaufmann _ et al . _ [ j. phys . b * 22 * , 2223 ( 1989 ) ] . we assess the range of applicability of this approach by studying the hydrogen atom , i.e. the simplest atom for which `` exact '' calculations on a grid can be performed . we notably study the effect of increasing the basis set cardinal number , the number of diffuse basis functions , and the number of gaussian pseudo - continuum basis functions for various laser parameters . our results show that the latter significantly improve the description of the low - lying continuum states , and provide a satisfactory agreement with grid calculations for laser wavelengths @xmath0 = 800 and 1064 nm . the kaufmann continuum functions therefore appear as a promising way of constructing gaussian basis sets for studying molecular electron dynamics in strong laser fields using time - dependent quantum - chemistry approaches .

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Proceed to summarize the following text: a liquid upon cooling undergoes first order phase transition and forms a crystal . however if the cooling rate is increased it can not crystallize and forms an amorphous glassy material @xcite . in addition to fast supercooling , there are other methods to favor glass formation over crystallization . in bulk metallic glass community the usual thumb rules are to at least have a two component mixture with negative enthalpy of mixing and a 12@xmath0 size ratio between the components @xcite . single component systems are known to crystallize in a fcc+hcp structure @xcite , thus multi - component systems are commonly used for making glasses . the negative enthalpy of mixing makes sure that the components remain in a mixed state and do not demix to form single component crystals , whereas the size ratio provides frustration in packing . although there is an array of experimental systems which form glasses , in computer simulation studies there is only a handful of systems known to be good glass former @xcite . note that most of the glass forming systems have global crystalline minima @xcite . thus depending on the barrier to crystallization it is just a matter of time for the systems to crystallize . with the increase in the available computational power some of the well known glass former like kob - anderson ( ka ) model and wahnstrom ( wa ) model are now found to crystallize @xcite . thus in order to design better glass formers we need to be able to estimate the glass forming ability ( gfa ) of these systems . in order to quantify gfa , first we need to understand the origin behind the stability against crystallization . this is an active field of research and different studies have attributed the gfa to different phenomena @xcite . the most popular among them is the theory of frustration first proposed by frank @xcite . according to him , the local liquid ordering is different from the crystalline order and this frustrates the system and decreases the rate of crystallization . it has also been argued that regions with locally favored structures ( lfs ) give rise to domains and are connected to the slow dynamics in the supercooled liquids @xcite . sometimes the lfs can also be related to the underlying crystalline structure @xcite . in some cases the lfs connected to crystal structure grows more than the one connected to the liquid structure @xcite . the locally favored structures can be different in different dimensions . there are lfs , like the icosahedral ordering , which can cause frustration in the euclidean space but tile the curved space @xcite . frustrations are not always structural but can also be energetic in nature @xcite . most binary equimolar mixtures form crystalline structures @xcite , where the crystal structure may vary according to the size ratio of the components . there are also some exceptions like the equimolar cuzr structure which is found to be a good glass former @xcite . however , when the composition of the mixtures are changed then it is usually found that close to the deep eutectic point many of them form glasses . one of the argument in favor of the deep eutectic point being a good glass forming zone is that the viscosity is highest at this point so kinetically it takes a longer time to form a crystal nucleus . however it has also been shown that the structural frustration between two different crystal structures can make this region a good glass former . this kind of phase diagram ( in temperature vs. composition space ) are often referred to as a v - shaped phase diagram where the bottom of the v is the glass forming region @xcite . in a recent work by some of us we have shown that even though all the systems at equimolar mixture undergo crystallization , as the composition of the larger size particles increases , the zone which forms cscl crystal at equimolar composition does not crystallize any more @xcite . it is already known from the study of energetics that the global free energy minima of these systems are cscl+fcc crystals @xcite . the well known ka glass former is one of the systems present in this more generic cscl zone and it has shown strong resistance towards crystallization even after being inserted with a cscl seed in the liquid @xcite . so far only in one study it has been reported to crystallize but in a structure which is different from that of the global minima @xcite . in the earlier study we have shown that in the cscl+fcc crystal structure the bigger a " particles need to have two different populations where there is a large difference in the order parameter ( coordination number and bond orientational order parameter ) of these two populations . according to us , this large difference in order parameter creates frustration . thus the stability against crystallization is attributed to the structural frustration between the cscl and the fcc crystal structure @xcite . in this present work we study a similar series of binary systems by changing the composition and also the inter species interaction length . many of the binary systems studied here are good glass formers and have a global minima which is cscl+fcc structure . thus according to our earlier study the structural frustration for these systems are similar . however , these different systems although share the same structural frustration are expected to have different glass forming ability . the goal of this work is to get a relative estimate of the gfa of different systems and then explore the origin behind their differences . our study shows that the free energy cost for cscl crystallization increases with the composition of the smaller particles . the system with lowest free energy cost also shows a pre - crystalline demixing in the liquid phase near the liquid / crystal interface . the demixing takes place due to the structural frustration between the cscl and fcc structures . upto a certain composition , the composition dependence of the free energy cost to create a crystal nucleus can be related to the composition dependence of this demixing entropy . our study of energetics shows that although in the whole range of composition the global minima is cscl+fcc crystal the driving force of crystallization in a certain region is the cscl crystal and in another region is fcc crystal . in the former region the system tends to demix and form cscl+fcc crystal and demixing frustrates the crystallization process . however , in the latter region we show that demixing does not play a crucial role . it is primarily the slow dynamics near eutectic point and lfs around the smaller b " particles which frustrate the crystallization process . the simulation details are given in the next section . in section iii we present the definition and method for evaluating different quantities , in section iv we have the results and discussion , and section v ends with a brief summary . the atomistic models which are simulated are two component mixtures of classical particles ( larger `` a '' and smaller `` b '' type ) , where particles of type _ i _ interact with those of type _ j _ with pair potential , @xmath1 , where r is the distance between the pair . @xmath1 is described by a shifted and truncated lennard - jones ( lj ) potential , as given by : @xmath2 where @xmath3 $ ] and @xmath4 . subsequently , we ll denote a and b types of particles by indices 1 and 2 , respectively . the different models are distinguished by different choices of lengths and composition parameters . length , temperature and time are given in units of @xmath5 , @xmath6 and @xmath7 , respectively . here we have simulated various binary mixtures with the interaction parameters @xmath5 = 1.0 , @xmath8 = 0.88 , @xmath9 = 1 , @xmath10 = 1.5 , @xmath11 = 0.5 , @xmath12 and the inter - species interaction length @xmath13 . we have simulated systems with different compositions , varying @xmath14 from 0.50 to 0.0 , where @xmath14 is the mole fraction of the smaller b type particles @xcite . the molecular dynamics ( md ) simulations have been carried out using the lammps package @xcite . we have performed md simulations in the isothermalisobaric ensemble ( npt ) using nos - hoover thermostat and nos - hoover barostat with integration timestep 0.005@xmath15 . the time constants for nos - hoover thermostat and barostat are taken to be 100 and 1000 timesteps , respectively . except for the liquid / crystal interface study where we use a rectangular box , all of the other studies are performed in a cubic box with periodic boundary condition . the free energy barrier calculations are done via biased monte carlo method . all the studies are performed at @xmath16 . bond orientational order ( boo ) parameter was first prescribed by steinhardt _ et al . _ to characterize specific crystalline structures @xcite . to characterize specific crystal structures we have calculated the locally averaged boo parameters ( @xmath17 ) of _ l_-fold symmetry as a 2__l__+1 vector,@xcite @xmath18 where @xmath19 here @xmath20 is the number of neighbours of the i - th particle and the particle i itself . @xmath21 is the local boo of the i - th particle . @xmath22 where @xmath23 are the spherical harmonics , @xmath24 and @xmath25 are spherical coordinates of a bond @xmath26 in a fixed reference frame , and @xmath27 is the number of neighbours of the _ i_-th particles . two particles are considered neighbours if @xmath28 , where @xmath29 is the first minimum of the radial distribution function ( rdf ) . for the liquids and the crystals the @xmath29 has been chosen as the first minima of the respective partial rdf of the a " type of particles . for the pure cscl crystal this comprises of 14 neighbours and for fcc 12 neighbours . in fig.[fig1 ] we plot the probability distribution of @xmath30 of the liquid at three different composition and also the same for pure cscl and fcc crystals . we note that at the level of this parameter all the three liquids can be clearly separated from the two different crystal forms . for the liquid at three different compositions @xmath31 at @xmath32 . we also plot the same for the cscl crystal made up of a " and b " type of particles and pure fcc made up of a " particles . ] we have calculated the relaxation times obtained from the decay of the overlap function @xmath33 , where @xmath34 . it is defined as @xmath35 the overlap function is a two - point time correlation function of local density @xmath36 . it has been used in many recent studies of slow relaxation @xcite . in this work , we consider only the self - part of the total overlap function ( i.e. neglecting the @xmath37 terms in the double summation ) . earlier it has been shown to be a good approximation to the full overlap function . so , the self part of the overlap function can be written as , @xmath38 again , the @xmath39 function is approximated by a window function @xmath40 which defines the condition of `` overlap '' between two particle positions separated by a time interval t : @xmath41 the time dependent overlap function thus depends on the choice of the cut - off parameter @xmath42 , which we choose to be 0.3 . this parameter is chosen such that particle positions separated due to small amplitude vibrational motion are treated as the same , or that @xmath43 is comparable to the value of the msd in the plateau between the ballistic and diffusive regimes . in order to calculate the crystallization rate and thus the glass forming ability we first determine the melting temperatures of the different crystals . the melting temperature is studied by calculating the temperature dependent growth / melting rate of the crystal and fitting them to a straight line . the temperature at which the growth rate cuts the temperature axis is the predicted melting temperature where the growth rate goes to zero @xcite . the simulations are done at @xmath16 . with the crystal at the center of the box and the crystal particles being pinned the liquid of 8000 particles is equilibrated at t=1.5 . the system is then quenched to the target lower temperatures and the crystal particles are unpinned . we then run a short equilibration of 1000 steps for the quenched system . depending on the temperature and the composition of the liquid the central seed either grows or melts . in the @xmath44 and @xmath45 systems we study the melting temperature of cscl crystal with an initial crystal seed of 432 particles . in the @xmath46 mixture we study the melting temperature of the pure fcc crystal comprising of 500 a " particles . the growth of the seed is monitored by cluster analysis where the @xmath30 is calculated for each particle and if the value of @xmath47 ( fig.[fig1 ] ) and it has a neighbour which is part of the existing cluster then it is included in the cluster . the cluster growth is monitored for about 100 - 500 @xmath48 , where @xmath48 is the temperature dependent @xmath49 relaxation time that varies across different systems . 5 - 10 independent runs are generated at each temperature by starting from the same initial configuration but randomized initial velocity . the growth rate is calculated by scaling the time w.r.t the corresponding @xmath48 . from the average growth / decay rate we approximate the melting temperature as the temperature where the predicted growth or decay rate is zero ( fig.[fig2 ] ) . the melting points obtained from fig.[fig2 ] is used to construct the composition dependent phase diagram reported in fig.[fig3 ] . system , the predicted melting temperature ( @xmath50 ) is 0.651 . ( b ) @xmath51 , @xmath52 . ( c ) @xmath46 , @xmath53 . ] ) . the melting temperature of the mixed cscl+fcc crystal and the distorted fcc crystal are obtained by step wise heating the system . ] we find that the @xmath44 mixture phase separates and forms a cscl+fcc crystal structure ( figs.[fig4]a ) . the @xmath51 mixture also shows similar tendency however the crystal growth rate is slower and within our simulation timescale the demixing is not complete . we also try to grow the cscl crystal in the @xmath46 mixture but we find that instead of cscl , fcc structure of a " particles grow around the initial seed ( fig.[fig4]b ) . this is similar to the observation reported earlier @xcite . when a fcc seed is inserted in the same mixture it continues to grow . in the above mentioned method it is not possible to calculate the melting temperature of the mixed cscl+fcc crystal as the growth of such crystal never happens . for this calculation at each composition ( @xmath54 ) we heat the mixed crystal ( cscl+fcc ) starting from temperature 0.2 - 0.3 and increase it up to 0.59 - 1.0 ( depending on the melting temperature of the crystal ) with temperature interval of 0.05 . closer to the melting temperature heating is done with 0.01 temperature interval . at each temperature equilibration is done for @xmath55 steps . the size of the initial crystal structure is in the range of 468 - 612 . the total number of particles are chosen in such a way that a perfect mixed crystal can be created . periodic boundary condition is applied in all directions . similar study is been done for the pure and distorted fcc crystal for @xmath56 systems . for @xmath57 we get pure fcc and for @xmath58 and @xmath59 the a " particles form fcc crystal but with distortion due to presence of the b " particles . in the @xmath46 system within our simulation run we could not form the fcc crystal . however as reported earlier in a mka2 model , if the interaction between the two species is reduced , then the system forms crystal @xcite . in a similar method by keeping the @xmath60 we first form a distorted fcc crystal of the @xmath46 system . once the crystal is formed we switch back to the larger inter species interaction of @xmath61 and study its melting . the melting of all the crystals happen instantaneously . the melting temperatures are reported in fig.[fig3 ] . in this section we perform a comparative study of the gibbs free energy ( potential of mean force ) of crystalline nucleation / growth in different binary mixtures using umbrella sampling technique with the reaction coordinate being the size of the largest crystalline cluster present in the system . the studies are performed at the same degree of undercooling at @xmath62 , where the melting temperatures used are those calculated by studying the temperature dependent growth / melting rate for the pure cscl and fcc crystals . a crystalline cluster is defined by a neighborhood criteria ( within a cut - off distance determined by the first minimum of the partial radial distribution of function of `` a''-type particles for respective systems ) of `` crystal - like '' particles ( with the criterion of @xmath63 ) . to grow the clusters we use a biased monte carlo approach , where we apply an external harmonic potential of the form @xmath64 , where @xmath65 is the force constant , @xmath66 is the number of particles in the largest cluster , and @xmath67 is the position of the bias window . we use @xmath68 for @xmath44 , and @xmath69 for @xmath70 and @xmath71 . we have used 5 - 7 umbrella windows ( depending on the system ) in the cluster size range of 15 - 35 . after equilibration , the data is collected for @xmath72 monte carlo steps per window and weighted histogram analysis method ( wham)@xcite is then used to compute the free energy as a function of the size of the largest cluster as reported in fig . [ fig5 ] . and @xmath45 we can grow the cscl cluster , whereas for @xmath46 we can only grow the fcc cluster . even with a initial small cscl seed the cluster that grows is of a " particles forming fcc lattice which is similar to that we find for melting study . ] , as obtained within each slab of width one @xmath5 as a function of the distance from the interface . the interface that has a " particles is taken and the plot is done for @xmath44 and @xmath51 . we find that for the former system where the initial rate of crystallization is higher the interface has higher concentration of b " particles compared to the bulk . thus there is pre - crystalline demixing in the liquid phase . ] while our calculations focus on the pre - critical region of the free energy surfaces , we can compare the relative free energy cost to form a crystalline nucleus of certain size as the composition of the system is varied . we observe that the free energy cost to grow a nucleus from 15 to 35 for all the systems are quite high ( in the range of 10 - 20 @xmath73 ) , which explains why all these systems are good glass formers . a comparative study of the cost of free energy shows that @xmath44 has a lower cost to grow a cscl crystal compared to @xmath51 . this explains the slow growth of the cscl crystal in the latter system which is observed during the melting study . we also try to grow cscl crystal for @xmath46 , which we do not observe during our simulation time . this implies that the free energy cost for cscl crystal growth in this system is even larger . however , similar to the melting study the crystal that grows around the initial cscl cluster in the @xmath46 system is made up of only `` a '' particles . next we study the free energy cost for fcc crystallization in @xmath46 system . we find that the free energy cost to grow a fcc crystal from 15 - 35 cluster size in @xmath46 system is lower than the free energy cost to grow a similar size range cscl crystal for @xmath51 . this implies that in the @xmath46 system the free energy cost for fcc crystallization is lower than the cscl crystallization . note that although we make this comparative statement we are unable to determine the free energy cost for growing a cscl crystal in the @xmath46 which leads us to believe that the cost must be very high . , that of the partially demixed pre - crystalline liquid @xmath74 and the difference between them @xmath75 plotted at different compositions . we also plot the @xmath76 where @xmath77 is the melting temperature of the mixed cscl+fcc crystal . the @xmath75 shows a non monotonic composition dependence with a maxima around @xmath46 . ] we next analyze the origin behind the difference in the free energy cost to grow a cscl crystal in different systems . in a recent study of crystallization in @xmath78 mixture it is found that the barrier to crystallization for the mixed system is about @xmath79 higher than the pure system @xcite . the @xmath80 and the @xmath81 have a small difference in their sizes and form fcc crystal structure . thus unlike structural frustration between the cscl and fcc crystal present in the systems studied here @xcite there exists no structural frustration in the @xmath78 system . however due to higher @xmath82 interaction the crystal nucleus for the @xmath78 system has a higher concentration of the @xmath80 molecules compared to that in the bulk . this leads to demixing in the system and the authors concluded that this demixing leads to higher barrier . in a separate study it is shown that the phase that nucleates easily is the one which has composition closer to the liquid @xcite . in this present study we note that in the cscl crystallization process , except for the equimolar mixture the composition of the nucleus is different from that of the liquid . the difference increases as we go towards smaller @xmath83 values . thus it is obvious that the growth of cscl crystal leads to demixing in the system . however , we would like to investigate if signature of demixing is present in the liquid which surrounds the crystal . in a recent study it has been shown that the liquid in the crystal / liquid interface shows some compositional ordering @xcite . in a similar spirit we now look at the crystal / liquid interface and investigate if the demixing takes place in the pre - crystalline liquid . for this study we perform @xmath84 calculation in a rectangular box where @xmath85 . initially the system consists of 432 cscl crystal particles ( equal amount of a " and b " particles ) and 864 liquid particles . the 010 layer of the crystal faces the liquid where the last layer of the crystal on one side has a " particles and on the other side has b " particles . the box length in the @xmath86 and @xmath87 direction is @xmath88 . the box length in the @xmath89 direction is @xmath90 . period boundary condition is applied in all directions . since we want to study the interface property it is important to not have a rugged interface thus the study is performed above the melting temperature of the pure cscl crystal in the respective liquid ( @xmath91 ) by pinning the crystal particles . although performed above the melting temperature while equilibrating the @xmath44 system we find the growth of a layer of particle on both sides of the crystal . in the analysis we consider these two layers , which are not pinned , to be part of the crystal . thus for this system after equilibration there are 504 crystal particles and 792 liquid particles . the liquid particles span over more than @xmath92 distance which makes it possible to study both the interfacial and bulk properties of the liquid . for the @xmath51 system an extra layer of a " particles grow on the surface which has b " particles facing the liquid . due to the scarcity of b " particles no extra b " layer grows on the other side . in this analysis we consider the surface where the extra layer of a " particle has grown . we calculate the fraction of b " particles , @xmath93 , within each slab of width @xmath94 , as a function of distance from the interface . the first point ( z=0 ) in this plot is taken within the crystal which for the both the systems show same value of @xmath93 . interestingly we find that for the @xmath44 system the concentration of the b " particles are higher at the interface and it gradually reaches the bulk value around z=4 . however for @xmath51 system the concentration of the b " particles are same at the interface and at the bulk . thus we show that the liquid which has a lower free energy cost for crystal growth also undergoes a pre - crystalline demixing in the liquid phase . similar to the earlier study @xcite we find that the liquid / crystal interface properties differ for apparently similar systems with different glass forming ability . thus we show that the process of crystallization requires demixing which takes place in the pre - crystalline liquid . we now analyze the role of demixing in the free energy barrier . note that the per particle mixing entropy in a liquid can be written as , @xmath95 where @xmath96 is the mole fraction of the components . to form cscl+fcc crystal the liquid needs to demix . we show here that the demixing takes place in a liquid state ( refer to fig . [ fig6 ] ) . although the demixing process happens step wise here we calculate the total effect of demixing . thus we consider that to form a cscl+fcc crystal , part of the liquid needs to form a equimolar mixture and the other part should have pure a " particles . thus the per particle mixing entropy in the pre - crystalline partially demixed liquid should be , @xmath97 the difference between these two entropies , @xmath98 , is the mixing entropy at per particle level that a liquid will loose in the process of partial demixing . @xmath75 as a function of @xmath83 is shown in fig . [ fig5 ] which shows a non - monotonic behaviour with a maximum around @xmath99 . note that this kind of non monotonic behaviour is obtained in the free energy barrier to crystallization for the @xmath78 mixture which as discussed earlier is attributed to the demixing process@xcite . thus our demixing entropy study can explain the increase in the free energy cost for cscl crystal growth with decrease in @xmath83 till it reaches a value of @xmath100 . however this study does not explain why in the @xmath46 system where cscl+fcc is the global minima the free energy cost for fcc crystllization is much lower than cost for the cscl crystallization , the latter being so high that an estimation of it is beyond the scope of the present study . in order to understand the origin behind lower free energy cost for fcc crystallization we analyze the role of different crystal structures in crystallization by studying the energetics . in fig.[fig8]a we plot the energy per particle of the liquid , the mixed crystal , the fcc crystal and the cscl crystal for different compositions at 0.8 times the melting temperature of their respective mixed crystals ( given in fig.[fig1 ] ) . this is the melting temperature which has been obtained by step wise heating the mixed crystal . we find that the energy of the mixed crystal is always lower than the supercooled liquid , which implies that the liquid is in a metastable state . the energy of the cscl crystal is always lower than the liquid . however for higher @xmath83 values the energy of the fcc crystal is above the liquid and at lower @xmath83 values although it becomes less than the liquid it is always higher than the cscl value . this would imply that the cscl crystal always drives the crystallization process . however this does not explain why both in the melting study and the free energy barrier calculation at @xmath46 although we can not grow cscl crystal we can grow fcc crystals . our subsequent analysis will explain this discrepancy . next we make an estimation of energy of the mixed crystal , @xmath101 , at the per particle level , at different compositions by assuming that 2@xmath83 of the crystal forms cscl and the rest forms fcc . @xmath102 here @xmath103 and @xmath104 are the energy of the cscl and fcc crystal respectively , at per particle level calculated for each system at their respective @xmath62 . @xmath105 and @xmath106 are the estimated contribution from the respective cscl and fcc crystal part of the mixed crystal again presented at the per particle level . note that the values of @xmath105 and @xmath106 take into account the fraction of the system which is in different crystal form . in this calculation we of course make some mistake by neglecting the surface energy . however we find that the value of energy per particle of the mixed crystal thus calculated is not too different from the value of the actual crystal ( fig.[fig8]b ) . these are again calculated at the same temperatures as reported in fig.[fig8]a . we now break up the contribution of the two components , the contribution from cscl and that from fcc and plot them separately . once we do that we find that although at higher @xmath83 values the cscl formation drives the crystallization at lower @xmath83 values it is the fcc crystallization which drives the crystallization . although the energy per particle of the fcc crystal is still lower than that of the cscl , the larger fraction of the fcc crystal wins over . a cross over happens just above @xmath46 . this explains why the system at @xmath46 , whose global minima is the mixed crystal , shows higher tendency towards fcc formation . however , this does not explain why the the crystallization process when driven by fcc formation has a lower free energy cost than when driven by cscl formation . in order to understand this , we study the coordination number between the b " particles , @xmath107 in the @xmath46 system , before and after crystallization . since we can not crystallize the @xmath46 system we study the crystallization of the mka2 model ( referred earlier in the melting temperature study ) which according to dyre and coworkers is similar in structure as the ka model but with a lower viscosity @xcite . confirming their conclusion we find that the lfs of the mka2 model appears quite similar to the ka model however the dynamics is orders of magnitude faster . we now analyze the @xmath107 as obtained in the mka2 system when it is in liquid form at @xmath108 and when it forms distorted fcc crystal around @xmath109 . these are plotted in fig . [ fig9 ] . for comparison we also plot the @xmath107 for the pure cscl+fcc crystal at @xmath108 . note that the probability distribution of @xmath107 in the cscl+fcc crystal should ideally have a peak at 6 but the peak is shifted to smaller value due to the presence of a large number of surface layer of b " particles . the study shows that to form distorted fcc crystal although there is an increase in the @xmath107 it is not as much as required for the cscl+fcc crystal . thus demixing in the distorted fcc is much weaker that cscl+fcc . analysis of the same kind for the @xmath110 system ( not shown here ) shows similar behaviour . thus on the right hand side of the crossover where cscl drives the crystallization there should be free energy barriers due to demixing . however on the left hand side the system can avoid or reduce the loss of mixing entropy by paying some energetic penalty to form distorted crystals . the lower energetic stability of the distorted fcc is evident from fig.[fig1 ] . we find that for @xmath46 the disordered fcc structure melts at a lower temperature compared to the mixed cscl+fcc structure . [ h ] as obtained in the mka2 liquid at @xmath108 , the distorted fcc crystal formed by the mka2 liquid at @xmath109 and cscl+fcc crystal at @xmath108 . the cscl+fcc is formed at the same composition as the mka2 liquid . the demixing required by the cscl+fcc liquid is much higher than the distorted fcc . ] in order to strengthen our argument that it is indeed the demixing that frustrates the cscl driven crystallization process and leads to high free energy cost , we present a study of a similar system . reported in a earlier study by some of us we have shown that for the nacl system ( @xmath111 ) the crystallization takes place not only at equimolar composition but also at smaller value of @xmath83 forming mixed nacl+fcc crystal @xcite . a similar energetic study of the @xmath111 system is shown in fig.[fig10]a . we find that energy of the nacl crystal is always lower than the fcc crystal ( fig.[fig9]a ) ) . a similar crossover is also obtained for this system where at higher @xmath83 values the crystallization is driven by nacl and at lower @xmath83 values it is driven by fcc ( fig . [ fig10]b ) . thus we should expect a similar crystallization problem in this system which appears not to be the case . although the nacl and cscl systems appear quite similar there are some basic differences . the cscl crystal is made up of two interpenetrating sc structures of a " and b " type of particles . thus in the cscl+fcc crystal the a " particles have two different population one which forms sc and the other which forms fcc structure . in an earlier work we had mentioned that this wide difference in the order parameter of the two population causes the frustration between the two structures@xcite . if we are away from the equimolar mixture the growth of a cscl will deplete the population of the b " particles in the neighbourhood which should promote the formation of fcc structure between the a " particles . however a unit cell of fcc is not compatible with the cscl structure thus to reduce the structural frustration the system sacrifices the mixing entropy and increase the concentration of the b " particles in the liquid near the cluster as seen in fig.[fig6 ] to form more cscl structures till finally it is devoid of any more `` b '' particles in the liquid . this is the reason we find ab " and a " rich zone separated in fig.[fig4]a . the nacl crystal on the other hand is compatible with a fcc crystal as both require the a " particles to form fcc structure with same lattice spacing . thus unlike cscl and fcc the nacl and fcc can grow in a seamless fashion and the system does not require any demixing which reduces the free energy barrier . a snapshot of the nacl+fcc structure is shown in fig.[fig11 ] which shows that there is no specific a " rich zone . [ h ] . the simulation is done at t=0.6 . the snapshot shows seamless formation of nacl and fcc structure with no demixing . ] the study of the energetics can also explain the glass forming ability of some systems which has been previously proposed by dyre and coworkers @xcite . in the above calculation if we decrease the interaction between the ab " particles then the contribution from the cscl in lowering the system energy will decrease and the crossover will happen at a higher @xmath83 value . thus the @xmath46 will show higher tendency of fcc crystallization as has been reported earlier @xcite . in the same system if we make the interaction between the a " particles repulsive then in a similar fashion the crossover will shift to lower @xmath83 values and this will imply that the @xmath46 system will still be driven by the cscl crystallization . since this will also require demixing thus the system will be a better glass former as reported earlier @xcite . we find that the loss of mixing entropy is maximum for @xmath46 . in the free energy study within the scope of our calculation we can not grow a cscl crystal and thus can not estimate the free energy cost to grow a cscl crystal in this system . which implies that free energy cost is high and w.r.t cscl formation the @xmath46 system is most frustrated and a better glass former . however in the free energy calculation and the study of energetics this system shows a tendency towards fcc crystallization . the fcc crystallization also has a free energy cost because although without clear demixing the system can crystallize in fcc structure the presence of the lfs centered around b " particles can frustrate this crystallization process . however the cost of free energy to form a fcc crystal in @xmath46 system is lower than the cost of free energy to form a cscl crystal in @xmath51 system . thus it is tempting to comment that the @xmath51 system is a better glass former . however the process of crystallization is not only dependent on the free energy barrier but also on the dynamics of the system . this is the reason the eutectic point is expected to be a better glass forming region and our study of dynamics shows that indeed the @xmath46 system is the slowest . for the study of the dynamics we calculate @xmath48 from overlap function at the respective 0.8@xmath77 . note these are the temperatures where the free energy calculations are done . we find that for the @xmath44 system @xmath112 at @xmath113 , for the @xmath51 system , @xmath114 at @xmath115 and for @xmath46 system @xmath116 at @xmath117 . thus according to the study of the dynamics the @xmath46 system is a better glass former . note that the mka2 model which undergoes crystallization differs from the ka model not in terms of the structure of the liquid but in terms of dynamics . the local structure around the smaller b " particles which actually frustrates the fcc crystallization is present even in the mka2 model . however the relaxation timescale of the mka2 model is orders of magnitude faster than the ka model . thus our study confirms that as stated earlier@xcite it is indeed the dynamics / viscosity of the system which makes ka model a good glass former . in this article we study the comparative glass forming ability of different binary systems . in an earlier study by some of us we have shown that binary systems which form cscl crystals in a equimolar mixture fails to crystallization if the mole fraction of the larger particles are increased @xcite . the well known ka model is one of the systems . thus the ka models stability against crystallization is more generic and is similar to systems which form equimolar cscl crystal . the global structure for these systems are a mixed form of cscl+fcc crystal @xcite . in the cscl+fcc crystal the bigger a " particles need to create two different population one which contributes towards the cscl formation and the other which contributes towards the fcc formation . the order parameters such a boo and coordination number of the a " particles are quite different in these two crystal form . thus the failure to crystallize has been attributed to the frustration between the cscl and fcc crystal structure . note that there is an array of systems which have similar frustration . however the glass forming ability of these systems although have not been calculated but is believed to be different . thus there should be more factors contributing to the glass forming ability . in this article we perform a comparative study of binary glass forming liquids all having good glass forming ability and similar global minima . the study has been performed by changing the composition . we find that the free energy cost to grow a cscl nucleus increases as we move away from an equimolar mixture . the study of the liquid at the liquid / crystal interface shows that the system which has lowest free energy cost to form a nucleus also shows a demixing near the crystal surface . we believe that the structural frustration between the cscl and fcc structure makes this demixing a prerequisite for crystallization . our calculation of the partial demixing entropy in the liquid state shows a non monotonic dependence on composition . it shows a maxima for @xmath46 system . we could show a connection between the change in free energy cost to create a crystal nucleus and the change in demixing entropy as a function of composition . our study shows that although the @xmath46 system is strongly frustrated against cscl crystallization , it has tendency towards fcc growth . we can justify this tendency of fcc growth from the study of the energetics . we show that in the composition range studied here there are two regions , one which is driven by the cscl crystallization and the other at lower @xmath83 values is driven by fcc crystallization . it is primarily in the former region that the structural frustration between the cscl and fcc structure leads to the requirement of demixing which eventually increases the free energy barrier and provides stability against crystallization . this point has been confirmed by studying a nacl+fcc system which naturally undergoes crystallization at all compositions . the study of the energetics of this system also shows two similar region . however unlike the cscl+fcc system , in the region where crystallization is driven by nacl , due to the compatibility of the nacl and fcc structure no demixing has been observed and the crystal grows in a seamless fashion . in the second region driven by fcc crystallization we show that demixing is not a stringent criteria and the stability against crystallization comes from the frustration caused by the presence of the b " particles with well defined lfs and also the systems proximity to eutectic point where the dynamics is slow . thus although we study three very similar glass former , which ideally belong to the same class of system and differ only in composition , we find that they do not share the same origin of stability against crystallization . we should also comment that our search of crystal structures is not exhaustive and the system which we claim to be a better glass former can crystallize in a different crystal form like the @xmath118 structure is found to be a low energy state of a system belonging to the same class where @xmath119@xcite . this system also is known to show resistance towards crystallization . the cuzr liquid which has a low energy cscl like structure is also a good glass former @xcite . note that in these two systems the composition of the crystal is identical to that of the liquid . thus even above @xmath120 it is not always demixing which provides stability against crystallization . 53 ifxundefined [ 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

we present a comparative study of the glass forming ability of binary systems with varying composition , where the systems have similar global crystalline structure ( cscl+fcc ) . biased monte carlo simulations using umbrella sampling technique shows that the free energy cost to create a cscl nucleus increases as the composition of the smaller particles are decreased . we find that the systems with comparatively lower free energy cost to form cscl nucleus exhibit more pronounced pre - crystalline demixing near the liquid / crystal interface . the structural frustration between the cscl and fcc crystal demands this demixing . we show that closer to the equimolar mixture the entropic penalty for demixing is lower and a glass forming system may crystallize spontaneously when seeded with a nucleus . this entropic penalty as a function of composition shows a non - monotonic behavior with a maximum at a composition similar to the well known kob - anderson ( ka ) model . although the ka model shows the maximum entropic penalty and thus maximum frustration against cscl formation , it also shows a strong tendency towards crystallization into fcc lattice of the larger a " particles which can be explained from the study of the energetics . thus for systems closer to the equimolar mixture although it is the requirement of demixing which provides their stability against crystallization , for ka model it is not demixing but slow dynamics and structural frustration caused by the locally favored structure around the smaller b " particles which make it a good glass former . although the glass forming binary systems studied here are quite similar , differing only in composition , we find that their glass forming ability can not be attributed to a single phenomena .

You are an expert at summarizing long articles. Proceed to summarize the following text: membrane hemifusion is a possible pathway ( see @xcite for an alternative view ) to the complete fusion of membranes @xcite . current theories associate the initiation of hemifusion with the formation of a contact zone between the membranes in which the two proximal monolayers are connected by a stalk - shaped neck . the stalk then expands and a region is formed ( region c in fig . [ fig : setup ] ) , in which the two distal monolayers form a single bilayer . in general , the energetic cost of the splay of the lipid chains in the stalk , prohibits its spontaneous expansion . however , the presence of additional , external forces ( e.g. pressure , surface tension gradients , electrostatic effects ) can lead to expansion of the stalk into a hemifusion region and to the growth of this zone . clear evidence for the existence of these two distinct pre - fusion stages , stalk formation and hemifusion , was found for peg mediated fusion of vesicles @xcite . a recent theoretical paper @xcite suggested that the flow of lipids from region b to region a can be caused by an increase of the surface tension in region a due to the presence ( in that region only ) of additional polymer in solution . the tension gradient between these regions induces a flow of lipids , that leads to the growth of region c. a different scenario , where hemifusion can be an alternative pathway to fusion was found in influenza hemagglutinin - mediated fusion @xcite . the initial local stalk may evolve to a fusion pore @xcite , or it may expand to hemifusion . in the latter case , no fusion occurs . in this paper , we predict the dynamics of the expansion of the initial stalk and its role in the growth of a mesoscopic hemifusion diaphragm . the nucleation of a stalk by thermal fluctuations was recently shown to be thermally accessible @xcite . a detailed description of the kinetics of this nucleation event ( that typically describes the formation of a stalk of several nanometers in extent ) is outside the scope of our work . instead , we focus on estimates of the conditions that facilitate stalk expansion into hemifusion . we discuss the implications of our theory on biological fusion mechanisms and on in - vitro experiments . in addition , we predict the growth of the hemifusion region ( e.g. from nanometers to microns ) as a function of time and discuss the physical parameters that can be used to control the time scale for hemifusion . this dynamic part is relevant mainly to in - vitro experiments , since biological fusion events generally remain at the microscopic scale of the stalk . if hemifusion is an intermediate state of fusion then it is important to contrast the time scales of hemifusion diaphragm expansion and pore formation , in order to determine the rate limiting step . @xcite predicted that pore expansion is exponential in time , with a time scale of @xmath0 sec , where @xmath1 is the membrane viscosity and @xmath2 is the surface tension difference ( both are estimated below ) . however , if pore nucleation is slow enough significant expansion of the hemifusion diaphragm can occur before pore formation . this is the case considered here , where we predict that the hemifusion diaphragm expands as the square root of time . our theoretical model is motivated by and consistent with the experiments described by @xcite , where two bilayers supported on mica surfaces were brought into contact in the presence of a peg - water solution . hemifusion , that eventually extended over a distance of @xmath3 was observed in a time of about 10 minutes , while the time it took the initial stalk to form was less then 3 minutes . this suggest that , at least in this experiment , the rate limiting step for hemifusion is the expansion of the fusion zone , as opposed to stalk formation . this paper presents a simple theoretical model relevant to this experimental system @xcite , and predicts the time dependence of hemifusion expansion . the overall time scale we find is comparable with the measurements of @xcite while the details of the predicted temporal dependence have yet to be tested experimentally . our theoretical model is illustrated in fig . [ fig : setup ] that is a simplification of the experimental system of @xcite wherein two bilayers deposited on mica cylinders are brought together in a solution of peg and water . the lipids of the distal monolayers are physisorbed on the mica ; this fixes their lateral density . from here on in this paper , the term lipid density relates to the lateral density of the proximal monolayers ( see fig . [ fig : setup ] ) . we assume that the lipids are in _ local _ equilibrium , so at a particular location @xmath4 , the free energy per lipid ( in the proximal monolayers ) @xmath5 , does not depend on the lipid microstate , but only on the lipid density @xmath6 . this assumption of local equilibrium is consistent with our results that predict an overall time scale for hemifusion expansion that is much larger than the local diffusion time of a single lipid molecule . the experimental system we consider is macroscopically cylindrically symmetric and we therefore assume cylindrical symmetry of all the physical quantities at mesoscopic length scales . this is justified because all flows ( of water and lipids ) are laminar , and there are no mechanisms that might induce angular fluctuations or instabilities . we distinguish between three regions , illustrated in fig . [ fig : setup ] : * * region a * - where the distance @xmath7 between the bilayers is typically much larger than the polymer correlation length @xmath8 @xcite . in this region , the outer lipid monolayer is in contact with the peg in the solution . the free energy per molecule in this region is given by @xmath9 , and is different ( in its functional form ) from the free energy @xmath10 of the monolayer in the absence of peg . * * region b * - where @xmath11 . for these values of @xmath7 , the peg density near the bilayers is negligible and our model assumes that there is no peg in contact with the bilayers in this region . the free energy per lipid in this zone is @xmath12 . in addition we assume that the distance between the mica surfaces is constant ( the mica surfaces in the experiment are deformed and flattened under pressure ) , and that this region is ring - shaped with an outer radius @xmath13 , and an inner radius @xmath14 . * * region c * - the region where the distal bilayers are in contact . the bilayers are langmuir - blodgett deposited in water , without peg , which is added later . the energy per lipid when the monolayers are in contact with water is @xmath10 and the proximal monolayers are initially langmuir - blodgett deposited with a density @xmath15 that minimizes @xmath16 . when peg is added , it induces an effective attraction between the polar heads @xcite , and changes the functional form of the energy as function of the lipid density to @xmath17 . the effect of lipid condensation in the presence of peg @xcite hasa been discussed in terms of the the dehydration of the bilayer by the peg @xcite . this dehydration affects the lipids in region a that are in microscopic proximity to the peg , but has no effect on the lipids in region b. in sec . [ sec : pressure ] we demonstrate that the osmotic pressure induced by the peg is too small to induce hemifusion . this stands in contrast to the surface tension effects that are the main focus of our work . if equilibrium could be reached , the lipid density in region a would tend to increase in the presence of peg . however , the number of lipids in the monolayers can not increase to any significant degree within the time scale of the experiments , since the concentration of lipids in the bulk solution is negligible and the number of lipids that can be transported from region b to region a is much smaller than that of region a. thus , the lipid density is unchanged and the energy per lipid in region a is now @xmath18 , with the derivative @xmath19 due to the induced head attraction . this condensation effect thus leads to a _ negative tension _ in the proximal monolayers that ideally would cause them to contract in extent . they can not do this without exposing the chains of the inner monolayers to the water and this is energetically prohibitive . the outer monolayers are therefore stressed and one way of relieving that stress is for additional lipid to enter this region ; this will allow the local lipid density to increase , while still covering the original area occupied by the outer monolayer . the peg concentration near the outer monolayers in region b is given by @xmath20 , where @xmath21 is the peg concentration near the outer monolayers in region a , @xmath8 is polymer correlation length , and @xmath7 is the distance between the bilayers in region b @xcite . since by the definition of region b , the bilayer spacing in that region is small , @xmath11 , we have @xmath22 and the peg concentration in region b is negligible ; we thus take this concentration to be zero . the energy per lipid in region b is _ initially _ given by @xmath23 , where @xmath15 is the lipid density in the absence of polymer . since the free energy per lipid , @xmath16 , is minimized when the density @xmath24 and the tension in region b initially vanishes , since either expansion or compression of the lipids will increase their energy . the tension gradient between regions a ( initially at negative tension ) and b ( initially at zero tension ) induces a flow of lipids from region b to region a. since region a is much larger than region b , we can treat it as a reservoir , and assume that even though lipid is flowing from region b to region a , the lipid density in region a is not changed from its initial value of @xmath15 . the system is a dynamical one and the chemical potential ( equivalent in our single component system to the free energy per lipid , @xmath16 ) is not constant in all of space at the mesoscopic scale ; this results in lipid flow and dynamics . however , since local equilibrium _ is _ maintained , we must have equal chemical potentials at any given point in the system . in particular , at the boundary between regions a and b , the chemical potentials of the lipids must be equal : @xmath25 , where @xmath26 is the lipid density at the edge of region b. we note that this equality of chemical potentials determines the lipid density at the boundary of region b , @xmath27 ; the functional form of the two free energies @xmath10 and @xmath17 are not the same , since in region a , the lipids are in contact with polymer . the initial lipid density in region b ( @xmath15 , which is the density at which the lipids self - assemble in water in the absence of polymer ) is higher than the lipid density at the ab boundary : @xmath28 . this inequality is a consequence of the fact that the tension at the boundary is negative , as shown in section [ sec : boundcond ] . more intuitively , the negative tension in region a tends to pull in additional lipids from the boundary region of region b into region a as explained above . this lipid flow reduces the lipid density at the boundary @xmath29 from @xmath15 to @xmath27 . in turn , the reduced lipid density at the boundary of regions a and b , ( @xmath30 ) induces a flow of lipids from the rest of region b towards the boundary . this is because the minimum energy state in region b is one where @xmath31 ; thus lipids from the entirety of region b flow to the boundary in an attempt to restore the lipid density there to values closer to @xmath15 . this flow , in turn , reduces the lipid density at the boundary between regions b and c ( the hemifusion region ) at @xmath32 , and lead to a negative tension that tends to expand region c. at the boundary of regions b and c , the lipid density is determined by a force balance between the membrane negative tension ( arising from the lipids flowing to the ab boundary ) , that tends to expand region c , and the force exerted by the boundary ring around region c that tends to shrink it . the main contribution to the energy of this ring is of the tilt of the lipid tails imposed by the toroidal geometry . this tilt is needed in order to form the three - way junction of the boundary ring cross section while avoiding an intra - membrane void , which has a much higher energetic cost @xcite . the energetic cost of the tilt can be considered through the related intra - membrane strain and the adjacent stress tensor @xcite . we assume that for @xmath33 the energetic cost @xmath34 for a cross section of the bc boundary ring is independent of @xmath14 . thus , the ring energy is given by @xmath35 . the force per unit length that the ring exerts on region b of the membrane tends to shrink region c and pull region b in the @xmath36 direction . this force ( per unit length ) is @xmath37 and tends to shrink the boundary ring ; that is , the expansion of region c is energetically costly . in local equilibrium , this force is balanced by the surface tension @xmath38 , which may be considered as a two dimensional lateral lipid pressure , in region b of the monolayer that tends to expand the ring : @xmath39 negative tension in region b tends to cause this region to contract and thus provides a force in the @xmath40 direction , balancing the force due to the bc boundary . in this section we derive the dynamics that govern the expansion of the hemifusion region and predict the flow of lipids within the monolayer as a function of the lipid density and of time . there are three local , dissipative forces that oppose any lipid motion . * the stress , or force per unit area due to the viscosity of the water that is moved along with the lipids is given by @xmath41 , where @xmath42 is the water velocity and @xmath43 erg s/@xmath44 is the viscosity of water . the stress is of order @xmath45 , where @xmath46 is the lipid velocity and @xmath7 is the spacing between the bilayers in region b. * the stress , or force per unit area due to the monolayer viscosity is given by @xmath47 , where @xmath1 is the monolayer friction coefficient @xcite . for a laminar flow we estimate this stress as @xmath48 ; that is , the relevant dimension is the size of region b in which there is monolayer flow . * the stress , or force per unit area that is due to the friction _ between _ the monolayers is given by @xmath49 , where @xmath50 is the friction coefficient . this stress depends only on the motion of the outer relative to the inner monolayer where there is no flow ; there is therefore no dependence of the length scale related to the geometry of the different regions . the friction between a dmpc monolayer and a supporting hts monolayer at @xmath51 is @xmath52erg s/@xmath53 , while for a supported ots monolayer the friction is @xmath54 erg s/@xmath53 @xcite . the experiments of @xcite were carried out at @xmath55 . it has been observed that the diffusion coefficient of a molecule in a dmpc monolayer increases about three folds when @xmath56 is increased from @xmath55 to @xmath57 @xcite , which suggest a corresponding decrease in @xmath50 . in this work we use an estimated value of @xmath58 erg s/@xmath53 . for dmpc bilayers at @xmath59 the bilayer viscosity is @xmath60 erg s/@xmath61 @xcite . the values relevant to the experiments of @xcite are @xmath62 cm and @xmath63 cm . with the estimates for the stress given above , we find that the frictional force due to relative motion of the two monolayers is much larger than either the lipid or water viscosity contributions to the stress . we thus neglect these latter two effects and predict the dynamics for a system where the only relevant dissipation is due to the relative friction between the monolayers . the lipid flow is induced by the tension gradient @xmath64 , and is opposed by the frictional @xmath65 . the force balance equation is @xmath66 in appendix [ ap : local ] we derive the lipid local dynamics using eq . [ eq : fb ] and the continuity equation . we consider the dynamics only to first order in the lipid density variations @xmath67 , which is known from experiments to be small . in @xcite a variation of @xmath68 was measured . to first order in @xmath69 the local dynamics has the form of a diffusion equation @xmath70 where @xmath71 is the harmonic spring constant of the monolayer . for a small density variation @xmath72 the surface energy cost is @xmath73 , and the related tension difference is @xmath74 . we note that the _ surface energy _ @xmath75 is the gibbs free energy per unit area , and is different then the _ surface tension _ @xmath76 , which has the thermodynamic role of the two dimensional pressure . we estimate @xmath77 using the phenomenological form @xmath78 where @xmath79 is the effective surface tension of the hydrocarbon - water interface @xcite . the second term in eq . [ eq : musigma ] accounts for the ( electrostatic ) effective head - group repulsion , while the first term represents the effective hydrocarbon - water repulsion . we note that this effective repulsion is smaller than the repulsion of the bare hydrocarbon - water interface , and has been estimated as @xmath80 erg/@xmath61 @xcite . from eq . [ eq : musigma ] we obtain @xmath81 erg/@xmath61 . for @xmath58 erg s/@xmath53 the effective diffusion constant is @xmath82 @xmath83sec . this quantity is larger than the actual , microscopic diffusion constant measured for free liquid bilayers above the gel transition , that are of the order of @xmath84 @xmath83sec @xcite . the einstein relation is not applicable in our case , since the flow ( that happens to scale like diffusion ) of the lipids from the high to low density regions is not due to the random motion of the molecules , but due to the tension gradient @xmath85 . indeed , for a characteristic molecular area @xmath86 @xmath61 we find that the related energy per molecule is @xmath87 . the boundary conditions for the lipid density were already discussed in section [ sec : thmod ] and we review them here for convenience . the local tension equilibrium at the boundary with region a determines the local lipid density @xmath27 at @xmath13 . in appendix [ ap : local ] we show that the tension in the monolayer is given by @xmath88 since the tension in region a is negative , from the tension equality at the boundary we see that @xmath89 is negative . moreover , because the function @xmath10 has a minimum at @xmath15 it is convex in a neighborhood of @xmath15 . if @xmath27 is in that neighborhood , then the condition @xmath90 yields that @xmath91 . the boundary ring near the hemifusion region at @xmath14 exerts a force that opposes hemifusion expansion ; this is because the boundary energy of the hemifusion region is increased as this region grows . this force is locally balanced by the negative tension in region b where lipids are flowing towards region a. as lipids pass from region b to a the lipid density in region b decreases ; the tension in region b , and in particular near its boundary with region c , becomes more negative and pulls on region c causing its expansion . the density of lipids in region b at the boundary @xmath14 is determined from the force balance eq . [ eq : fbeq ] . using eq . [ eq : pl ] we may write eq . [ eq : fbeq ] as @xmath92 before the flow begins , the initial lipid density in region b is @xmath15 , which implies that @xmath93 . for this value of the lipid density there is zero tension in region b , the stalk does not expand and hemifusion does not develop . due to the tension gradient between region b and a , lipids flow out of region b and a negative tension is built up . if at a certain time the lipid density at @xmath32 is low enough so that eq . [ eq : bc1 ] is satisfied , the stalk begins to expand . after the flow of lipids is initiated , lipids are removed from region b as they flow towards region a and the lipid density in region b is lower than @xmath15 . the lipid density in region b can not , however , be smaller than the value of @xmath27 , because when @xmath94 the free energies per lipid in regions a and b are equal , and the flow stops . thus , we require @xmath95 in all of region b if there is to be flow and stalk expansion that leads to hemifusion . at an early time after the stalk formation , while the stalk does not expand , the lipid density in all of region b approaches the equilibrium density profile @xmath96 . using eq . [ eq : bc1 ] , the condition for the stalk to begin to expand with a finite amount of time is : @xmath97 where @xmath98 and @xmath99 is the radius of the stalk . in our model , we consider the process for @xmath14 much larger than the molecular size @xmath99 that characterizes the size of the stalk . the tilt energy @xmath34 is in general positive . from eqs . [ eq : pl ] and [ eq : inicond ] , for @xmath100 we have @xmath101 . since we consider all quantities only to first order in @xmath102 we use the approximation @xmath103 . in appendix [ ap : global ] we use the integral continuity equation , that expresses the conservation the lipid number in the system , in order to obtain a dynamic equation for the hemifusion radius @xmath14 . in appendix [ ap : adiabatic ] we show that the time scale that governs the local dynamics is much faster then the rate of change of @xmath14 . we use an adiabatic approximation in order to solve the dynamics . first , we fix @xmath14 and find the asymptotic ( @xmath104 ) lipid density profile @xmath105 we use this density profile to obtain the dependence of the hemifusion radius @xmath14 on the time @xmath106 to find : @xmath107 this predicts an approximately square root dependence of the hemifusion region size on time ( with logarithmic corrections ) . the same temporal dependence was obtained by @xcite under the assumption of constant lateral lipid density . however , their result is quantitatively different from ours since they have considered the monolayer viscosity as the main dissipative force , while we have showed that it is negligible compared to the friction @xmath50 . from eq . [ eq : taur ] we find that the time it takes the hemifusion region to evolve from the initial stalk radius @xmath108 to a final radius of @xmath109 is @xmath110 . with @xmath111 and @xmath112 @xmath83sec , we predict that the time for expansion of the hemifusion zone to a scale of @xmath113 m is @xmath114 sec . this is consistent with the experiment of @xcite where a time of @xmath115 sec was measured . the time @xmath116 found here can also be derived ( up to a numerical factor ) from a simple scaling argument , that does not depend on the specific details of our model . as hemifusion is initiated , the tension difference between the bulk ( at @xmath13 ) and the hemifusion front ( at @xmath14 ) is @xmath117 . when @xmath118 the average tension gradient is @xmath119 . for a fully damped flow with a friction coefficient @xmath50 the average lipid velocity is @xmath120 . the hemifusion front ( bc boundary ) advances with the velocity @xmath121 of the lipids near it . the time to advance a distance of @xmath13 with a velocity @xmath122 is @xmath123 . the change in the monolayer surface energy due to the presence of peg in region a is @xmath124 , where @xmath23 is the free energy per lipid in the absence of peg , and @xmath125 is the free energy per lipid in the presence of peg . since we have defined @xmath27 by the condition @xmath126 , we can expand @xmath16 around its minimal value @xmath24 , and find that to lowest order in @xmath102 the surface energy difference @xmath127 and the tension difference @xmath2 induced by the peg are @xmath128 in @xcite a change of @xmath129 in lipid density was deduced from the measured thinning of the bilayer . using the value @xmath130 erg/@xmath61 we estimate @xmath131 erg/@xmath61 ; @xmath132 erg/@xmath61 . initiation of stalk expansion is relevant not only to events of mesoscopic fusion , but also to in - vivo fusion events , where a fusion pore is formed soon after stalk expansion . in many cases of biological interest , the fusion process is regulated by fusion proteins that promote stalk formation and expansion . one hypothesized bio - molecular mechanism that promote expansion is the penetration of hydrophobic fusion protein domains into the membrane and its subsequent destabilization @xcite . the protein domains may increase the membrane surface energy by inducing an effective attraction of the hydrophobic head groups , similar to the effect of peg @xcite ; they may also penetrate the membrane , increasing the intra - membrane tension . our theory suggests that the former mechanism , which work to increase in @xmath127 , may be more effective energetically than the latter , which increases @xmath2 . that is , for a given change in lipid density , @xmath102 , a smaller energy is involved ( eq . [ eq : tendiff ] ) . snare proteins that promote exocytosis in nerve synapses are thought to induce stalk expansion through a conformational change by which the protein pull on the stalk to widen it @xcite . another possible cause for stalk expansion is calcium ions induced membrane tension @xcite . we conclude from our theory that the latter mechanism may be more effective energetically . in section [ sec : boundcond ] we found that in order for expansion of the hemifusion region to occur , the driving force due to the negative tension in region b must be large enough to overcome the tendency of the boundary of region c to shrink . we thus deduced that the normalized lipid density at @xmath13 must obey @xmath133 from this condition , we estimate the minimum stalk radius @xmath99 for which the lateral tension in the monolayer can induce expansion . the energy of the lipid tails tilt at the hemifusion front is estimated by @xcite as @xmath134 erg / cm . for the values of @xmath77 and @xmath102 given above , we find that the mechanism described here is sufficient to cause hemifusion for @xmath135 nm , which is of the order of the typical radius of a thermally nucleated stalk @xcite . note that if @xmath102 vanishes ( that is , no polymer is present in region a ) hemifusion will not be initiated for any finite ( reasonable ) stalk radius . experiments have demonstrated that hemifusion may be caused by sufficiently large normal pressure @xcite or by negative pressure in the water layer @xcite . we shall now determine the conditions under which pressure induced in region b can in and of itself ( i.e. with no surface tension effects as induced by the added polymer ) cause hemifusion expansion by forcing water to flow out of the contact zone . we do this by using the simplifying assumption that the water in region b is under a constant pressure @xmath136 , where @xmath137 is the normal pressure on the bilayers and @xmath138 is the osmotic pressure induced by the solute in the bulk . the finite thickness of the water layer in region b ( whose thickness is on the order of a nanometer ) is always maintained because of hydration forces : the water molecules are organized around the polar head groups of the lipids in order to partially cancel their electric dipole ; removing the water layer would increase the free energy because of the energetic cost of these electric dipoles whose normal components , in general , point to the same direction due to the hydrophobic nature of the lipid layer . thus the water flow out of region b and into region a is possible only by the expansion of region c. the energy ( per unit area ) difference associated with a pressure difference of @xmath139 is @xmath140 , where @xmath7 is the distance between the two proximal monolayers . this should be compared with the energy difference @xmath127 associated with the free energy gradient in the monolayer . in the experiment of @xcite that yield @xmath141 erg/@xmath61 , which is of the same order of @xmath127 . nevertheless , we show below that the external normal pressure has only a minor effect on the pressure in the monolayer and on its density . we will thus show that under the experimental conditions of @xcite , the external pressure is _ insufficient _ to cause hemifusion expansion . in the experiment of @xcite the applied normal pressure is @xmath142 atm and the osmotic pressure is @xmath143 atm , so the total pressure between the bilayers is @xmath144 atm . we now estimate the contribution of this pressure to the lipid density variation in the experiment . for a fluid membrane , the relation between the tension @xmath38 - the two dimensional pressure in the membrane - to the three dimensional pressure @xmath139 , is @xmath145 , where @xmath146 is the thickness of the monolayer . in order to induce the observed density variation @xmath147 the tension needed is @xmath148 erg/@xmath61 . for @xmath149 nm the pressure required to induce such tension is 5 atm much larger than the actual pressure in the experiment . thus , the contribution of the normal and the osmotic pressures to the density variation is negligible compared with the surface tension effects due to the peg - lipid interactions that result in densification of the lipids . this result underscores the point made in sec . [ sec : init ] : changes in the pressure are much less effective than surface energy variation for the initiation of stalk expansion . we now estimate the pressure @xmath139 needed to initiate hemifusion , without a lipid density gradient ( that is , with @xmath150 ) . the radial force per unit length on the boundary at @xmath14 due to the external normal pressure is @xmath151 from eq . [ eq : fbc ] , the condition for spontaneous fusion is @xmath152 . for the values given above , we require @xmath153dyne/@xmath154 atm . experimental results in different conditions are within that range . the pressure needed for the hemifusion of bilayers directly supported on mica ( with no added polymer or other mechanisms that give rise to lipid density gradients ) was found by @xcite to be @xmath155 atm . @xcite used a surface forces apparatus to apply pressure on dmpc bilayers supported on polymer layers . the polymer layer allowed the bilayers some lateral conformational freedom , thus permitting more freedom for the adjustment of stalk shape and size @xcite . in that case , where the stalk geometry could easily adjust , the cost for forming the stalk was reduced and hemifusion was observed at a much lower pressure of @xmath156 atm . in the experiment of @xcite the pressure @xmath157 atm is too low to be the driving force for hemifusion . pressure in itself is not enough to cause hemifusion , but it is sometimes necessary . @xcite showed that the amount of pressure needed for hemifusion is directly related to the lipid density near the contact area . in that experiment , two bilayers were brought into contact using a surface forces apparatus . when ca@xmath158 ions were introduced , there was a phase separation in the bilayers . the density of lipids in the bilayer regions that were brought into contact was characterized by the hydrophobic adhesion energy . when thinner regions were brought together ( characterized by adhesion energy of @xmath159 erg/@xmath61 ) they either hemifused spontaneously , or required only a small amount of pressure ( @xmath160 atm ) to induce hemifusion . for denser bilayers ( @xmath161 erg/@xmath61 ) a pressure of @xmath162 atm was required for hemifusion . @xcite induced negative osmotic pressure on the water layer between the bilayers by lowering the relative humidity of the environment of dphpc . at 80% humidity the lipids were at the lamellar phase . as the relative humidity was decreased the water were expelled from between the bilayers by the osmotic pressure and the lamella were connected by stalks , directly observed by x - ray diffraction . in this experiment the dehydration was due to negative pressure of the water layer induced by the reduced relative humidity , and not by normal pressure , but the physical effect of the two is similar . in this paper we used a model based on lipid density gradients induced by surface energy variation that occur far from the hemifusion zone , to predict the the conditions for the initiation of hemifusion by stalk expansion and the dynamics of mesoscopic hemifusion . our theory was motivated by the experiments of @xcite . however , the quantitative scheme presented here can be generalized to any system of two lipid bilayers initially connected by a stalk , where a perturbation in region a , mesoscopically far from the stalk , causes tension in the membrane in that region . for example one can apply our results to tension induced by the electrostatic interactions caused by calcium ions @xcite , tension induced by laser tweezers @xcite , or the effective tension induced by the attraction of oppositely charged bilayers @xcite . we have compared the effect of the friction of the two monolayers , the the water viscosity and the intra - monolayer viscosity on the two dimensional lipid motion and showed that the friction dominates . thus , the lipid dynamics depend on the friction and not on hydrodynamics . this means that the spacing between the two layers is irrelevant for the lipid dynamics . experiments similar to those of @xcite can test the predictions of the model for the time scales as functions of the lipid density and friction as well as the value of the driving force due to the tension induced in region a. one could vary each of the parameters @xmath102 ( the relative change in lipid density ) , @xmath77 ( related to the induced tension ) and @xmath50 ( the interlayer friction ) independently , and measure the hemifusion radius @xmath163 , the final radius @xmath13 and the time to complete the process @xmath116 as functions of these parameters . in particular , the friction @xmath50 can be varied independently of @xmath77 by changing the composition of the distal bilayers while maintaining the same composition of the proximal bilayers . the friction can be varied by changing the interactions between the chains that is responsible for most of the friction , via chain length changes or temperature changes @xcite . once an empirical , temporal profile for the hemifusion expansion , @xmath163 , is measured for systems with known parameters , one can use the same experiment to estimate the effective diffusion constant for the lipid flow , @xmath164 , for _ different _ lipid bilayers . one can easily vary the lipid density at the boundary , @xmath27 by changing the polymer ( or calcium ions ) concentration since the density @xmath27 is determined by the equality of the chemical potentials of the lipids exposed to the polymer and those exposed only to the water . the static part of our theory deals with the initial conditions required for stalk expansion . we have evaluated the necessary density variation @xmath165 and demonstrated that the related surface energy @xmath166 is much smaller then the surface tension @xmath167 . this result is not surprising , since it is a general result of a first order expansion around an energetic minimum . still , it does give a new insight regarding biological fusion mechanisms . it suggest mechanisms working through the change of the surface energy @xmath127 are much more effective than mechanisms that exert force or normal pressure on the stalk . the predicted dependence of stalk expansion on the lipid density can be tested by measuring the critical density @xmath102 at which stalk expansion occurs . the results may serve to learn more about the stalk structure and energetics . we expect that near the end of the process of hemifusion expansion , when @xmath168 experimental results may differ from our predictions , since the density profile of the polymer ( or calcium ions in the case of @xcite ) may vary in a gradual manner around @xmath13 ; in our theory we assumed a sharp ( step function ) decrease of the polymer density at @xmath13 . we also expect a deviation from our theory when the radius @xmath163 of the hemifusion region is close to its initial , molecular stalk radius @xmath99 , due to microscopic details of the lipid structure in the stalk . we distinguish between hemifusion induced by surface tension gradients , which we consider in our model , and hemifusion induced by pressure . hemifusion may be induced by normal pressure on the bilayers @xcite or by dehydration which induces negative pressure in the water layer between them @xcite . we showed that this pathway to hemifusion requires much more energy ( per unit area ) than fusion that is induced by surface tension gradients . we have shown that the induced pressure @xmath139 in the experiment of @xcite can not be the primary direct cause of hemifusion . still , pressure does play an important role in stalk formation . it may also effect stalk expansion through its effect on the lipid tilt energy @xmath34 and on the initial stalk radius @xmath99 . we bring here the full calculation of the local lipid dynamics . note that though in our final result we leave only the terms linear in @xmath69 , one may also calculate in the same framework the non linear terms in the case @xmath69 is not small . the force balance equation is @xmath169 and the continuity equation is @xmath170 writing the energy per lipid as @xmath10 the surface tension is @xmath171 where @xmath172 is the a macroscopic area and @xmath173 is the number of lipids in this area . from eqs . [ eq : a_fb ] , [ eq : a_cont ] and [ eq : a_pl ] we have @xmath174 to first order in the density variation @xmath69 , eq . [ eq : a_lpdyn ] has the form @xmath175 where @xmath71 . in sec . [ sec : boundcond ] we consider the boundary conditions for the lipid density . in order to fully predict the dynamics of hemifusion expansion , we also need to determine the flow at the boundaries . for this we use the integral form of the continuity equation : @xmath176 the left hand side of eq . [ eq : a_bc2 ] describes the rate of change of the lipid number in region b while the right hand side gives the flow of lipids through the boundary @xmath13 . we assume cylindrical symmetry , so @xmath177 . from eqs . [ eq : a_fb ] and [ eq : a_pl ] we obtain @xmath178 we now use eq . [ eq : a_eqmot ] to calculate the left hand side of eq . [ eq : a_bc2 ] : @xmath179 if we take only terms linear in @xmath69 , eq . [ eq : a_bc2 ] gives : @xmath180 equations [ eq : a_bc3 ] and [ eq : a_eqmot ] along with the boundary conditions completely determine the time evolution of the monolayers to first order in @xmath69 . from these equations we can calculate @xmath163 , and predict the temporal profile of hemifusion expansion . we write these two equations using dimensionless variables and scale the spatial variables so that they are of order of unity , in order to get an estimate of the time scales . the natural spatial scale is the final size of the hemifusion region , @xmath13 . we thus define : @xmath181 , @xmath182 , @xmath183 as well as two time variables : a `` fast '' time @xmath184 at which the local lipid flow occurs , and a `` slow '' ( since @xmath102 is small ) time @xmath185 which is the scale over which the hemifusion region expands . [ eq : a_bc3 ] and [ eq : a_eqmot ] become @xmath186 @xmath187 since all the variables that appear on the right hand side of eqs . [ eq : a_nsc1 ] and [ eq : a_nsc2 ] are of order unity , the units of @xmath188 and @xmath189 suggest the time scales of the processes described by the equations . for @xmath190 we have @xmath191 , which implies that we can use an adiabatic approximation : the local lipid flow occurs quickly so that the lipid density is instantaneously given by the asymptotic equilibrium solution of eq . [ eq : a_nsc1 ] for @xmath192 . we then use this solution to determine the slower time evolution of the hemifusion radius @xmath14 from eq . [ eq : a_nsc2 ] . at asymptotically long times , both sides of eq . [ eq : a_nsc1 ] vanish . the adiabatic density profile reached is @xmath193 plugging this solution into eq . [ eq : a_nsc2 ] we obtain @xmath194 the solution of this equation is implicitly given by @xmath195 ben - shaul , a. 1995 . molecular theory of chain packing , elasticity and lipid - protein interaction in lipid bilayers . handbook of biological physics , vol . r. lipowsky , and e. sackman , editors . elsevier , amsterdam . chernomordik , l. v. , v. a. frolov , e. leikina , p. bronk , and j. zimmerberg . the pathway of membrane fusion catalyzed by influenza hemagglutinin : restriction of lipids , hemifusion , and lipidic fusion pore formation . 140:13691382 . kuhl , t. , y. guo , j. l. alderfer , a. d. berman , d. leckband , j. israelachvili , and s. w. hui . 1996 . direct measurement of polyethilene glycol induced depletion attraction between lipid membranes . .

one approach to the understanding of fusion in cells and model membranes involves stalk formation and expansion of the hemifusion diaphragm . we predict theoretically the initiation of hemifusion by stalk expansion and the dynamics of mesoscopic hemifusion diaphragm expansion in the light of recent experiments and theory that suggested that hemifusion is driven by intra - membrane tension far from the fusion zone . our predictions include a square root scaling of the hemifusion zone size on time as well as an estimate of the minimal tension for initiation of hemifusion . while a minimal amount of pressure is evidently needed for stalk formation , it is not necessarily required for stalk expansion . the energy required for tension induced fusion is much smaller than that required for pressure driven fusion .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently , a great deal of attention has been payed to wireless sensor networks whose nodes sample a physical phenomenon ( hereinafter referred to as field ) , i.e. , air temperature , light intensity , pollution levels or rain falls , and send their measurements to a central processing unit ( or _ sink _ node ) . the sink is in charge of reconstructing the sensed field : if the field can be approximated as bandlimited in the time and space domain , then an estimate of the discrete spectrum can be obtained . however , the sensors measurements typically represent an irregular sampling of the field of interest , thus the sink operates based on a set of field samples that are not regularly spaced in the time and space domain . the reasons for such an irregular sampling are multifold . ( i ) the sensors may be irregularly deployed in the geographical region of interest , either due to the adopted deployment procedure ( e.g. , sensors thrown out of an airplane @xcite ) , or due to the presence of terrain asperities and obstacles . ( ii ) the transmission of the measurements from the sensors to the central controller may fail due to bad channel propagation conditions ( e.g. , fading ) , or because collisions occur among the transmissions by sensors simultaneously attempting to access the channel . in this case , although the sample has been collected by the sensor , it will not be delivered to the central controller . ( iii ) the sensors may enter a low - power operational state ( sleep mode ) , in order to save energy @xcite . while in sleep mode , the nodes neither perform sensing operations nor transmit / receive any measurement . ( iv ) the sensors may be loosely synchronized , hence sense the field at different time instants . clearly , sampling irregularities may result in a degradation of the reconstructed signal @xcite . the work in @xcite investigates this issue in the context of sensor networks . other interesting studies can be found in @xcite and @xcite , just to name a few , which address the perturbations of regular sampling in shift - invariant spaces @xcite and the reconstruction of irregularly sampled images in presence of measure noise @xcite . in this work , our objective is to evaluate the performance of the field reconstruction when the coordinates in the @xmath0-dimensional domain of the field samples , which reach the sink node , are randomly , independently distributed and the sensors measurements are noisy . we take as performance metric the mean square error ( mse ) on the reconstructed field . as a reconstruction technique , we use linear filtering and we adopt the filter that minimizes the mse ( i.e. , the lmmse filter ) @xcite . the matrix representing the sampling system , in the following denoted by @xmath2 , results to be a @xmath0-fold vandermonde matrix matrix @xmath5 is vandermonde if its @xmath6th entry , @xmath7 can be written as @xmath8 , @xmath9 . ] . by drawing on the results in @xcite , we derive both the moments and an expression of the limiting spectral distribution ( lsd ) of @xmath3 , as the size of @xmath2 goes to infinity and its aspect ratio has a finite limit bounded away from zero . then , by using such an asymptotic model , we approximate the mse on the reconstructed field through the @xmath4-transform @xcite of @xmath3 , and derive an expression for it . we apply our results to the study of network scenarios of practical interest , such as sensor sensor deployments with coverage holes , communication in presence of a fading channel , massively dense networks @xcite , and networks using contention - based channel access techniques @xcite . the rest of the paper is organized as follows . section [ sec : related ] reviews previous work , while section [ sec : system ] describes the system model under study . in section [ sec : preliminaries ] , we first provide some useful definitions and introduce our performance metric , then we recall previous results on which we build our analysis . in section [ sec : results - vandermonde ] , we derive asymptotic results concerning the moments and the lsd of @xmath3 . such results are applied to different practical scenarios in section [ sec : applications ] . finally , section [ sec : conclusions ] concludes the paper . in the context of sensor networks , several works @xcite have studied the field reconstruction at the sink node in presence of spatial and temporal correlation among sensor measurements . in particular , in @xcite the observed field is a discrete vector of target positions and sensor observations are dependent . by modeling the sensor network as a channel encoder and exploiting some concepts from coding theory , the network capacity , defined as the maximum value of the ratio of the target positions to the number of sensors , is studied as a function of the noise , the sensing function and the sensor connectivity level . the paper by dong and tong @xcite considers a dense sensor network where a mac protocol is responsible to collect samples from network nodes . the work analyzes the impact of deterministic and random data collection strategies on the quality of field reconstruction . as a performance measure , the maximum of the reconstruction square error over the sensed field is employed , as opposed to our work where the mean square error is considered . also , in @xcite the field is a gaussian random process and the sink always receives a sufficiently large number of samples so as to reconstruct the field with the required accuracy . the problem of reconstructing a bandlimited field from a set of irregular samples at unknown locations , instead , has been addressed in @xcite . there , the field is oversampled by irregularly spaced sensors ; sensor positions are unknown but always equal to an integer multiple of the sampling interval . different solution methods are proposed , and the conditions for which there exist multiple solutions or a unique solution are discussed . differently from @xcite , we assume that the sink can either acquire or estimate the sensor locations and that the coordinates of the sampling points are randomly located over a finite @xmath0-dimensional domain . as for previous results on vandermonde matrices , in @xcite ryan and debbah considered a vandermonde matrix @xmath2 with @xmath10 and complex exponential entries , whose phases are i.i.d . with continuous distribution . under such hypothesis , they obtained the important results that , given the phases distribution , the moments of @xmath3 can be derived once the moments for the case with uniformly distributed phases are known . also , a method for computing the moments of sums and products of vandermonde matrices , for the non - folded case ( i.e. , @xmath10 ) , has recently appeared in @xcite ; further insights on the extremal eigenvalues behavior , still for the case of non - folded vandermonde matrices , can be found in @xcite . moreover , in @xcite it has been shown that the lsd of @xmath3 converges to the marenko - pastur distribution @xcite when @xmath2 is @xmath0-fold vandermonde with uniformly distributed phases and @xmath11 . note that , with respect to previous studies on vandermonde matrices with entries that are randomly distributed on the complex unit circle , in this work we obtain results on the lsd of @xmath3 where the entries of @xmath2 have phases drawn from a _ generic continuous distribution_. by relying on the results in @xcite , we show that such an lsd can be related to that of @xmath3 when the phases of @xmath2 are _ uniformly _ distributed on the complex unit circle . we also provide some numerical results that show the validity of our analysis . to our knowledge , these results have not been previously derived . we then apply them to the study of several practical scenarios in the context of sensor networks , although our findings can be useful for the study of other aspects of communications as well @xcite . we consider a network composed of @xmath12 wireless sensors , which measure the value of a spatially - finite physical field defined over @xmath0 dimensions , ( @xmath13 ) . we denote by @xmath14 the hypercube over which the sampling points fall , and we assume that the sampling points are i.i.d . randomly distributed variables , whose value is known to the sink node . note that this is a fair assumption , as one can think of sensor nodes randomly deployed over the geographical region that has to be monitored , or , even in the case where the network topology is intended to have a regular structure , the actual node deployment may turn out to be random due to obstacles or terrain asperities . in addition , now and then the sensors may enter a low - operational mode ( hence become inactive ) in order to save energy , and they may be loosely synchronized . all the above conditions yield a set of randomly distributed samples of the field under observation , in both the time and the space domain @xcite . by truncating its fourier series expansion , a physical field defined over @xmath0 dimensions and with finite energy can be approximated in the region @xmath15 as @xcite @xmath16 where @xmath17 is the approximate one - sided bandwidth ( per dimension ) of the field , @xmath18\tran$ ] is a vector of integers , with @xmath19 , @xmath20 . the coefficient @xmath21 is a normalization factor and the function @xmath22 maps uniquely the vector @xmath23 over @xmath24 $ ] . @xmath25 denotes the @xmath26-th entry of the vector @xmath27 of size @xmath28 , which represents the approximated field spectrum , while the real vectors @xmath29 , @xmath30 represent the coordinates of the @xmath0-dimensional sampling points . in this work , we assume that @xmath29 , @xmath30 , are i.i.d . random vectors having a generic continuous distribution @xmath31 , @xmath32 . in the specific case where @xmath29 are i.i.d with i.i.d . entries @xmath33 , uniformly distributed in @xmath34 , we denote the distribution of @xmath29 by @xmath35 . the coordinates of the @xmath0-dimensional sampling points , however , are known to the sink node , because _ ( i ) _ either sensors are located at pre - defined positions or their position can be estimated through a localization technique @xcite , and _ ( ii ) _ the sampling time is either periodic or included in the information sent to the sink . now , let @xmath36^{\rm t}$ ] be the values of the samples at @xmath37 $ ] , respectively . following @xcite , we can write the vector @xmath38 as a function of the field spectrum : @xmath39 where @xmath2 is the @xmath40 @xmath0-fold vandermonde matrix with entries @xmath41 randomly distributed on the complex circle of radius @xmath42 , and @xmath43 is the ratio of the rows to the columns of @xmath2 , i.e. , @xmath44 in general , the entries of @xmath27 can be correlated with covariance matrix @xmath45 $ ] . however , in the following , we restrict our attention to the class of fields characterized by @xmath45= \sigma^2_a{{\bf i}}$ ] . in the case where the sensor measurements , @xmath46\tran$ ] , are noisy , then the relation between the sensor samples and the approximated field spectrum can be written as : @xmath47 where @xmath48 is a @xmath12-size , zero - mean random vector representing the noise . here , we assume a white noise , i.e. , with covariance matrix @xmath49 = \sigma^2_n{{\bf i}}_{m}$ ] . note that the additive white noise affecting the sensor measurements may be due to quantization , round - off errors or quality of the sensing device . in this section , we report some definitions and previous results that are useful for our study . let us consider an @xmath50 non - negative definite random matrix @xmath51 , whose eigenvalues are denoted by @xmath52 . the average empirical cumulative distribution of the eigenvalues of @xmath51 is defined as @xmath53 $ ] , where the superscript @xmath54 indicates that we refer to a system with size @xmath17 and @xmath55 is the indicator function . if @xmath56 converges as @xmath57 , then @xmath58 . the corresponding limiting probability density function , or limiting spectral distribution ( lsd ) , is denoted by @xmath59 . the @xmath4-transform of @xmath51 is given by : @xmath60 = { \mathbb{e}}\left[\frac{1}{n}\sum_{i=1}^n\frac{1}{\gamma\lambda_{{{\bf a}},i } + 1}\right]\ ] ] where @xmath61 is the normalized matrix trace operator and @xmath62 is a non - negative real number . if @xmath63 converges as @xmath57 , then the corresponding limit is @xmath64 $ ] @xcite , where @xmath65 is the generic asymptotic eigenvalue of @xmath51 , whose distribution is @xmath66 , and the average is computed with respect to @xmath65 @xcite . next , consider the matrix @xmath2 as defined in ( [ def : v_multifold ] ) and that the lmmse filter is used for field reconstruction . then , the estimate of the unknown vector @xmath27 in ( [ eq : p ] ) , given @xmath67 and @xmath2 , is obtained by computing @xmath68 { \mathbb{e}}[{{\bf p}}{{\bf p}}\herm]^{-1 } { { \bf p}}$ ] . through easy computations and using the sherman - morrison - woodbury identity , we can obtain the mse as @xmath69 = \eta^{(n)}_{{{\bf v}}{{\bf v}}\herm}\left(\frac{\gamma}{\beta_{n , m}}\right ) \label{eq : mse_n}\ ] ] where @xmath70 denotes the signal - to - noise ratio on the sensor measurements , and we employed the definition of the @xmath4-transform given in ( [ eq : eta - t ] ) . next , we approximate the mse of the finite size system in ( [ eq : p ] ) through an asymptotic model , which assumes the size of @xmath2 to grow to infinity while the ratio of its number of rows to its number of columns tends to a finite limit , @xmath71 , greater than zero , i.e. , we assume @xmath72 indeed , in our recent works @xcite it was shown that this asymptotic model provides a tight approximation of the mse of the finite size system , already for small values of @xmath17 and @xmath12 . under these conditions , we therefore define the asymptotic expression of the mse as @xcite : @xmath73 if the limit exists . vandermonde matrices have been studied in a number of recent works @xcite . specifically , @xcite considered the case where the vectors @xmath29 are i.i.d . , for @xmath30 , and their entries , @xmath74 are i.i.d . random variables with uniform distribution in @xmath34 . the work there studied the eigenvalue distribution of @xmath3 for both finite and infinite ( i.e. , @xmath75 ) matrix size . although an explicit expression of such lsd is still unknown , @xcite provided an algorithm to compute its moments of any order in closed form . indeed , as @xmath76 with @xmath77 having a finite limit @xmath78 , in @xcite it was shown that the @xmath79-th moment of the generic asymptotic eigenvalue of @xmath3 , denoted by @xmath80 , is given by @xmath81 where @xmath82 represents the distribution of @xmath80 . moreover , @xmath83 is the set of partitions of the set @xmath84 in @xmath85 subsets , and @xmath86 $ ] , @xmath87 is a rational number that can be analytically computed from @xmath88 following the procedure described in @xcite . the subscript @xmath89 in @xmath90 and @xmath82 indicates that a uniform distribution of the entries of @xmath29 is considered in the matrix @xmath2 . in @xcite it was also shown that when @xmath91 , with @xmath77 having a finite limit @xmath78 , the eigenvalue distribution @xmath82 converges to the marcenko - pastur law @xcite . a similar result @xcite also applies when the vectors @xmath29 ( @xmath30 ) are independent but not i.i.d . , with equally spaced averages . more recently , ryan and debbah in @xcite considered @xmath10 and the case where the random variables @xmath92 , @xmath30 , are i.i.d . with continuous distribution @xmath93 , @xmath94 . under such hypothesis , it was shown that the asymptotic moments of @xmath3 can be written as @xmath95 where the terms @xmath96 depend on the phase distribution @xmath93 and are given by @xmath97 for @xmath98 . the subscript @xmath99 in @xmath100 indicates that in the matrix @xmath2 the random variables @xmath92 have a generic continuous distribution @xmath93 . note that for the uniform distribution we have @xmath101 , for all @xmath85 . the important result in ( [ eq : moments_1d ] ) states that , given @xmath71 , if the moments of @xmath3 are known for uniformly distributed phases , they can be readily obtained for any continuous phase distribution @xmath93 . in this work , we extend the above results by considering a sampling system defined over @xmath102 dimensions with nonuniform sample distribution , where samples may be irregularly spaced in the time and spatial domains , as it occurs in wireless sensor networks . being our goal the estimation of the quality of the reconstructed field , we aim at deriving the asymptotic mse ( i.e. , @xmath103 ) . we start by considering a generic continuous distribution , @xmath104 , @xmath32 of the samples measured by the sensors over the @xmath0-dimensional domain . we state the theorem below , which gives the asymptotic expression of the generic moment of @xmath3 , for @xmath105 . [ th:1 ] let @xmath2 a @xmath0-fold @xmath106 vandermonde matrix with entries given by ( [ def : v_multifold ] ) where the vectors @xmath29 , @xmath30 , are i.i.d . and have continuous distribution @xmath104 . then , for @xmath76 , with @xmath77 having a finite limit @xmath78 , the @xmath79-th moment of @xmath3 is given by @xmath107 where @xmath108 and the terms @xmath109 are defined as in @xcite . the proof is given in appendix [ app : th1 ] . using theorem [ th:1 ] and the definition of @xmath96 , it it possible to show the theorem below , which provides the lsd of @xmath3 . [ th:2 ] let * @xmath2 be a @xmath0-fold @xmath106 vandermonde matrix with entries given by ( [ def : v_multifold ] ) where the vectors @xmath29 , @xmath30 , are i.i.d . and have continuous distribution @xmath104 , @xmath32 * @xmath110 be the set where @xmath104 is strictly positive , i.e. , @xmath111 * the cumulative density function @xmath112 defined denotes the measure of the set @xmath113 for @xmath114 and let @xmath115 be its corresponding probability density function . then , the lsd of @xmath3 , for @xmath76 with @xmath77 having a finite limit @xmath78 , is given by @xmath116 the proof can be found in appendix [ app : th2 ] . from theorem [ th:2 ] , the corollary below follows . [ cor - scal ] consider @xmath31 such that @xmath117 @xmath118 . then , let us denote by @xmath119 a scaled version of this function , so that @xmath120 where @xmath121 . it can be shown that @xmath122 where @xmath123 . the proof can be found in appendix [ app : cor - scal ] . as an example of the result given in corollary [ cor - scal ] , consider that a unidimensional ( @xmath10 ) sensor network monitors the segment @xmath124 $ ] . due to terrain irregularities and obstacles , nodes are deployed with uniform distribution only in the range @xmath125 $ ] ( with @xmath126 ) . we therefore have @xmath127 for @xmath128 and 0 elsewhere . moreover , @xmath129 . the expression of @xmath130 is given by ( [ eq : f - scal ] ) , by replacing @xmath10 and the subscript @xmath99 with the subscript @xmath89 . this result is well supported by simulations as shown in figures [ fig : beta08_x08 ] and [ fig : beta02_x05 ] . in the plots , we compare the asymptotic empirical spectral distribution ( aesd ) @xmath131 and @xmath132 instead of the lsds @xmath133 and @xmath134 since an analytic expression of @xmath134 is still unknown . however , in @xcite it is shown that , already for small values of @xmath17 , the aesd @xmath132 appears to rapidly converge to a limiting distribution . figure [ fig : beta08_x08 ] refers to the case @xmath135 and @xmath136 . the solid and dashed lines represent , respectively , the functions @xmath137 and @xmath138 , for @xmath139 . note that the probability mass of @xmath140 at @xmath141 is not shown for simplicity . similarly , figure [ fig : beta02_x05 ] shows the case @xmath142 and @xmath143 . as evident from these plots , the match between the two functions is excellent for any parameter setting , thus supporting our findings . since we are interested in evaluating the mse , taking into account the result in ( [ eq : mseinf ] ) , we now apply the definition of the @xmath4-transform to ( [ eq : th2 ] ) . the corollary below immediately follows . [ cor2 ] the @xmath4-transform of @xmath3 is given by @xmath144 hence , the asymptotic mse on the reconstructed field , defined in ( [ eq : mseinf ] ) , is given by @xmath145 the proof can be found in appendix [ app : cor2 ] . in ( [ eq : eta ] ) , in order to avoid a heavy notation we referred to @xmath146 as @xmath147 when the phases of the entries of @xmath2 follow a generic random continuous distribution , while @xmath148 refers to the case where the phases are uniformly distributed . since @xmath149 and @xmath150 , the integral in the right hand side of ( [ eq : eta ] ) is positive , then @xmath151 . it follows that the mse is lower - bounded by the measure of the total area where the probability of finding a sensor is zero . this clearly suggests that , in order to obtain a good quality of the field reconstructed at the sink node , this area must be a small fraction of the region under observation . next , we observe that , in the case of massively dense networks where the number of sampling sensors is much larger than the number of harmonics considered in the approximated field , i.e. , @xmath152 , an interesting result holds : [ cor1 ] let @xmath110 be the set where @xmath31 is strictly positive ; then @xmath153 the proof can be found in appendix [ app : cor1 ] . thus , as evident from corollary [ cor1 ] , for the limit of @xmath154 , the lsd of @xmath3 is the density of the density of the phase distribution @xmath104 . furthermore , for massively dense networks , we have : [ cor3 ] let @xmath110 be the set where @xmath31 is strictly positive ; then @xmath155 the proof can be found in appendix [ app : cor3 ] . the result in ( [ eq : cor3.2 ] ) shows that even for massively dense networks @xmath156 is the minimum achievable @xmath157 , when an area @xmath110 can not covered by sensors . here , we provide examples of how our results can be used in wireless sensor networks to investigate the impact of a random distribution of the coordinates of the sampling points on the quality of the reconstructed field . in particular , we first consider a wireless channel affected by fading , and then the effects of contention - based channel access . we consider a wireless sensor network whose nodes are uniformly distributed over a geographical region . without loss of generality , we assume a square region of unitary side ( @xmath158 , @xmath159 ^ 2 $ ] ) , where the sink is located at the center and has coordinates @xmath160 . through direct transmissions , the sensors periodically send messages to the sink , including their measurements . at every sample period , a sensor message is correctly received at the sink if its signal - to - noise ratio ( snr ) exceeds a threshold @xmath161 . the communication channel is assumed to be affected by slow fading and to be stationary over the message duration . let @xmath0 be the distance between a generic sensor and the sink . then , the signal to noise ratio at the receiver is given by @xmath162 where @xmath163 is a circularly symmetric gaussian complex random variable representing the channel gain , and @xmath164 is the signal to noise ratio in the absence of fading and when the sensor - sink distance is @xmath10 . the probability that a message is correctly received at the sink is given by @xmath165 with @xmath166 and @xmath167 being the cumulative density function of @xmath168 . the probability density @xmath169 corresponding to sensors at distance @xmath170 , @xmath171 from the sink and successfully sending a message is then given by @xmath172 where @xmath173 , is the density representing the sensor deployment ( recall that nodes are assumed to be uniformly distributed in the region hence their density is constant and equal to 1 ) . using ( [ eq : snr ] ) , we obtain : @xmath174 where @xmath175 in order to compute ( [ eq : eta ] ) , we need the function @xmath176 , i.e. , the density of @xmath169 . note that @xmath169 is circularly symmetric with respect to @xmath160 . let @xmath177 be the value of density of the sampling points at distance @xmath178 from the sink . then , from ( [ eq : f_xi ] ) we obtain @xmath179 , thus the network area where the density is lower than @xmath177 is given by @xmath180 for @xmath181 , i.e. , @xmath182 . for @xmath183 , it is possible to show that @xmath184 in conclusion , @xmath185 and @xmath186 since @xmath187 , then the asymptotic mse can be obtained by computing @xmath188 for @xmath189 db , i.e. , for different values of the snr threshold @xmath161 . ] figure [ fig : g ] shows the density @xmath115 for @xmath189 db . note that @xmath166 , thus for a fixed @xmath164 ( i.e. , the signal to noise ratio at distance @xmath190 in the absence of fading ) the parameter @xmath191 is proportional to the snr threshold @xmath161 . in particular , as @xmath161 decreases , the probability that a message successfully reaches the destination increases and , thus , the spatial distribution of correctly received samples , @xmath192 , tends to the uniform distribution @xmath193 . as a consequence , the density of @xmath169 , i.e. , @xmath115 , for @xmath194 and 5 db is concentrated close to @xmath195 . however , for high values of @xmath161 , messages originated from sensor nodes located far from the sink are successfully received with low probability . thus , @xmath115 shows a significant probability mass around @xmath196 . ) and in presence ( @xmath104 ) of fading , as the signal to noise ratio on the sensor measurements varies . ] figure [ fig : mse ] shows the effect of the fading channel on the mse of the reconstructed field ( dashed lines ) , and compares the obtained results with the mse obtained in absence of fading ( solid lines ) . the plot considers different values of @xmath71 , namely , @xmath197 , and @xmath198 db . the mse is plotted versus the signal to noise ratio on the sensor measurements , @xmath62 . the curves have been obtained by numerically computing ( [ eq : eta_xi_2 ] ) , where @xmath115 is given by ( [ eq : g_xi ] ) and @xmath199 is replaced by @xmath200 , with @xmath201 . recall that the analytic expression of the lsd @xmath82 is unknown , hence in the numerical results we considered the aesd @xmath202 instead . we observe that for low values @xmath71 , in spite of the presence of fading , the sink node still receives a large number of samples from the sensors , hence the degradation of the mse shown in figure [ fig : mse ] is negligible . on the contrary , for @xmath203 ( i.e. , for a larger value of the ratio of the number of harmonics composing the approximated field to the number of sensors ) , the reconstruction performance degrades significantly and this is particularly evident in presence of high values of @xmath62 . in presence of fading , with @xmath198 db and @xmath187 . the curves obtained for different values of @xmath71 are compared with the density @xmath204 . ] in the case of massively dense networks , the lsd of @xmath3 is given by ( [ eq : cor1 ] ) and from ( [ eq : cor3.2 ] ) we know that the mse tends to 0 as @xmath205 . this result is confirmed by the plot in figure [ fig : dense ] , which shows the aesd @xmath206 , for @xmath187 , @xmath198 db , and @xmath201 . the behavior of such a function is compared with the density @xmath204 as @xmath71 varies . we note that , as @xmath71 decreases , the matching between @xmath206 and @xmath204 improves , and the latter represents an excellent approximation already for @xmath207 , as predicted by the result in ( [ cor1 ] ) . in environmental monitoring applications , it is often desirable to vary the resolution level with which the field measurements are taken over the region under observation , depending on the field variations and the interest level of the different locations @xcite . it follows that the number of samples generated by the sensors network ( i.e. , the offered traffic load ) varies in the spatial domain . to represent such a scenario , we consider a wireless sensor network whose nodes are uniformly deployed over a square region . we also identify @xmath208 areas , @xmath209 @xmath210 , each corresponding to a different value of the offered traffic load . as often assumed in the literature ( see e.g. , @xcite ) and widely applied in the practice , the network is divided into clusters and a hierarchy of clusters is created . more specifically , at the first hierarchical layer , layer 1 , the sensors are grouped into clusters , each of which is controlled by a cluster - head . the cluster - head is in charge of handling all traffic packets it receives from the nodes . at a given layer @xmath211 of the hierarchy , the cluster - heads are grouped into clusters on their turn and forward the traffic to their parent cluster - head . at the highest layer , layer @xmath212 , we have only one cluster whose cluster - head coincides with the sink node . without loss of generality , we assume that the cluster at the @xmath212-th layer is composed of @xmath208 cluster - heads , each handling the traffic generated within one of the @xmath208 areas defined above . as for the medium access control ( mac ) layer , we consider that the nodes implement the ieee 802.15.4 standard specifications for wireless sensor networks @xcite . in particular , all nodes within a cluster are in radio visibility of each other and use the slotted carrier - sense multiple - access / collision avoidance ( csma / ca ) technique @xcite . this is a contention - based scheme and transmissions may fail if two or more sensors access the channel at the same time . inter - cluster interference is instead avoided by assigning different frequency channels to neighboring clusters . we consider that packets , whose transmission fails , are discarded . in order to derive the probability that a packet transmission fails within a cluster due to collision , we use the markov chain model presented in @xcite . we denote by @xmath213 the average number of sensors belonging to the generic cluster at the @xmath214-th layer of the hierarchical architecture , in area @xmath209 ( @xmath210 and @xmath215 ) . similarly , we define @xmath216 as the average traffic load per node , again within the generic cluster at the @xmath214-th layer , in area @xmath209 . then , we set the size of the packet payload to 32 bytes , and the value of the other parameters as in @xcite . under this setting , we compute the value of the collision probability within the generic cluster at layer @xmath214 , in area @xmath209 , as a function of @xmath213 and @xmath216 , i.e. , @xmath217 @xcite . furthermore , we observe that at the generic layer @xmath214 , with @xmath218 , a node , which acts as cluster - head at layer @xmath219 in area @xmath209 , will have a traffic load equal to @xmath220 $ ] . it follows that the probability that a packet is successfully delivered to the corresponding @xmath214-layer cluster - head within area @xmath209 ( @xmath210 ) can be obtained as @xmath221 . then , the probability that a measurement generated by a sensor located in @xmath209 ( @xmath210 ) is successfully delivered to the sink is given by : @xmath222 next , denoting by @xmath223 the measure of @xmath209 , we define @xmath224 as the normalized probability that a message is successfully delivered to the sink . then , the spatial density of the sensors successfully sending their message is as follows : @xmath225 the density of @xmath169 is therefore given by @xmath226 and the asymptotic mse is given by @xmath227 ) and the case where all measurements successfully reach the sink ( @xmath35 ) . the mse is shown as a function of @xmath71 and for different values of signal - to - noise ratio ( @xmath228 , @xmath229 , @xmath230 , @xmath231 , @xmath232 , @xmath233 ) . ] ) and the case where all measurement transmissions are successful ( @xmath35 ) . the mse is shown as @xmath71 varies and for different values of signal - to - noise ratio ( @xmath228 , @xmath229 , @xmath234 , @xmath235 , @xmath236 , @xmath237 ) . ] figures [ fig : lambda2 ] and [ fig : lambda1 ] show the impact of collisions due to the contention - based channel access , on the quality of the reconstructed field . in particular , they compare the mse of the reconstructed field when collisions are taken into account ( @xmath104 ) with the one obtained in the idealistic case where all messages ( measurements ) sent by the sensors successfully reach the sink ( @xmath35 ) . the results refer to a square region of unitary side , where there are four areas of equal size ( @xmath238 , @xmath239 ) but corresponding to different resolution levels in the measurements collection ( i.e. , they are characterized by different traffic loads ) ; the number of hierarchical levels is set to @xmath228 . we set @xmath230 , @xmath231 , @xmath232 , @xmath233 in figure [ fig : lambda2 ] , and a higher traffic load in figure [ fig : lambda1 ] , i.e. , @xmath234 , @xmath235 , @xmath236 , @xmath237 . looking at the plots , we observe that both @xmath71 and @xmath62 have a significant impact of the obtained mse , with the mse increasing as @xmath71 grows and smaller values of @xmath62 are considered . most interestingly , by comparing the two figures , we can see that as the traffic load , hence the collision probability , increases , the performance derived taking into account the contention - based channel access significantly differs from the idealistic one . furthermore , the latter effect is particularly evident as @xmath62 increases , since the higher the signal - to - noise ratio , the more valuable the samples sent by the sensors toward the sink . we studied the performance of a wireless network whose nodes sense a multi - dimensional field and transfer their measurements to a sink node . as often happens in practical cases , we assumed the sensors to be randomly deployed over ( the whole or only a portion of ) the region of interest , and that their measurements may be lost due to fading or transmission collisions over the wireless channel . we modeled the sampling system through a multi - folded vandermonde matrix @xmath2 and , by using asymptotic analysis , we approximated the mse of the field , which the sink node reconstructs from the received sensor measurements with the @xmath4-transform of @xmath3 . our results clearly indicate that the percentage of region where sensors can not be deployed must be extremely small if an accurate field estimation has to be obtained . also , the effect of losses due to fading or transmission collisions can be greatly mitigated provided that a suitable value for the ratio between the number of harmonics approximating the field bandwidth and the number of sensors is selected . the @xmath79-th moment of the asymptotic eigenvalue distribution of @xmath3 can be expressed as @xcite @xmath240\ ] ] where @xmath241 is the normalized matrix trace operator . the matrix power can be expanded as a multiple sum over the entries of @xmath2 : @xmath242\ ] ] where @xmath243 , @xmath244 are integer indices and @xmath245 , @xmath246\tran$ ] , @xmath247 are the indices identifying the rows of @xmath2 . since , @xmath248 for @xmath249 and the elements of @xmath250 are i.i.d . , we have that @xmath251 \non\ ] ] where the index @xmath252 is to be considered modulo @xmath79 , i.e. , @xmath253 . as for the sum over the indices @xmath254 we note that any choice of @xmath255\tran$ ] induces a partition @xmath88 of the set , @xmath256 in @xmath85 subsets @xmath257 , @xmath258 , under the equality relation @xcite . in the following , we denote by @xmath83 the set of partitions of @xmath259 in @xmath85 subsets , @xmath258 . since there are @xmath260 possible vectors @xmath261 inducing a given partition @xmath87 , we can write the @xmath79-th moment as @xmath262 \non & = & \lim_{n , m \rightarrow \infty}\sum_{k=1}^p\beta_{n , m}^{p - k}\sum_{{\boldsymbol{\omega}}\in \omega_{p , k } } \frac{{\mathbb{e}}\left[\phi_{{\boldsymbol{\omega } } } ( { { \bf x}}_1,\ldots,{{\bf x}}_k ) \right]}{n^{d(p - k+1 ) } } \label{eq : moment2}\end{aligned}\ ] ] where @xmath263 , @xmath264 , and @xmath265 is the index of the subset of @xmath259 containing @xmath252 . recall that @xmath266 and that @xmath77 . moreover , since the vectors @xmath250 are i.i.d . , we removed the dependence on the subscript @xmath267 . following the same steps as in ( * ? ? ? * appendix h ) , we compute the limit @xmath268}{n^{d(p - k+1 ) } } & = & \lim_{n \rightarrow \infty}\int_{{{\cal h}}^k}f_x({{\bf x}}_1)\cdots f_x({{\bf x}}_k)\frac{\phi_{{\boldsymbol{\omega}}}({{\bf x}}_1,\ldots,{{\bf x}}_k)}{n^{d(p+1-k ) } } { { \rm\,d}}{{\bf x}}_1 \cdots { { \rm\,d}}{{\bf x}}_k \non & = & \lim_{n \rightarrow \infty}\int_{{{\cal h}}^k}f_x({{\bf x}}_1)\cdots f_x({{\bf x}}_k)\prod_{j=1}^d \frac{f_{{\boldsymbol{\omega}}}(x_{1j},\ldots , x_{kj})}{n^{p+1-k } } { { \rm\,d}}x_{1j}\cdots { { \rm\,d}}x_{kj } \nonumber\end{aligned}\ ] ] we then define @xmath269 $ ] where @xmath270 $ ] , for @xmath271 and we integrate first with respect to the variables @xmath272 obtaining @xmath268}{n^{d(p - k+1 ) } } & = & \lim_{n \rightarrow \infty}\int_{[-\frac{1}{2},\frac{1}{2}]^{(d-1)k } } g_{{\boldsymbol{\omega}}}({{\bf y}}_1,\ldots,{{\bf y}}_k)\prod_{j=2}^d \frac{f_{{\boldsymbol{\omega}}}(x_{1j},\ldots , x_{kj})}{n^{p - k+1 } } { { \rm\,d}}x_{1j}\cdots { { \rm\,d}}x_{kj } \nonumber\end{aligned}\ ] ] where @xmath273^{k}}\frac{f_{{\boldsymbol{\omega}}}(x_{11},\ldots , x_{k1})}{n^{p - k+1}}f_x([x_{11 } , { { \bf y}}_1])\cdots f_x([x_{k1 } , { { \bf y}}_k]){{\rm\,d}}x_{11}\cdots { { \rm\,d}}x_{k1 } \label{eq : g_omega}\ ] ] in ( * ? ? ? * appendix h ) it was shown that , because of the properties of @xmath274 , @xmath275 where @xmath276 for any @xmath277 . this means that the integral in ( [ eq : g_omega ] ) can be limited to the @xmath278 on the diagonal where @xmath279 . therefore @xmath280^{k}}\frac{f_{{\boldsymbol{\omega}}}(x_{11},\ldots , x_{k1})}{n^{p - k+1 } } f_x([x_{k1 } , { { \bf y}}_1])\cdots f_x([x_{k1 } , { { \bf y}}_k]){{\rm\,d}}x_{11}\cdots { { \rm\,d}}x_{k1 } \non & = & \int_{[-\frac{1}{2},\frac{1}{2 } ] } \prod_{h=1}^k f_x([x_{k1 } , { { \bf y}}_h ] ) \lim_{n \rightarrow \infty}\left(\int_{[-\frac{1}{2},\frac{1}{2}]^{k-1}}\frac{f_{{\boldsymbol{\omega}}}(x_{11},\ldots , x_{k1})}{n^{p - k+1}}\prod_{h=1}^{k-1}{{\rm\,d}}x_{h1}\right ) { { \rm\,d}}x_{k1 } \non & = & v({\boldsymbol{\omega } } ) \int_{[-\frac{1}{2},\frac{1}{2 } ] } \prod_{h=1}^k f_x([x_{k1 } , { { \bf y}}_h ] ) { { \rm\,d}}x_{k1 } \end{aligned}\ ] ] note that the limit @xmath281^{k-1}}\frac{f_{{\boldsymbol{\omega}}}(x_{11},\ldots , x_{k1})}{n^{p - k+1}}\prod_{h=1}^{k-1}{{\rm\,d}}x_{h1}\ ] ] does not depend on @xmath282 and the coefficient @xmath283 $ ] is described in @xcite . next , iterating this procedure by integrating over the variables , @xmath284 , @xmath285 we finally get @xmath286}{n^{d(p - k+1 ) } } & = & v({\boldsymbol{\omega}})^d \int_{[-\frac{1}{2},\frac{1}{2}]^d } f_x(x_{k1 } , \ldots , x_{kd})^k { { \rm\,d}}x_{k1 } \cdots { { \rm\,d}}x_{kd } \non & = & v({\boldsymbol{\omega}})^d \int_{{{\cal h } } } f_x({{\bf x}}_k)^k { { \rm\,d}}{{\bf x}}_k \non & = & v({\boldsymbol{\omega}})^d i_k\end{aligned}\ ] ] where we defined @xmath287 . it follows that @xmath288 which proves the theorem . note that when the entries of @xmath289\tran$ ] are independent with continuous distribution @xmath290 such that @xmath291 , we have @xmath292 with @xmath293 } f_{x_j}(x)^k{{\rm\,d}}x$ ] . from theorem [ th:1 ] and the definition of @xmath96 , we have that @xmath294 next , we define the set @xmath110 where @xmath104 is strictly positive as @xmath295 note that for @xmath296 the contribution to the integral in ( [ eq : moment_psi ] ) is zero . thus , @xmath297 where for any @xmath298 , @xmath299 is the @xmath79-th moment of @xmath3 when the phases are uniformly distributed in @xmath15 and the ratio @xmath300 is given by @xmath301 note also that ( [ eq : moments_psi2 ] ) holds for @xmath302 since , by definition , the zero - th moment of any distribution is equal to 1 . expression ( [ eq : moments_psi2 ] ) allows us to write the moments of @xmath3 for any distribution @xmath104 , given the moments for uniformly distributed phases . likewise , it is possible to describe the lsd of @xmath3 , for any continuous @xmath104 , in terms of the lsd obtained for uniformly distributed phases . indeed , let us denote the laplace transform of @xmath303 by @xmath304 if it exists . then , whenever the sum converges @xmath305 since @xmath306 for any distribution , using ( [ eq : moments_psi2 ] ) we obtain : @xmath307 where @xmath308 is the measure of @xmath110 and @xmath309 is the laplace transform of @xmath82 . by using the properties of the laplace transform and by taking its inverse , we finally get @xmath310 we can rewrite the second term of ( [ eq : distribution_psi ] ) by defining the cumulative density function @xmath311 for @xmath114 . by using the corresponding probability density function , @xmath115 , and lebesgue integration , we can rewrite ( [ eq : distribution_psi ] ) as in ( [ eq : th2 ] ) . from the result in ( [ eq : th2 ] ) and from the assumption @xmath117 @xmath118 ( i.e. , @xmath187 ) , we have @xmath312 then , from the definition of @xmath119 given in ( [ eq : fxi_scaled ] ) it follows that @xmath313 and , by consequence , @xmath314 . therefore , from ( [ eq : f - scal ] ) we have : @xmath315 from the expression of the moments given in theorem [ th:1 ] and the results in @xcite , it is easy to show that for uniformly distributed phases , we have : @xmath316 for any @xmath317 . it immediately follows that @xmath318 where @xmath319 is the dirac s delta function . by applying this result to ( [ eq : th2 ] ) , we get @xmath320 by using the definition of the @xmath4-transform and the result in ( [ eq : th2 ] ) , we obtain : @xmath321 \non & = & \int_0^{\infty } \frac{1}{\gamma z+1}f_{\lambda , x}(d,\beta , z ) { { \rm\,d}}z \non & = & \int_0^{\infty } \frac{1-|{{\cal a}}|}{\gamma z+1}\delta(z ) + |{{\cal a}}| \int_0^{\infty}\frac{g_x(y)}{y } \int_0^{\infty } \frac{1}{\gamma z+1}f_{\lambda , u}\left(d,\frac{\beta}{y},\frac{z}{y}\right){{\rm\,d}}z { { \rm\,d}}y \non & = & 1-|{{\cal a}}|+ |{{\cal a}}| \int_0^{\infty } g_x(y ) \int_0^{\infty } \frac{1}{\gamma y z+1}f_{\lambda , u}\left(d,\frac{\beta}{y } , z\right){{\rm\,d}}z { { \rm\,d}}y \non & = & 1-|{{\cal a}}|+ |{{\cal a}}| \int_0^{\infty } g_x(y ) \eta_u\left(d,\frac{\beta}{y } , \gamma y\right ) { { \rm\,d}}y .\end{aligned}\ ] ] then , by considering that @xmath322 , the expression of the asymptotic mse immediately follows . in appendix [ cor1 ] we have shown that @xmath323 . from the definition of the @xmath4-transform , it follows that @xmath324 and @xmath325 . thus , from ( [ eq : eta ] ) we have : @xmath326 where we defined @xmath327 . as a consequence , @xmath328 m. perillo , z. ignjatovic , and w. heinzelman , `` an energy conservation 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environmental monitoring is often performed through a wireless sensor network , whose nodes are randomly deployed over the geographical region of interest . sensors sample a physical phenomenon ( the so - called field ) and send their measurements to a _ sink _ node , which is in charge of reconstructing the field from such irregular samples . in this work , we focus on scenarios of practical interest where the sensor deployment is unfeasible in certain areas of the geographical region , e.g. , due to terrain asperities , and the delivery of sensor measurements to the sink may fail due to fading or to transmission collisions among sensors simultaneously accessing the wireless medium . under these conditions , we carry out an asymptotic analysis and evaluate the quality of the estimation of a @xmath0-dimensional field ( @xmath1 ) when the sink uses linear filtering as a reconstruction technique . specifically , given the matrix representing the sampling system , @xmath2 , we derive both the moments and an expression of the limiting spectral distribution of @xmath3 , as the size of @xmath2 goes to infinity and its aspect ratio has a finite limit bounded away from zero . by using such asymptotic results , we approximate the mean square error on the estimated field through the @xmath4-transform of @xmath3 , and derive the sensor network performance under the conditions described above .

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Proceed to summarize the following text: the methods used to calculate the energy and the lifetime of a resonance state are numerous @xcite and , in some cases , has been put forward over strong foundations @xcite . however , the analysis of the numerical results of a particular method when applied to a given problem is far from direct . the complex scaling ( complex dilatation ) method @xcite , when the hamiltonian @xmath5 allows its use , reveals a resonance state by the appearance of an isolated complex eigenvalue on the spectrum of the non - hermitian complex scaled hamiltonian , @xmath6 @xcite . of course in an actual implementation the rotation angle @xmath7 must be large enough to rotate the continuum part of the spectrum beyond the resonance s complex eigenvalue . moreover , since most calculations are performed using finite variational expansions it is necessary to study the numerical data to decide which result is the most accurate . to worsen things the variational basis sets usually depend on one ( or more ) non - linear parameter . for bound states the non - linear parameter is chosen in order to obtain the lowest variational eigenvalue . for resonance states things are not so simple since they are embedded in the continuum . the complex virial theorem together with some graphical methods @xcite allows to pick the best numerical solution of a given problem , which corresponds to the stabilized points in the @xmath7 trajectories @xcite . other methods to calculate the energy and lifetime of the resonance , based on the numerical solution of complex hamiltonians , also have to deal with the problem of which solutions ( complex eigenvalues ) are physically acceptable . for example , the popular complex absorbing potential method , which in many cases is easier to implement than the complex scaling method , produces the appearance of nonphysical complex energy stabilized points that must be removed in order to obtain only the physical resonances @xcite . the aforementioned issues explain , at some extent , why the methods based only in the use of real @xmath2 variational functions are often preferred to analyze resonance states . these techniques reduce the problem to the calculation of eigenvalues of real symmetric matrices @xcite . of course , these methods also have its own drawbacks . one of the main problems was recognized very early on ( see , for example , the work by hol@xmath8ien and midtdal @xcite ) : if the energy of an autoionizing state is obtained as an eigenvalue of a finite hamiltonian matrix , which are the convergence properties of these eigenvalues that lie in the continuum when the size of the hamiltonian matrix changes ? but in order to obtain resonance - state energies it is possible to focus the analysis in a global property of the variational spectrum : the density of states ( dos)@xcite , being unnecessary to answer this question . the availability of the dos allows to obtain the energy and lifetime of the resonance in a simple way , both quantities are obtained as least square fitting parameters , see for example @xcite . despite its simplicity , the determination of the resonance s energy and width based in the dos is far from complete . there is no a single procedure to asses both , the accuracy of the numerical findings and its convergence properties , or which values to pick between the several `` candidates '' that the method offers @xcite . recently , pont _ et al _ @xcite have used _ finite size scaling _ arguments @xcite to analyze the properties of the dos when the size of the hamiltonian changes . they presented numerical evidence about the critical behavior of the density of states in the region where a given hamiltonian has resonances . the critical behavior was signaled by a strong dependence of some features of the density of states with the basis - set size used to calculate it . the resonance energy and lifetime were obtained using the scaling properties of the density of states . however , the feasibility of the method to calculate the resonance lifetime laid on the availability of a known value of the lifetime , making the whole method dependent on results not provided by itself . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . for each basis - set size , @xmath3 , used , there are @xmath3 variational eigenvalues @xmath10 . each one of these eigenvalues can be used , at least in principle , to compute a dos , @xmath11 , resulting , each one of these dos in an approximate value for the energy , @xmath12 , and width , @xmath13 , of the resonance state of the problem . if the variational problem is solved for many different basis - set sizes , there is not a clear cut criterion to pick the `` better '' result from the plethora of possible values obtained . this issue will be addressed in section [ model ] . in this work , in order to obtain resonance energies and lifetimes , we calculate all the eigenvalues for different basis - set sizes , and present a recipe to select adequately certain values of @xmath3 , and one eigenvalue for each @xmath3 elected , that is , we get a series of variational eigenvalues @xmath14 . the recipe is based on some properties of the variational spectrum which are discussed in section [ some - properties ] . the properties seem to be fairly general , making the implementation of the recipe feasible for problems with several particles . actually , because we use scaling properties for large values of @xmath3 , the applicability of the method for systems with more than three particles could be restricted because the difficulties to handle large basis sets . the set of approximate resonance energies , obtained from the density of states of a series of eigenvalues selected following the recipe , shows a very regular behaviour with the basis set size . this regular behaviour facilitates the use of finite size scaling arguments to analyze the results obtained , in particular the extrapolation of the data when @xmath15 . the extrapolated values are the most accurate approximation for the parameters of the resonance state that we obtain with our method . this is the subject of section [ recipe ] , where we present results for models of one and two particles . following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths in section [ golden - rule ] . using the scaling function that characterizes the behaviour of the approximate energies as a guide , it is possible to find a very good approximation to the resonance width since , again , the data generated using our prescription seems to converge when @xmath4 . finally , in section [ discusion ] we summarize and discuss our results . when one is dealing with the variational spectrum in the continuum region , some of its properties are not exploited to obtain more information about the presence of resonances , usually the focus of interest is the stabilization of the individual eigenvalues . the stabilization is achieved varying some non - linear variational parameter . if @xmath9 is the inverse characteristic decaying length of the variational basis functions , then the spectrum of the kinetic energy scales as @xmath16 , moreover , for potentials that decay fast enough , the spectrum of the whole hamiltonian _ also _ scales as @xmath16 for large ( or small ) enough values of @xmath9 ( see appendix ) . this is so , since the variational eigenfunctions are @xmath2 approximations to plane waves _ except _ when @xmath9 belongs to the stabilization region . when @xmath9 belong to the stabilization region of a given variational eigenvalue , say @xmath17 , then @xmath18 ( where @xmath0 is the resonance energy ) and the variational eigenfunction @xmath19 has the localization length of the potential well . we intend to take advantage of the changes of the spectrum when @xmath9 goes from small to large enough values . the variational spectrum satisfies the hylleras - undheim theorem or variational theorem : if @xmath3 is the basis set size , and @xmath20 is the @xmath21 eigenvalue obtained with a variational basis set of size @xmath3 , then @xmath22 actually , since the threshold of the continuum is an accumulation point , then for small enough values of @xmath9 and a given @xmath23 there is always a @xmath24 such that @xmath25 for the kinetic energy variational eigenvalues , and for fixed @xmath26 , and @xmath27 , if the ordering given by equation ( [ ordering ] ) holds for some @xmath9 then it is true for all @xmath9 . of course this is not true for a hamiltonian with a non zero potential that support resonance states . so , we will take advantage of the variational eigenvalues such that for @xmath9 small enough satisfies equation ( [ ordering ] ) but , for @xmath9 large enough @xmath28 despite its simplicity , the arguments above give a complete prescription to pick a set of eigenstates that are particularly affected by the presence of a resonance . choose @xmath3 and @xmath29 arbitrary , and then look for the smaller values of @xmath30 and @xmath27 such that the two inequalities , equations . ( [ ordering ] ) and ( [ reverse ] ) are fulfilled . so far , all the examples analyzed by us show that if the inequalities are satisfied for some @xmath30 and @xmath27 then they are satisfied too by the eigenvalues @xmath31 , for @xmath32 . to illustrate how our prescription works we used two different model hamiltonians . the first model , due to hellmann @xcite , is a one particle hamiltonian that models a @xmath33-electron atom . the second one is a two particle model that has been used to study the low energy and resonance states of two electrons confined in a semiconductor quantum dot @xcite . the details of the variational treatment of both models will be kept as concise as possible . the one particle model has been used before for the determination of critical nuclear charges for @xmath33-electron atoms @xcite , it also gives reasonable results for resonance states in atomic anions @xcite and continuum states @xcite . the interaction of a valence electron with the atomic core is modeled by a one - particle potential with two asymptotic behaviours . the potential behaves correctly in the regions where electron is far from the atomic core ( @xmath34 electrons and the nucleus of charge @xmath35 ) and when it is near the nucleus . the hamiltonian , in scaled coordinates @xmath36 , is @xmath37 where @xmath38 and @xmath39 is a range parameter that determines the transition between the asymptotic regimes , for distances near the nucleus @xmath40 and in the case @xmath41 the nucleus charge is screened by the @xmath34 localized electrons and @xmath42 . another advantage of the potential comes from its analytical properties . in particular this potential is well behaved and the energy of the resonance states can be calculated using complex scaling methods . so , besides its simplicity , the model potential allows us to obtain the energy of the resonance by two independent methods and check our results . the two particle model that we considered describes two electrons interacting via the coulomb repulsion and confined by an external potential with spherical symmetry . we use a short - range potential suitable to apply the complex scaling method . the hamiltonian @xmath5 for the system is given by @xmath43 where @xmath44 , @xmath45 the position operator of electron @xmath46 ; @xmath47 and @xmath48 determine the range and depth of the dot potential . after re - scaling with @xmath47 , in atomic units , the hamiltonian of equation ( [ hamiltoniano ] ) can be written as @xmath49 where @xmath50 . the variational spectrum of the two particle model , equation ( [ hamil2part ] ) , and all the necessary algebraic details to obtain it , has been studied with great detail in reference @xcite so , until the end of this section , we discuss the variational solution of the one particle model given by equation ( [ hamil ] ) . the discrete spectrum and the resonance states of the model given by equation ( [ hamil ] ) can be obtained approximately using a real @xmath51 truncated basis set @xmath52 to construct a @xmath53 hamiltonian matrix @xmath54 . we use the rayleigh - ritz variational method to obtain the approximations @xmath55 @xmath56 for bound states this functions are variationally optimal . the functions @xmath57 are @xmath58 and @xmath59 are the associated laguerre polynomials of @xmath60 order and degree @xmath61 . the non - linear parameter @xmath9 is used for eigenvalue stabilization in resonance analysis @xcite . note that @xmath9 plays a similar role that the finite size of the box in spherical box stabilization procedures @xcite , as stated by kar _ et . @xcite . resonance states are characterized by isolated complex eigenvalues , @xmath62 , whose eigenfunctions are not square - integrable . these states are considered as quasi - bound states of energy @xmath0 and inverse lifetime @xmath1 . for the hamiltonian equation ( [ hamil ] ) , the resonance energies belong to the positive energy range @xcite . using the approximate solutions of hamiltonian ( [ hamil ] ) we analyze the dos method @xcite that has been used extensively to calculate the energy and lifetime of resonance states , in particular we intend to show that 1 ) the dos method provides a host of approximate values whose accuracy is hard to assess , and 2 ) if the dos method is supplemented by a new optimization rule , it results in a convergent series of approximate values for the energy and lifetime of resonance states . the dos method relies on the possibility to calculate the ritz - variational eigenfunctions and eigenvalues for many different values of the non - linear parameter @xmath9 ( see kar and ho @xcite ) . the localized dos @xmath63 can be expressed as @xcite @xmath64 since we are dealing with a numerical approximation , we calculate the energies in a discretization @xmath65 of the continuous parameter @xmath9 . in this approximation , equation ( [ densidad_sin_suma ] ) can be written as @xmath66 where @xmath67 is the @xmath30-th eigenvalue of the @xmath53 matrix hamiltonian with @xmath68 and @xmath39 fixed . in complex scaling methods the hamiltonian is dilated by a complex factor @xmath69 . as was pointed out long time ago by moiseyev and coworkers @xcite , the role played by @xmath9 and @xmath70 are equivalent , in fact , our parameter @xmath9 corresponds to @xmath71 . besides , the dos attains its maximum at optimal values of @xmath9 and @xmath0 that could be obtained with a self - adjoint hamiltonian without using complex scaling methods @xcite . so , locating the position of the resonance using the maximum of the dos is equivalent to the stabilization criterion used in complex dilation methods that requires the approximate fulfillment of the complex virial theorem @xcite . the values of @xmath72 and @xmath73 are obtained performing a nonlinear fitting of @xmath63 , with a lorentzian function , @xmath74}.\ ] ] one of the drawbacks of this method results evident : for each pair @xmath75 there are several @xmath11 , and since each @xmath11 provides a value for @xmath76 and @xmath77 one has to choose which one is the best . kar and ho @xcite solve this problem fitting all the @xmath11 and keeping as the best values for @xmath0 and @xmath1 the fitting parameters with the smaller @xmath78 value . at least for their data the best fitting ( the smaller @xmath78 ) usually corresponds to the larger @xmath30 . this fact has a clear interpretation , if the numerical method approximates @xmath0 with some @xmath79 , where @xmath3 is the basis set size of the variational method , a large @xmath30 means that the numerical method is able to provide a large number of approximate levels , and so the continuum of positive - energy states is `` better '' approximated . in a previous work @xcite we have shown that a very good approximation to the energy of the resonance state is obtained considering just the energy value where @xmath11 attains its maximum . we denote this value as @xmath76 . figure [ prefig1 ] shows the approximate resonance energy @xmath76 for different basis set size @xmath3 , where @xmath29 is the index of the variational eigenvalue used to calculate the dos . we used the values @xmath80 and @xmath81 corresponding to the ones used before @xcite in the analysis of @xmath82 resonances . the figure [ prefig1 ] also shows the value calculated using complex scaling . it is clear that the accuracy of all the values shown is rather good ( all the values shown differ in less than 6@xmath83 ) , and that larger values on @xmath84 provide better values for the resonance energy . these facts are well known from previous works , _ i.e. _ almost all methods to calculate the energy of the resonance give rather stable and accurate results for @xmath0 . however , the practical importance of this fact is reduced : these are uncontrolled methods , so the accuracy of the values obtained from the dos can not be assessed ( without a value independently obtained ) and these values do not seem to converge to the value obtained using complex scaling when @xmath3 is increased and @xmath84 is kept fixed . there is another fact that potentially could render the whole method useless : for small or even moderate @xmath29 , the values @xmath76 become _ unstable _ ( see figure [ prefig1 ] ) when @xmath3 is large enough . this last point has been pointed previously @xcite . in the problem that we are considering is rather easy to obtain a large number of variational eigenvalues in the interval where the resonances are located , allowing us to calculate @xmath76 up to @xmath85 , but this situation is far from common see , for example , references @xcite . so far we have presented only results about the behaviour of the one particle hamiltonian , from now on we will discuss both models , equations ( [ hamil ] ) and [ hamil2part ] . it is known that the variational eigenvalues @xmath17 do not present crossings when they are calculated for some fixed values of @xmath3 , _ i.e _ the variational spectrum is non - degenerate for any finite hamiltonian matrix . as a matter of fact the avoided crossings between successive eigenvalues in the variational spectrum are the watermark of a resonance . an interesting feature emerges when the variational spectrum for many different basis set sizes @xmath3 are plotted together versus the parameter @xmath9 . besides the places where @xmath86 attains its minimum value , which correspond to the stabilization points , there are some gaps which correspond to crossings between eigenvalues obtained with different basis set sizes , see figure [ prefig2 ] . moreover , the crossings corresponds to eigenvalues with different index @xmath29 , and are the states that satisfy the inequalities equations ( [ ordering]),and ( [ reverse ] ) . it is worth to remark that the main features shown by figure [ prefig2 ] are independent of the number of particles of the hamiltonian and the particular values of the threshold of the continuum . figure [ bundle2p ] shows the behaviour of the variational eigenvalues obtained for the two particle hamiltonian equation ( [ hamil2part ] ) . in this case the ionization threshold is not the asymptotic value of the potential , but it is given by the energy of the one particle ground state . the resonance state came from the two - particle ground state that becomes unstable and enters into the continuum of states when the quantum dots becomes `` too small '' to accommodate two electrons . for more details about the model , see reference @xcite . the left panel of figure [ prefig3 ] shows the behaviour of the maximum value of the dos , @xmath87 , for the one particle hamiltonian , obtained for different basis - set sizes and fixed @xmath29 ( in this case @xmath85 ) , and the @xmath88 obtained choosing a `` bundle '' of states that are linked by a crossing , these states have @xmath89 and @xmath90 respectively . from our numerical data , the maximum value of the dos scales with the basis - set size following two different prescriptions . for @xmath29 fixed , @xmath91 , with @xmath92 , while when the pair @xmath93 is chosen from the set of pairs that label a bundle of states @xmath94 , with @xmath95 . in particular , for @xmath85 we get that @xmath96 , and @xmath97 when @xmath98 . of course we can pick sets of states that are not related by a crossing . for instance , we also picked sets with a simple prescription as follows : choose a given initial pair @xmath99 and form a set of states with the states labeled by @xmath100 and so on . figure [ prefig3](a ) shows two examples obtained choosing @xmath101 , @xmath102 and @xmath103 and @xmath104 both with @xmath105 . quite interestingly , the data in figure [ prefig3 ] show that the scaled maxima of the dos for a bundle and two different sets seem to converge to the _ same _ value when @xmath106 , but only for the bundle the scaling function is @xmath107 . the advantage obtained from picking those eigenvalues @xmath17 that belong to a given bundle is still more evident when the corresponding dos and @xmath0 are calculated . the right panel of figure [ prefig3 ] shows the @xmath0 obtained from the dos whose maxima are shown in the left panel . it is rather evident that these values now seem to converge , besides , the extrapolation to @xmath4 results in a more accurate approximate value for @xmath0 . in contradistinction , the values for @xmath0 corresponding to a fixed index @xmath29 ( the values shown in the figure [ prefig3 ] correspond to @xmath85 ) do not seem to converge anywhere close to the value obtained using complex rotation . figure [ er2par ] shows the resonance energies obtained from the bundles of states shown in figure [ bundle2p ] for the two - particle model . since the numerical solution of this model is more complicated than the solution of the one - particle model the number of approximate values is rather reduced . however , it seems that the data also supports a linear scaling of @xmath76 with @xmath108 . many real algebra methods to calculate resonance energies use a golden rule - like formula to calculate the resonance width . in this section we will use the formula and stabilization procedure proposed by tucker and truhlar @xcite that we will describe briefly for completeness . this projection formula seems to work better for one - particle models . for two - particle models its utility has been questioned @xcite , so to analyze the width of the resonance states of the quantum dot model we fitted the corresponding dos using equation ( [ lorentz ] ) . the method of tucker and truhlar @xcite is implemented by the following steps . choose a basis @xmath109 where @xmath9 is a non - linear parameter . diagonalize the hamiltonian using up to @xmath3 functions of the basis . look for the stabilization value @xmath110 and its corresponding eigenfunction @xmath111 which are founded for some value @xmath112 . define the projector @xmath113 where @xmath114 is the normalized projection of @xmath111 onto the basis @xmath115 for any other @xmath9 . diagonalize the hamiltonian @xmath116 in the basis @xmath115 , again as a function of @xmath9 , and find a value @xmath117 of @xmath9 such that @xmath118 where @xmath119 denotes eigenvalue @xmath84 of the projected hamiltonian for the scale factor @xmath117 , and @xmath120 is the corresponding eigenfunction . with the previous definitions and quantities , the resonance width @xmath1 is given by @xmath121 where @xmath122.\ ] ] despite some useful insights , the procedure sketched above does not determine all the intervening quantities , for instance there are many solutions to equation ( [ seg - estabili ] ) and , of course , the stabilization method provides several good candidates for @xmath111 and @xmath112 . we are able to avoid some of the indeterminacies associated to the tucker and truhlar procedure using a bundle of states associated to a crossing , so @xmath111 and @xmath112 are given by any of the eigenfunctions associated to a bundle and @xmath123 comes from the stabilization procedure . then we construct projectors @xmath124 where @xmath19 is one of the variational eigenstates that belong to a bundle of states . with the projectors @xmath125 we construct hamiltonians @xmath126 , and find the solutions to the problem @xmath127 since there is not an a priori criteria to choose one particular solution of equation ( [ tercera - estabili ] ) we show our numerical findings for several values of @xmath84 . figure [ prefig4 ] shows the behaviour of the resonance width calculated with equation ( [ golden2 ] ) , where we have used @xmath128 as @xmath111 and @xmath129 , where @xmath130 and @xmath131 . despite that the different sets corresponding to different values of @xmath84 do not converge to any definite value , for @xmath3 large enough all the sets scale as @xmath132 , with @xmath133 . since the resonance energy scales as @xmath107 , at least when a bundle of states with a crossing is chosen to calculate approximations ( see figure [ prefig3 ] ) , we suggest that the right scaling for @xmath1 is given by @xmath134 . of course for a given basis size , particular variational functions , stabilization procedures and so on , we can hardly expect to find a proper set of @xmath1 whose scaling law would be @xmath107 . instead of this we propose that the data in the right panel can be fitted by @xmath135 then the best approximation for the resonance width is obtained fitting the curve and selecting the @xmath136 as @xmath137 for @xmath138 the closest value to one . as pointed in reference @xcite , the projection technique to calculate the width of a resonance can be implemented if a suitable form of the projection operator can be found . as this procedure is marred by several issues we used the dos method to obtain the approximate widths of a resonance state of the two particle model . figure [ gama2par ] shows the widths calculated associated to the energies shown in figure [ er2par ] , the parameters of the hamiltonian are exactly the same . there is no obvious scaling function that allows the extrapolation of the data but , even for moderate values of @xmath3 , it seems as the data converge to the value obtained using complex scaling . in this work we analyzed the convergence properties of real @xmath51 basis - set methods to obtain resonance energies and lifetimes . the convergence of the energy with the basis - set size for bound states is well understood , the larger the basis set the better the results and these methods converge to the exact values for the basis - set size going to infinite ( complete basis set ) . this idea is frequently applied to resonance states . the increase of the basis - set size in some commonly used methods does not improve the accuracy of the value obtained for the resonance energy @xmath0 , as showed in figure [ prefig1 ] . this undesirable behavior comes from the fact that the procedure is not variational as in the case of bound states . moreover , the exact resonance eigenfunction does not belong to the hilbert space expanded by the complete basis set . in this work we presented a prescription to pick a set or bundle of states that has linear convergence properties for small width resonances . this procedure is robust because the choice of different bundles results in very similar convergence curves and energy values . in fact , in the method described here , the pairs @xmath93 of the bundles play the role of a second stabilization parameter together with the variational parameter @xmath9 . of a second we tested the method in others one and two particle systems and the general behavior of them is the same . the results are very good in all cases leading to an improvement in the calculation of the resonance energies . nevertheless we have to note that the method could no be applied in cases where two or more resonance energies lie very close because the overlapping bundles . the lifetime calculation is more subtle . the use of golden - rule - like formulas , as we applied here , always give several possible outcomes for the width @xmath139 , corresponding to different pseudo - continuum states @xmath140 . the projection technique , equation ( [ golden2 ] ) , is not the exception and it is not possible to select _ a priori _ which value of @xmath141 is the most accurate . the linear convergence of the dos with basis - set size suggests that the scaling in the lifetime value , in accordance with the energy scaling , should be linear . regrettably , the projection method gives discrete sets of values which can not be tuned to obtain an exact linear convergence . our recipe is to choose the set @xmath139 whose scaling is closest to the linear one , then the best estimation for the resonance width is obtained from extrapolation . many open questions remain on the analysis of the different convergence properties of resonance energy and lifetime . the method presented here to obtain the resonance energy from convergence properties works very well , but the appearance of bundles in the spectrum is not completely understood . even there is not a rigorous proof , the numerical evidence supports the idea that the behaviour of the systems studied here is quite general . in this appendix we give arguments that support our assumptions on the scaling of the eigenenergies with the basis - set parameter @xmath9 . we present our argument for one body hamiltonians , but it is straightforward to generalize to more particles with pair interactions decaying fast enough at large distances . 1 . let @xmath142 be an @xmath143 matrix with all its matrix elements having the form @xmath144 , where @xmath145 , then if @xmath146 the eigenvalues of @xmath142 scales with @xmath147 : @xmath148=0\;\rightarrow det[a(1)-\frac{\lambda(\eta)}{f(\eta)}\,i]=0 \rightarrow \lambda(1)=\frac{\lambda(\eta)}{f(\eta ) } \,.\ ] ] 2 . let @xmath149 @xmath143 be symmetric matrices with @xmath150 , and @xmath151 the eigenvalues of @xmath149 and @xmath152 respectively , in nondecreasing order , then , by the minimax principle @xcite consider a spherical one - particle potential with compact support , @xmath154 if @xmath155 , and finite , @xmath156 ( both conditions could be relaxed , but we adopt them for simplicity ) . let the basis - set functions be of the form @xmath157 with @xmath158 and @xmath159 , then the coefficients take the form @xmath160 . and then , by equation ( [ asa ] ) , all the eigenvalues of the kinetic energy have the same scaling with @xmath16 . we have to show that , in both limits , @xmath162 and @xmath163 , for all the potential matrix elements hold @xmath164 , and then , by the wielandt - hoffman theorem @xcite , the eigenenergies are a perturbation of the eigenvalues of the kinetic energy . a u hazi and h s taylor , phys . rev . a * 14 * , 2071 ( 1976 ) . a ferrn , o osenda and p serra , phys . a * 79 * , 032509 ( 2009 ) . h feschbach , ann . ny * 19 * , 627 ( 1962 ) n moiseyev , c corcoran , phys . rev . a * 20 * , 814 ( 1979 ) . m e fisher , in _ critical phenomena _ , proceedings of the 51st enrico fermi summer school , varenna , italy , m. s. green , ed . ( academic press , new york 1971 ) . s e tucker and d. g. truhlar , j.chem . * 86 * , 6251 ( 1987 ) . f r manby and g doggetty , j. phys . b : at . mol . . phys . * 30 * , 3343 ( 1997 ) j. h. wilkinson , _ the algebraic eigenvalue problem _ , oxford university press , london , ( 1965 ) .

the resonance states of one- and two - particle hamiltonians are studied using variational expansions with real basis - set functions . the resonance energies , @xmath0 , and widths , @xmath1 , are calculated using the density of states and an @xmath2 golden rule - like formula . we present a recipe to select adequately some solutions of the variational problem . the set of approximate energies obtained shows a very regular behaviour with the basis - set size , @xmath3 . indeed , these particular variational eigenvalues show a quite simple scaling behaviour and convergence when @xmath4 . following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths . using the scaling function that characterizes the behaviour of the approximate energies as a guide , it is possible to find a very good approximation to the actual value of the resonance width .

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Proceed to summarize the following text: it is the object of relativistic heavy - ion collisions to create and study strongly interacting matter excited beyond its hadronic phase @xcite . the existence of such a phase [ the quark gluon plasma ( qgp ) ] has been predicted by lattice qcd calculations @xcite which exhibit a sudden rise in the scaled pressure and entropy density as the temperature is raised just beyond @xmath0 mev . detailed models of nuclear reactions had predicted that the energy deposition in the center - of - mass frame should be sufficient to cause temperatures at mid - rapidity in central collisions of gold nuclei to reach upwards of @xmath1 mev @xcite . these predictions have been confirmed by the experimental results of the four relativistic heavy - ion collider ( rhic ) detector collaborations , which have set a lower bound of about @xmath2 gev/@xmath3 on the energy density at a time @xmath4 fm / c in central au+au collisions @xcite . according to lattice calculations , such energy densities should place the excited matter firmly in the deconfined region . the fact that various lattice observables in the excited phase assume values close to those expected for a free gas of quarks and gluons has let to the picture that beyond @xmath5 , the degrees of freedom in chromodynamic matter are quasiparticles that carry the quantum numbers of quarks and gluons . however , experimental results from the four rhic detectors have cast doubts on this picture : the observed , large , elliptic and radial flow exhibited by the produced matter have led to the speculation that the produced matter may be strongly interacting @xcite . these findings have given rise to phenomenological models in terms of bound states of quarks and gluons @xcite . such approaches , however , do not fare well in comparison with lattice susceptibility calculations @xcite . such lattice comparisons do not reveal any information regarding the gluon sector of the plasma . as a result , efforts to elucidate the nature of the gluon structure have taken a phenomenological turn @xcite _ e.g. _ , there have been recent attempts to probe this structure through jet correlations @xcite . here , we explore a novel means of probing the gluonic structure of the produced matter is proposed : through its possible electromagnetic signature . lepton pairs and real photons occupy a privileged status as they suffer essentially no final state interaction @xcite after their initial production . thus , their emission rates have the potential to provide direct insight into the nature of the medium and its interactions . to this end , we focus on the electromagnetic signatures of a series of pure glue processes , where the final rates are directly proportional to the gluon density of the produced matter . gluons do not carry electric charge , yet their interactions may generate electromagnetic signatures if the _ medium _ is itself electrically charged . in partonic matter at equilibrium , this is achieved by the introduction of a non - vanishing charge chemical potential or a net asymmetry between the quark and anti - quark populations . this leads to a violation of furry s theorem @xcite and the appearance of diagrams where two gluons may fuse through a quark loop to form a photon see fig . [ gg - gamma ] . the possibility of such rates was first pointed out in ref . @xcite , the dilepton rates from such processes in different scenarios was expounded upon in ref . @xcite . because of restrictions imposed by yang s theorem @xcite , dilepton rates from such processes become appreciable only at high transverse momentum of the dilepton pair or in the case where the incoming gluons are themselves massive . the current work , in some ways an extension of these efforts will focus on the photon signature , which does not suffer from either of these constraints . a large number of previous and even recent photon production calculations from an electrically charged qgp have neglected the baryon chemical potential @xmath6 ( as well as other chemical potentials ) in the plasma @xcite : in these cases , the photon production rate only depends on the temperature . it is now known that the central region at the cern sps and even rhic is not just heated vacuum @xcite , but actually displays a finite baryon density or an asymmetry between baryon and anti - baryon populations . consequently , the baryon chemical density and thus @xmath6 in the qgp , does not vanish . in this case , the photon production rate ( from the deconfined sector ) is a function of both temperature @xmath7 and quark chemical potential @xmath8 of the qgp . in this treatment , isospin symmetry of two flavors is imposed , _ i.e. _ , both @xmath9 and @xmath10 quarks will be assumed to be massless and have chemical potentials @xmath11 . as can be immediately demonstrated , such a plasma is globally electrically charged . the earliest estimates of photon production rates from electrically charged plasmas , in ref . @xcite , had pointed out that given an energy density @xmath12 , the photon production rate will decrease strongly with increasing chemical potential of the medium . however , such calculations only include processes which are non - vanishing at @xmath13 . the calculation of rates from these channels have been rigorously carried out in ref . @xcite , in the hard - thermal - loop effective theory @xcite at one loop . rates at two loops in the htl theory , at vanishing chemical potential , were first presented in ref . @xcite , where it was demonstrated that the two - loop rates from bremsstrahlung processes actually dominate over rates at leading order in the coupling . all order resummed rates for bremsstrahlung processes which includes the landau - pomeranchuck migdal suppression from multiple scattering have been presented in ref . the two loop rates have been extended to finite chemical potential and chemical non - equilibrium in ref . the effects of dynamical evolution of the quark gluon plasma on space - time integrated photon yields from such two loop rates were presented recently in ref . @xcite . it is the object of the current work to extend this line of inquiry and present rates for hard photon production from processes which arise solely at finite chemical potential . as this is the first such attempt , we focus on establishing the basic processes and will pursue phenomenological applications elsewhere . in the following section , the calculation of the matrix element of the one - loop gluon - gluon - photon vertex in the htl formalism is presented . in sect . iii , the matrix element is incorporated into the hard photon production rate . in sect . iv , numerical estimates of photon rates from these processes are presented and compared with the leading order rates of ref . @xcite . a comparison with the resummed rates of ref . @xcite can not be performed as yet as these calculations have not been extended to finite baryon density . we summarize and outline future directions in sect . at zero temperature and at finite temperature and zero charge density , diagrams in qed that contain a fermion loop with an odd number of photon vertices ( e.g. fig . [ gg - gamma ] ) are canceled by an equal and opposite contribution originating from the same diagram with fermion lines running in the opposite direction ( furry s theorem @xcite ) . this statement can also be generalized almost unchanged to qcd , for processes with two gluons and an odd number of photon vertices . a physical perspective is obtained by noting that all these diagrams are encountered in the perturbative evaluation of green s functions with an odd number of gauge field operators . at zero ( finite ) temperature , in the well defined case of qed , the focus lies on quantities such as @xmath14 ( @xmath15 $ ] ) under the action of the charge conjugation operator @xmath16 . the photon , being charge conjugation negative , leads to @xmath17 . in the case of the vacuum @xmath18 , we note that @xmath19 , as the vacuum is uncharged . as a result @xmath20 at a temperature @xmath7 , the corresponding quantity to consider is @xmath21 where @xmath22 and @xmath8 is a chemical potential . here , however , @xmath23 , where @xmath24 is a state in the ensemble with the same number of antiparticles as there are particles in @xmath25 and vice - versa . if @xmath26 _ i.e. _ , the ensemble average displays zero density , inserting the operator @xmath27 as before , one obtains , @xmath28 the sum over all states will contain the mirror term @xmath29 , with the same thermal weight . as a result , summing over all states in the ensemble gives , @xmath30 and furry s theorem still holds . however , if @xmath31 ( @xmath32 unequal number of particles and antiparticles ) then @xmath33 the mirror term in such a case is @xmath34 , with a different thermal weight due to the fact that the net charge of the state @xmath35 is different and hence is weighted differently by the chemical potential . as a result , the thermal expectation of an odd number of gauge field operators is non - vanishing , @xmath36 and furry s theorem no longer holds . one may say that the medium , being charged , manifestly breaks charge conjugation invariance and these green s functions are thus finite , leading to the appearance of new processes in a perturbative expansion . two comments are in order : the eventual evaluation of the photon rate will appear to depend on the net baryon density ( as opposed to the net quark density ) , as , in the model of the plasma adopted , the net baryon density is carried equally by the up and down flavors . as pointed out in ref . @xcite , this is not the only means by which a finite charge density may be achieved . the processes of fig . [ gg - gamma ] are also affected by the constraints imposed by yang s theorem which states that a spin one particle may not decay or be formed by two identical massless vectors @xcite . the processes outlined in the following are computed within a thermalized medium , where the presence of longitudinal gluon excitations leads to a breaking of the symmetry which underlies yang s theorem . the incorporation of thermal gluon masses and self - energies in perturbation theory has to be done carefully , owing to issues arising from color gauge invariance . in this work , the calculation is carried out in the gauge invariant resummed theory of hard - thermal - loops @xcite , where one assumes the temperature @xmath37 and as a result the coupling constant @xmath38 . effective propagators and vertices involving soft @xmath39 momenta are obtained by integrating out the hard @xmath40 modes . this allows for a well defined perturbative expansion of the photon production amplitude . the feynman diagrams corresponding to the leading contributions ( in coupling ) to the new channel of photon production are those of fig . [ gg - gamma ] , with two gluons and a photon attached to a quark loop @xcite . such a process does not exist at zero temperature , or even at finite temperature and vanishing chemical potential . at non - zero density , this leads to two new sources for photon production : the fusion of gluons to form a photon ( @xmath41 ) and the decay of a massive gluon into a photon and a softer gluon ( @xmath42 ) . the full , physical , matrix element is obtained by summing contributions from both diagrams which have fermion number running in opposite directions , @xmath43 . the amplitudes corresponding to these two diagrams may be expressed in the imaginary time formalism as : @xmath44 { ( q+k)_\alpha q_\beta(q+k - p)_\gamma\over ( q+k)^2 q^2(q+k - p)^2}\nonumber \\ t_2^{\mu\rho\nu}(p , p - k , k ) & = & t\sum_{q_0 } \int { d^3q\over ( 2\pi)^3 } eg^2 { \delta_{ab}\over 2 } { \mathrm tr}[\gamma^\mu\gamma^\gamma\gamma^\rho\gamma^\beta\gamma^\nu\gamma^\alpha ] { ( q - k)_\alpha q_\beta(q - k+p)_\gamma\over ( q - k)^2 q^2(q - k+p)^2}.\end{aligned}\ ] ] the masses of quarks have been omitted as the momenta of the quarks is considered hard @xmath40 in the htl expansion . in the imaginary time formalism , the zeroth components of four momentum are discrete matsubara frequencies , @xmath45 where integers @xmath46 , @xmath47 and @xmath48 in the above expression range from @xmath49 to @xmath50 , and @xmath8 is the quark chemical potential . it may be easily demonstrated @xcite , using the the properties of the @xmath51 matrices , that at zero chemical potential these two amplitudes cancel each other , consistent with the qcd generalization of furry s theorem @xcite . the sum over the matsubara frequencies may be conveniently performed using the non - covariant propagator method of ref . @xcite . here , one defines a time - three - momentum propagator @xmath52 , as @xmath53 in the above equation , @xmath54 represents the real energy of the particle from its three momentum and not the conjugate the imaginary time @xmath55 . the explicit expression of the imaginary time quark propagator is given by @xmath56{e^{-\tau ( { se}\mp\mu ) } } , \end{aligned}\ ] ] where @xmath57 + 1})}$ ] are fermi - dirac distribution functions . performing the summation of the matsubara frequency @xmath58 leads to the expression for the amplitude : @xmath59 \sum_{s_1s_2s_3 } \frac{s_1s_2s_3}{8e_{q+k}e_qe_{q+k - p } } \nonumber \\ & & { \tiny \frac{(q+k)_{s_1\alpha}{q_{s_2\beta } } { ( q+k - p)_{s_3\gamma } } } { i\omega_p - s_1e_{q+k}+s_3e_{q+k - p } } \left(\frac{{\delta\widetilde{n}}(e_{q+k})-{\delta\widetilde{n}}({e_q } ) } { { { { i\omega}}_k}-{s_1e_{q+k}}+{s_2}{e_q}}- \frac{{\delta\widetilde{n}}(e_{q+k - p})-{\delta\widetilde{n}}({e_q } ) } { { i\omega_k-{{i\omega } } _ p}-{s_3e_{q+k - p}}+{s_2}{e_q}}\right)}.\end{aligned}\ ] ] in the above equation , @xmath60 , and @xmath61 . the further evaluation of the photon production amplitude is carried out in the htl approximation for the quark loop . in this limit , the photon and gluon momenta are considered soft @xmath62 , and the quark momenta are hard @xmath63 , where @xmath7 is the temperature and @xmath64 is the effective coupling constant in the medium . the quark lines which carry a component of the external gluon energies are taylor expanded as follows , @xmath65 where @xmath66 and @xmath67 . using the above approximation , allows for a factorization of the quark angular integral . performing the integration over the magnitude of the quark momentum @xmath68 analytically , leads to the expression , @xmath69 \nonumber \\ & & \left\{-{\hat{q}_{+\alpha}\hat{q}_{+\beta}\hat{q}_{+\gamma } \over p\cdot\hat{q}_+ } \left({\vec{k}^2-(\vec{k}\cdot\hat{q})^2\over k\cdot\hat{q}_+ } + { \vec{k}{'^2 } -(\vec{k}'\cdot\hat{q})^2\over k'\cdot\hat{q}_+}\right ) - { k \hat{\mathcal{k}}_{\alpha}\hat{q}_{+\beta}\hat{q}_{+\gamma } - k'\hat{q}_{+\alpha}\hat{q}_{+\beta}\hat{\mathcal{k}'}_{\gamma } \over p\cdot\hat{q}_+}\left({\vec{k}\cdot\hat{q}\over k\cdot\hat{q}_+}-{\vec{k}'\cdot\hat{q}\over k'\cdot\hat{q}_+}\right ) \right . \nonumber \\ & & \left . \over p\cdot\hat{q}_+ } { [ \vec{k}^2-(\vec{k}\cdot\hat{q})^2 ] - [ \vec{k}{'^2}-(\vec{k}'\cdot\hat{q})^2]\over p\cdot\hat{q}_+}\left({\vec{k}\cdot\hat{q}\over k\cdot\hat{q}_+ } -{\vec{k}'\cdot\hat{q}\over k'\cdot\hat{q}_+ } \right ) \right . \nonumber \\ & & \left . - { \hat{q}_{+\alpha}\hat{q}_{+\beta}\hat{q}_{+\gamma } \over p\cdot\hat{q}_+ } \left({\vec{k}\cdot\hat{q}\over k\cdot\hat{q}_+ } { \vec{k}^2-(\vec{k}\cdot\hat{q})^2\over k\cdot\hat{q}_+ } + { \vec{k}'\cdot\hat{q}\over k'\cdot\hat{q}_+ } { \vec{k}{'^2}-(\vec{k}'\cdot\hat{q})^2\over k'\cdot\hat{q}_+ } \right ) + { 2\hat{q}_{+\alpha}\hat{q}_{+\beta}\hat{q}_{+\gamma } \over p\cdot\hat{q}_+ } \left({(\vec{k}\cdot\hat{q})^2\over k\cdot\hat{q}_+}+{(\vec{k}'\cdot\hat{q})^2\over k'\cdot\hat{q}_+ } \right ) \right . \nonumber \\ & & \left . - \hat{q}_{-\alpha}\hat{q}_{+\beta}\hat{q}_{+\gamma } { \vec{k}'\cdot\hat{q}\over k'\cdot{q}_+ } - \hat{q}_{+\alpha}\hat{q}_{-\beta}\hat{q}_{+\gamma } { \vec{p}\cdot\hat{q}\over p\cdot\hat{q}_+ } - \hat{q}_{+\alpha}\hat{q}_{+\beta}\hat{q}_{-\gamma } { \vec{k}\cdot\hat{q}\over k\cdot\hat{q}_+ } \right\},\end{aligned}\ ] ] where @xmath70 and @xmath71 is the differential solid angle of the quark momentum @xmath68 . the first line of the above equation , demonstrates explicitly that the amplitude is directly proportional to the chemical potential @xmath8 . as a result , the contribution to the photon production rate from this channel will grow quadratically with increasing chemical potential if the temperature of the medium is held constant . the remnant angular integral over @xmath72 is nontrivial and is performed numerically . the possibility of radiation or absorption of a space - like gluon by an on - shell quark induces a long distance enhancement in a small part of phase space in each of the two diagrams separately . such contributions are diminished by the destructive interference between the two diagrams and thus do not contribute to any eventual long distance enhancement in the rate of photon production from this channel . including all contributions , leads to the survival of only the imaginary part of the amplitude in this sector . the resulting expression is , @xmath73 \left(-{\mu\omega\over 8k}\right){i\pi } \nonumber\\ & & \left\ { -{1\over p\cdot\hat{\bar{q } } } \left ( k\hat{\mathcal{k}}_{\alpha } \hat{\bar{q}}_{+\beta } \hat{\bar{q}}_{+\gamma } - k'\hat{\bar{q}}_{+\alpha } \hat{\bar{q}}_{+\beta } \hat{\mathcal{k}'}_{\gamma } \right)-{[\vec{k}^2-(\vec{k}\cdot\hat{\bar{q}})^2 ] - [ \vec{k}{'^2}-(\vec{k}'\cdot\hat{\bar{q}})^2]\over ( p\cdot\hat{\bar{q}}_+)^2}\hat{\bar{q}}_{+\alpha}\hat{\bar{q}}_{+\beta } \hat{\bar{q}}_{+\gamma}\right.\nonumber\\ & & \left . + { k\over p\cdot\hat{\bar{q}}}\left(\hat{\mathcal{q}}_{\alpha } \hat{\bar{q}}_{+\beta } \hat{\bar{q}}_{+\gamma } + \hat{\bar{q}}_{+\alpha } \hat{\mathcal{q}}_{\beta } \hat{\bar{q}}_{+\gamma } + \hat{\bar{q}}_{+\alpha } \hat{\bar{q}}_{+\beta } \hat{\mathcal{q}}_{\gamma}\right ) - \hat{\bar{q}}_{+\alpha } \hat{\bar{q}}_{+\beta } \hat{\bar{q}}_{-\gamma } \right\},\end{aligned}\ ] ] where @xmath74 and @xmath75 are given by @xmath76 the above equations represent the central result of this effort . while this been derived in a thermalized environment , it may be easily generalized to moderate departures from equilibrium . this will remain the subject of a future effort , where the above matrix element will be used to study the photon signature emanating from the gluon sector of different models of the qgp . in the subsequent sections , the above amplitude will be applied to compute the photon production rate from gluon fusion and decay in the simplest model of a qgp , a plasma of quasi - particle quarks and gluons in complete thermal and chemical equilibrium . in the preceding sections , the origin and derivation of the amplitude of photon production from two gluons through a quark triangle has been outlined . such an amplitude may be used for different processes such as the fusion of gluons to form a photon or the decay of a gluon into a photon and a gluon of lower energy . the production or absorption rate of photons from a dense medium is related to the imaginary part of the photon self - energy in the medium . the choice of self - energy depends on the process of interest . in what follows , the focus will lie on the imaginary part of the photon self - energy of fig . [ photon - sf ] . in the case of a medium in complete thermal and chemical equilibrium , the thermal photon emission rate @xmath77 is related to the discontinuity or the imaginary part of the retarded photon self - energy @xmath78 at finite temperature @xmath7 through the relation @xcite , @xmath79 where @xmath80 and @xmath48 are the energy and momentum of the photons . this formula is valid to all orders of strong interactions , but only to @xmath81 in the electromagnetic interactions , as the photons , once produced , will tend to escape from the matter without further interaction . the photon self energy from fig . [ photon - sf ] may be expressed formally as , @xmath82 where @xmath83 is the effective photon - gluon - gluon vertex evaluated in the last section , and @xmath84 is the effective gluon propagator , after summing up all the htl contributions to the self - energy of the gluon . in the coulomb gauge , the propagator is given by @xcite @xmath85 in the above equation , @xmath86 , @xmath87 is the transverse projection tensor and @xmath88 specifies the rest frame of the medium ; the explicit expressions for @xmath89 and @xmath90 can be found in ref . it is convenient to define the transverse and longitudinal gluon propagators @xmath91 and @xmath92 . in the complex @xmath93 plane , these propagators exhibit a discontinuity or cut from @xmath94 to @xmath47 ; in addition , they have poles at @xmath95 , which give the two dispersion relations for the longitudinal and transverse modes of the gluons in the medium . in the interest of completeness , these are plotted in fig . [ dispersion - relation ] . for transverse gluons and longitudinal gluons in a quark gluon plasma , where @xmath96 is the thermal gluon mass.,width=302 ] in the plot , the upper branch is the dispersion relation for transverse excitation modes , and the lower branch is for the the longitudinal one . the solid line represents the light cone . the dispersion relations admit a thermal gluon mass at the intercept @xmath97 given as @xmath98 @xcite . in order to calculate the thermal photon differential production rate , we evaluate the imaginary part or the discontinuity of the photon self - energy , which involves evaluating its various cuts . in the interest of a physical interpretation of the various cuts , the polarization tensor @xmath99 may be expanded as a product of polarization vectors as @xmath100 , where the @xmath101-axis is chosen as the direction of the photon momentum @xmath102 . then the effective propagator may be formally written as @xmath103 where we set @xmath104 and @xmath105 , @xmath106 . the entire expression for the rate , may be expressed in a factorized form @xmath107 , where @xmath108 is the matsubara frequency of the gluon and @xmath109 is the frequency of the external photon . the remaining sum over @xmath108 and the discontinuity across the real @xmath109 is achieved by means of the identity , @xcite , @xmath110 where @xmath111 and @xmath112 are the distribution functions of gluons , and @xmath113 and @xmath114 are the spectral functions of @xmath115 and @xmath116 . the spectral function @xmath117 for @xmath118 is defined as @xmath119 . employing the above formula , the discontinuity in the photon self - energy may be expressed in a kinetic form , @xmath120 where the matrix element @xmath121 , and @xmath122 and @xmath123 are the spectral functions of the gluon propagators @xmath124 and @xmath125 . @xmath126 + \beta_{t , l}(k_0,k)\theta(k^2-k_0 ^ 2).\end{aligned}\ ] ] the spectral functions @xmath127 contain contributions from the poles @xmath128 with residue @xmath129 as well as from the branch cuts @xmath130 . the product of two @xmath127 functions give three types of contribution : pole - pole , pole - cut , and cut - cut . the pole - pole terms represent the process involving two quasi - particles with dispersion relations displayed in fig . [ dispersion - relation ] . the terms from the cuts represents the processes involving space - like gluons from the medium , _ i.e._. , gluons which are intermediate states of a scattering process . in this first effort , the focus will lie on the hard photon production rate , _ i.e. _ , photons with momenta @xmath131 . this requires that at least one of the gluons in fig . [ photon - sf ] to be hard . while in the usual htl prescription , such particles receive suppressed contributions from hard loops , a component of the htl self - energy which produces a thermal gluon mass is retained . the cut - cut contribution with two space - like gluons is dominant only in the region where both gluon momenta are soft and may be ignored in this effort . as a result , the two main contributions to the hard photon rate computed in this paper emanate form the pole - pole and pole - cut terms . in the subsequent section , numerical estimates of the photon rate from a set of production scenarios is provided and compared to the corresponding rate from the leading process of hard photon production from compton scattering and pair annihilation . in this section , numerical results for the hard photon production rate from a plasma with a finite charge density will be presented . the calculation is performed for two massless flavors of quarks with @xmath132 . in such a plasma , the strong coupling constant @xmath133 . the strange sector has been ignored in this calculation . in fig . [ t=200 ] , the photon production from our new channel is compared with the contribution from the leading order qcd processes of quark anti - quark annihilation and quark gluon compton scattering . vertex in a hot and dense medium with temperature @xmath134 compared with the contribution from qcd annihilation and compton processes.,width=377 ] the photon differential rate from qcd annihilation and compton processes at finite temperature and chemical potential is parameterized as in ref.@xcite by @xmath135 where the dimensionless quality @xmath136 is fitted to be @xmath137 for @xmath138 and @xmath139 for @xmath140 . one may immediately note from fig . [ t=200 ] that the contribution from the new channels presented in this paper , the @xmath141 vertex , to the photon production is much smaller than the qcd annihilation and compton processes at low chemical potential . however , with increasing chemical potential at a fixed temperature , the photon production rate from the @xmath141 vertex tends to increase at a swifter rate than qcd annihilation and compton contribution . this may be understood from the fact that the matrix element corresponding to the @xmath141 vertex in fig . [ gg - gamma ] is proportional to the chemical potential as we see from eq . [ amplitude ] . this leads to the conclusion that in baryon - rich matter such as that produced in low energy collisions of heavy ions or in the core of neutron stars , where the chemical potential of the medium is very large , the new channel from @xmath141 vertex will assume significance in comparison to the leading order rates . in the above estimates , the chemical potential and temperature are held fixed separately . in ref . @xcite , it is shown that the photon production rate from qcd annihilation and compton processes have a strong dependence on increasing chemical potential @xmath8 of the medium if the energy density of the medium is fixed . if the energy density were held constant , the temperature @xmath7 and @xmath8 are related to each other by the equation of state ( eos ) . in what follows , an estimate of hard photon production from a medium with fixed energy density is presented and compared with the leading order rates . the equation of state is derived from the phenomenological mit bag model @xcite , where the energy density is given as , @xmath142 in the above equation , the constants are given as : @xmath143 and the bag constant @xmath144 is fixed to be @xmath145 mev@xmath146 . if @xmath7 is made dependent of @xmath8 is this way , then both rates will decrease strongly with increasing chemical potential because of the decreasing of the temperature @xmath7 of the medium . such a drop is even more pronounced for the case of the leading order rates . at rhic , one expects a maximum energy density of about @xmath147 gev/@xmath3 , and the average energy density will be smaller than this value . we pick a conservative estimate of @xmath148 gev/@xmath3 , which corresponding to @xmath149 mev at zero chemical potential . the results of the leading order rate and that from the two gluon channel is presented in fig . [ epsilon=1.8 ] . vertex in a hot and dense medium with energy density fixed , @xmath150 , compared with the contribution from qcd annihilation and compton processes.,width=377 ] in the plot , we raise the chemical potential @xmath8 from @xmath151 to @xmath152 . as is clearly evident , the photon production rate from our new channel decreases with increasing chemical potential with fixed energy density , showing a similar dependence to the qcd annihilation and compton processes . it would appear that with energy density fixed , the photon production from our new channel has a much stronger dependence on the temperature than on the chemical potential and has not exceeded the photon production rate from qcd annihilation and compton processes in the range of energies explored . it should be recalled that the above statement is true of only the hard photon production rate and may not hold in an extension to soft photons where cut - cut contributions will contribute . this is left as the subject for a future effort . a determination of the degrees of freedom prevalent in the early dense matter created in a heavy - ion collision remains an outstanding question in the study of excited strongly interacting matter . while lattice susceptibilities allow for constraints on the flavor sector of the produced matter , phenomenological explorations remain the sole method to determine the structure of the gluon sector . as a contribution to this on - going effort we have presented the hard photon signature of the gluon sector . while the gluonic sector may not itself consist of quasi - particle gluons , the production of hard photons through the gluon fusion channel outlined in this paper , similar to the production of large mass drell - yan pairs , will be sensitive to the gluon structure functions of such matter . the matrix element for the conversion of a gluon pair into a photon is vanishing in the vacuum and is non - zero solely in the presence of a charge density in the medium . the amplitude for the production of a photon from two gluons in a thermal environment at finite chemical potential was presented in sec . this was incorporated into the photon self - energy in a thermalized plasma in sec . expressions for the photon production rate from the fusion or decay of in - medium gluons in such a scenario were derived subsequently . phenomenological estimates of the photon production rate from such equilibrium channels have tended to be suppressed compared to the leading order rates for realistic values of temperature and chemical potential . such estimates however do not constrain the photon production rate from jet plasma interaction channels @xcite . neither do they limit the rates of photons produced form such channels in non - equilibrium scenarios , such as in the early plasma where the gluon populations far exceed those of the quarks . the results of sec . iii , which have been cast in the form of a kinetic theory , may be easily extended away from the equilibrium scenarios where they have been derived and applied to the above mentioned situations . yet another application of such rates is to photon production in neutron stars where the chemical potential far exceeds the temperature and many of the conventional channels are pauli blocked . estimates of the photon production rate from two gluons in such diverse scenarios will be presented in upcoming efforts . this work is supported in part by the natural sciences and engineering research council of canada , and by the u.s . department of energy under grant no . de - fg02 - 05er41367 and under contract no . de - ac03 - 76sf00098 .

a new channel of direct photon production from a quark gluon plasma ( qgp ) is explored . this process appears at next - to - leading - order in the presence of a charge asymmetry in the heated matter and may be effectively described as the bremsstrahlung of a real photon from a thermal gluon . the photon production from this new mechanism is calculated in the effective theory of qcd at high temperature . the results show that the photon production rate may not as big as the annihilation and compton scattering at low baryon density , but could become important in baryon - rich matter .

You are an expert at summarizing long articles. Proceed to summarize the following text: recently fully microscopic calculations of nuclei with @xmath4 have become feasible @xcite . the @xmath0be nucleus is such a system of special interest , as it allows tests of theories of interaction of composite particles @xcite . the properties of low energy continuum of @xmath0be are of particular importance in this connection . however , the corresponding experimental data on the low energy photodisintegration of @xmath0be are not in mutual agreement ( see fig . 1 ) . in the present work we develop a semi microscopic model to describe the process , and we analyze the experimental data with its help . the model accounts simultaneously for both resonant and non resonant contributions to the cross section . an estimation of the reliability of various data is obtained and a theoretical photodisintegration cross section is derived . we also calculate the reaction rates of the reaction @xmath0be@xmath5 and the reverse reaction for astrophysical conditions . these reaction rates are of relevance in the high entropy bubble in type ii supernovae , an astrophysical site that has been suggested for the r process @xcite . the baryonic matter in this bubble is dominated in the beginning by @xmath6particles , neutrons , and protons . the abundance distribution shifts then to higher masses through the recombination of the free @xmath6particles , neutrons , and protons . this generates the so called @xmath6process leading to the formation of massive isotopes ( @xmath7 ) . the reaction path in the @xmath6process is mainly determined by requirements of nuclear statistical equilibrium and depends also on the reaction rates of the various recombination paths bridging the mass 5 and 8 gaps . it has been shown that there are three principal reaction paths from @xmath8he to @xmath9c : + ( i ) @xmath8he(2@xmath6,@xmath10)@xmath9c + ( ii ) @xmath8he(@xmath6 n,@xmath10)@xmath0be(@xmath6,n)@xmath9c + ( iii ) @xmath8he(2n,@xmath10)@xmath11he(@xmath6,n)@xmath0be(@xmath6,n)@xmath9c . it was shown in refs . @xcite that the triple alpha process ( i ) can be neglected compared to the reaction sequence ( ii ) via @xmath0be under r process conditions in the @xmath6process . also the reaction path ( iii ) via @xmath11he can be neglected for this scenario @xcite . this is true even if the reaction rate of @xmath8he(2n,@xmath10)@xmath11he is strongly enhanced @xcite , because then @xmath11he is also destroyed very effectively through photodissociation . therefore , for the @xmath6 and r process the reaction @xmath8he(@xmath6 n,@xmath10)@xmath0be plays a key role in bridging the unstable mass gaps at @xmath12 and @xmath13 . the reaction rates of @xmath8he(@xmath6 n,@xmath10)@xmath0be and the reverse photodisintegration of @xmath0be were estimated in ref . @xcite from the experimental photodisintegration cross section . however , ref . @xcite did not include information on which experimental data their estimate was based . in view of the astrophysical relevance of these reactions we recalculate in the present work the rates of the first step of the reaction ( ii ) above . the same problem is also addressed in ref . these authors obtain the resonant contribution to the @xmath0be@xmath14be cross section from the breit wigner formula for the first excited state of @xmath0be with the parameters taken from ref . @xcite . and @xmath15 parameters of the resonance used in ref . @xcite seem to be incorrect . the resonant properties of the @xmath16 state of @xmath0be will be considered in our future work . ] in order to calculate the non resonant contribution they introduce a single particle potential with the depth chosen to reproduce the ground state , calculate both ground and final state continuum wave functions in this potential , and multiply the cross section obtained by the shell model spectroscopic factor . they then add this cross section constructively or destructively to the resonant cross section to establish possible upper and lower bounds for the reaction rates . this procedure has certain shortcomings : a resonant contribution to the cross section should not emerge as an addition to the dynamic model used , since a correct quantum mechanical model should necessarily contain such a contribution itself , along with the non resonant contribution and an interference term . besides , the potential wells used for the ground state and continuum state should in fact be different : an additional spin orbit potential , for example , should be present in the ground p state as compared to the continuum s state . in our model we use a three - body specification of the @xmath0be bound state , and a semimicroscopic continuum wave function which describes the essential scattering degrees of freedom at low relative energies . in sect . 2 this model is formulated . in sect . 3 the results for the @xmath0be@xmath17 cross section are given . in sect . 4 the astrophysical rates for @xmath3be@xmath2 and the reverse reaction are calculated . the relevant experimental data on the low energy @xmath0be@xmath17 cross section are presented in fig . 1 . the available data are those in @xcite obtained with bremsstrahlung photons and those in @xcite obtained from @xmath10radiation from radioactive isotopes . the peak at very low energy exhibited by the data of ref . @xcite is not confirmed by other groups , and may arise from discrepancies caused by neutron energy loss in the targets @xcite . the radioactive isotope techniques normally provide more reliable results due to the absence of difficulties with the energy resolution . however , the cross section can be determined only for a few discrete photon energies with this method . this drawback will be cured below by the use of an appropriate theoretical model . our strategy will be thus to analyze the radioactive isotope data . we shall consider the range of energies up to 0.5mev above threshold . we need to obtain the ground state and continuum wave functions ( wf ) and calculate the transition matrix element . we start with the three body @xmath18 representation of the @xmath0be system . within this representation the wf in the c.m . system is the @xmath19 relative motion function times the intrinsic wfs of the two @xmath6particles . since a predominant contribution to the transition matrix element comes from distances large compared to the size of the @xmath6particle additional antisymmetrization may be disregarded . system @xcite , for example . ] the intrinsic @xmath6 particle wfs then will drop out from the calculation . in the following we shall refer to the three particle relative motion function as to the wf of the system . let us denote and @xmath20 the distance between the @xmath6 particles and that from their center of mass to the neutron , respectively . the ground state is @xmath21 , and its wf is of the form @xmath22_l\chi_{1/2}]_j . \label{eq : gswf}\ ] ] here @xmath23 is the neutron spin function , and the brackets @xmath24 $ ] stand for angular momentum coupling . because of the pauli principle and parity requirements @xmath25 is even , and @xmath26 is odd . the wf in eq . ( [ eq : gswf ] ) was obtained from the three body schrdinger equation with @xmath27 and @xmath28 potentials reproducing the observed two body phase shifts . these potentials , along with some details of the calculation , are listed in the appendix . practically an exact solution to the @xmath18 bound state problem is obtained , but one can not get the experimental binding energy with these `` bare '' interparticle interactions . a possible reason is that the two @xmath6particles may distort each other in the @xmath18 bound state compared to the pure @xmath29 case . this may lead to a change in the @xmath29 interaction . to obtain a reasonable ground state wf the strength of the attractive central component of the @xmath29 potential is reduced by 8% in our calculation . this leads to values of 1.50mev and 2.48fm for the binding energy and charge radius of @xmath0be , sufficiently close to the experimental values of 1.5736mev and ( @xmath30)fm . coming to the continuum wave function , we note first that the @xmath31be resonance produced in the reaction may be safely treated as a stable particle for the purposes of our calculation . this is because its width of 7kev is extremely narrow on a nuclear scale . the @xmath27 continuum wave function , taken at the resonance energy and normalized to unity in the interior region , decreases practically to zero in the coulomb barrier region as shown in fig . 2 ( solid line ) . this function represents the wf of the resonance extremely well and is taken as the @xmath31be `` bound state '' wave function . second , we argue that photodisintegration of @xmath0be proceeds entirely into the @xmath31be@xmath32 channel . indeed , one can estimate that , at small energies considered , the three fragment @xmath18 disintegration channel is strongly suppressed due to the threshold regime . the experimental data also strongly supports the absence of this channel @xcite . at the same time , fragment @xmath8he@xmath33he channel is still closed and ineffective due to the broad width of @xmath34he . thus our cross section starts from the @xmath31be@xmath32 threshold of 1.6654mev . the cross section in the region between this threshold and the @xmath19 threshold of 1.5736mev is known to be tiny @xcite and will be disregarded . as in the previous work ( e.g. @xcite ) we confine ourselves to an s wave relative @xmath35be motion , i.e. , with @xmath16 continuum states . predictions of the above dynamic @xmath19 model for the photodisintegration of @xmath0be depend crucially on the position of the excited state of @xmath0be with respect to the threshold . preliminary three body calculations gave us a peak lying too high in energy , and too broad . this could not be used for a reasonable fit to the data , so in the following we shall formulate an alternative representation of the continuum wf . we shall seek it in the form @xmath36 here @xmath37 is the intrinsic wave function of @xmath31be calculated with the same @xmath27 potential as for the ground state , and @xmath38 is the @xmath35be relative motion function , where for large @xmath39 the normalization @xmath40 is used . generally speaking , the true continuum wave function differs from eq . ( [ eq : cwf ] ) not only in the @xmath35be interaction region but also in the outer region . however , at energies in the vicinity of the long living excited state of @xmath0be the representation ( [ eq : cwf ] ) should approximately be valid in the outer region . indeed , the decay of the long living state into the three body @xmath19 channel is inhibited due to the threshold regime . due to the approximate validity of the wf in eq . ( [ eq : cwf ] ) in the outer region one obtains a correct energy dependence of the cross section when using this wf . for small energies the main energy dependence of the transition matrix element appears as a factor in the continuum wf and is determined by an outer part of the wf , i.e. the phase shift . besides , one can see below that just the outer region ( where @xmath38 takes the form of eq . ( [ eq : sin ] ) ) gives the biggest contribution to the transition matrix element . one can therefore hope that a wf of the form of eq . ( [ eq : cwf ] ) suffices for the fitting purpose in the whole energy range considered . we seek the relative motion function @xmath38 as a solution to the relative motion schrdinger equation with some potential whose parameters are chosen from a fit of the theoretical cross section to the data . taking into account that the s wave @xmath28 repulsion and the p wave @xmath28 attraction have comparable ranges one can assume a smooth attractive potential . the woods saxon family @xmath41 will be adopted below as a good representative . consider now the representation of the photodisintegration cross section in our model . the matrix element of the dipole transition operator @xmath42 between the wfs in eq . ( [ eq : gswf ] ) and eq . ( [ eq : cwf ] ) has to be calculated . after integrating over @xmath43 the matrix element reduces to the overlap between the @xmath35be relative motion functions , namely the scattering function entering eq . ( [ eq : cwf ] ) and the `` effective bound state wf '' @xmath44_{j=3/2}\ ] ] obtained from eq . ( [ eq : gswf ] ) : @xmath45 using eqs . ( [ eq : cwf ] ) , ( [ eq : eff ] ) the cross section is of the same form as in the single particle case . the total ( s wave ) photodisintegration cross section is calculated in the simple form @xmath46 here @xmath47 is the @xmath35be reduced mass , and @xmath48 is the excitation energy @xmath49 that will be denoted as @xmath50 below . in case of a single particle description of the process , i.e. , the `` valence neutron '' model , eq . ( [ eq : crs ] ) is valid with the bound state wf normalized to unity , while in our case ( see below ) @xmath51 it is implied here and in eq . ( [ eq : crs ] ) that the @xmath31be and @xmath0be ground state wave functions are normalized to unity . the function @xmath52 is shown in fig . 2 ( dashed line ) . in ref . @xcite the data at the energies up to 185kev above threshold ( @xmath53 mev ) from refs . @xcite were reproduced at a qualitative level within the following framework . the valence neutron model with woods saxon type potentials was used . the results obtained in this way were multiplied by some constant factors less than unity , so called `` reduction factors '' , to approach the experimental cross section . to obtain the continuum wave function the depth of the potential was varied while the radius and diffuseness parameters were taken the same as for the central component of the potential in the bound state calculation . two fits were found , one with a reduction factor of 0.53 leading to a weakly bound @xmath31be@xmath32 @xmath16 state , and the other one with a reduction factor of 0.31 leading to a virtual state . the first possibility was once preferred in view of the results of ref . @xcite . in that work a two channel @xmath31be@xmath32 model of the ground state of @xmath0be was introduced to cure the single particle description @xcite of photodisintegration . the @xmath31be subsystem was allowed to be in the ground and first excited state , and that led to the reduction factor of 0.5 or 0.6 in the cross section depending on the assumptions . in ref . @xcite the reduction factor of 0.56 was found for that model . in contrast to ref . @xcite our results below definitely testify to a virtual @xmath31be@xmath32 @xmath16 state . this is probably due to a more realistic treatment of the ground state of @xmath0be in our model . in fact the reduction factors obtained in the two channel model of the ground state of @xmath0be @xcite should be used in conjunction with the channel coupling n@xmath54be dynamics , instead of using @xcite single channel dynamics . in ref . @xcite the same data were fitted with the line shape @xmath55 . this shape was derived incorrectly from the breit wigner cross section under the assumption that the @xmath16 level of @xmath0be is a bound or virtual @xmath31be@xmath32 state . in ref . @xcite the data of ref . @xcite were fitted with a one level r matrix approximation . the fit leads to a complex energy resonant state @xcite and the real part of the complex energy proves to be negative . first we shall analyze the data of refs . @xcite ( full circles , full squares , and full diamond in fig . 1 ) . a search of the parameters @xmath56 , @xmath39 , and @xmath57 of the potential ( [ eq : ws ] ) giving an acceptable fit to the data is performed . several local minima of the quantity @xmath58 in the space of the parameters are found . one of them is provided by @xmath59 these values seem to be very reasonable . for this set the @xmath60(degrees of freedom ) value equals 0.62 . another one is obtained with the parameters @xmath61 giving @xmath62 . several other minima also exist with sizable higher but still acceptable @xmath63 values . in fig . 3 the photodisintegration cross sections obtained with the parameters ( [ eq:1 ] ) and ( [ eq:2 ] ) are shown as the solid and dashed curve , respectively . the two cross sections prove to be quite close to each other . to clarify partially the reason for this we note that the biggest contribution to the matrix element of eq . ( [ eq : int ] ) comes from the distances beyond the range of the potential . the distances larger than 5 fm in eq . ( [ eq : int ] ) provide 60 - 70% contribution to the cross section , and at such distances the wave functions deviate from the asymptotic ones , eq . ( [ eq : sin ] ) , by not more than 10% ( except for regions in the vicinity of zeros ) . the asymptotic wave functions are determined by the phase shifts i.e. , predominantly by the scattering length @xmath57 and the effective range @xmath64 . therefore the procedure is equivalent to some degree to fitting @xmath57 and @xmath64 values . once this is done , the cross sections are not very dependent on the particular set of the potential parameters . the @xmath57 and @xmath64 values are @xmath65fm and @xmath66fm , respectively , for the set ( [ eq:1 ] ) , and @xmath67fm and @xmath68fm for the set ( [ eq:2 ] ) . all the other above mentioned sets of potential parameters giving local minima to @xmath63 lead to very similar @xmath57 and @xmath64 values . however considerable changes in the scattering wf inside the potential can influence the results , see the next paragraph . the following way to interpolate between the data has also been tried . let us denote by @xmath69 the cross section obtained in case when the right hand side of eq . ( [ eq : sin ] ) is used as a continuum wave function for all @xmath39 values . this cross section has been calculated taking @xmath70 from the effective range expansion with the @xmath57 and @xmath64 values given by the potential ( [ eq:1 ] ) . let us represent @xmath71 as @xmath72 and fit @xmath73 to experiment . it is hoped that , in contrast to @xmath71 , the factor @xmath73 behaves in a smooth way and thus can be reliably obtained from an interpolation procedure . indeed , the behavior of both @xmath71 and @xmath74 can be approximately described by the resonant factor @xmath75 times a slowly varying function . even a fit with @xmath76 proves to provide a sufficiently low @xmath63 value . the cross section obtained with this @xmath77 is shown in fig . 3 as the dotted curve . presumably this procedure provides less accurate results than the previous one . of course , the energy dependence @xmath75 is not accurate enough in the whole energy range , as a comparison with the exact solution for the potential ( [ eq:1 ] ) shows . hence @xmath73 should include an energy dependence , but this could not be determined because of experimental uncertainties . next we applied the procedure to the data of ref . @xcite ( open circles in fig . 1 ) . three local minima with acceptable @xmath63 values are found . however the parameters of the potential corresponding to all of them : @xmath78 prove to be rather unrealistic . there exists one more difference between these potentials and those in eqs . ( [ eq:1 ] ) and ( [ eq:2 ] ) . the latter potentials , as well as the other potentials ( [ eq : ws ] ) providing a good fit to the same data , support one deeply bound state and one state close to being bound . on the contrary , the potentials ( [ fu ] ) support only one very weakly bound state . an existence of one deeply bound @xmath79 level in the neutron mean field in the @xmath0be nucleus , or , equivalently , one node in the low energy scattering wave function inside the potential , seems to be natural from the shell model point of view . we think this point of view is sufficient to establish the correct number of nodes for the neutron motion inside the woods saxon potential , even for such a clusterized system . in the @xmath6particle oscillator model of @xmath0be @xcite , for example , the first allowed neutron @xmath79state contains a substantial admixture of the nodeless @xmath80 function , but this leads not to a disappearance but only to a shift of the node . if one admits that the state considered is a mixture of @xmath80 and @xmath81 oscillator functions then there exists just one node located within the distance of 3 fm from the origin . therefore we conclude that in the region where various data sets differ from each other the older radioactive isotope data are preferable . we also note that the cross section we obtain with potentials ( [ eq:1 ] ) and ( [ eq:2 ] ) for the highest energies considered , being lower than the fitted datum of ref . @xcite , agrees well with the bremsstrahlung jacobson data @xcite . the @xmath0be@xmath82 reaction rate per nucleus per time unit is calculated via the usual averaging the elementary photodisintegration cross section @xmath83 with the approximate , or wien distribution for the photon density , @xmath84 where @xmath85 mev . the rate of the reverse reaction ( the number of reactions per time unit per unit volume ) is @xmath86 where @xmath87 and @xmath88 are numbers of particles per unit volume . the reaction constant @xmath89 is obtained from eq . ( [ lam ] ) using the reverse ratio @xmath90 @xcite : @xmath91 here @xmath92 is avogadro s number , @xmath93 is the temperature in @xmath94 k , @xmath95 with @xmath96 mev , and it is implied that the quantities ( [ lam ] ) and ( [ 3 ] ) are given in sec@xmath97 and cm@xmath11 sec@xmath97 mole@xmath99 , respectively . use of the wien distribution instead of the exact , or planck , one , i.e. @xmath100^{-1}\rightarrow\exp(-e_\gamma / kt)$ ] , allows application of the above listed simple reverse ratio theory . for temperatures of @xmath101 and 10 , for example , it gives the reaction constant ( [ lam ] ) with relative errors of 1% and 5.4% , respectively . for @xmath102mev the cross section @xmath103 obtained in the preceding section with the potential ( [ eq:1 ] ) is used . for @xmath104 from 2.2mev up to 5mev the jacobson bremsstrahlung data @xcite are used . the former energy region provides 96% and 62% contribution to the cross section for @xmath105 and 5 , respectively . the contribution from energies @xmath104 higher than 5mev reaches 0.3% and 6.7% for @xmath101 and 8 , respectively . the values of the rate ( [ 3 ] ) obtained can be represented by the fit @xmath106 \left(1+\sum_{n=1}^7a_nt_9^n\right)^{-1 } \label{fit}\ ] ] with @xmath107 the fit reproduces our @xmath108 values with the accuracy better than 1% at any @xmath93 in the range @xmath109 . in eq . ( [ fit ] ) @xmath110/kt$ ] . the factor @xmath111 $ ] represents the asymptotic behavior of the @xmath31be formation contribution to the rate when @xmath93 tends to zero . in the table our values for the three body reaction rate ( [ 3 ] ) are compared with those of ref . @xcite and those of ref . @xcite where constructive or destructive interference between the resonant and non resonant contributions at energies above the resonance energy was assumed . summarizing , we have constructed a semi microscopic model for the low energy photodisintegration of the @xmath0be nucleus and have analyzed the experimental data with its help . our analysis supports the older radioactive isotope data . the theoretical cross section we derived may be compared with future microscopic calculations of the process . we have calculated the astrophysical rates for the reaction @xmath3be@xmath2 and the reverse reaction . our new reaction rates agree at @xmath112 with the ones given in ref . they are somewhat smaller ( larger ) for lower ( higher ) temperatures than @xmath112 . the reaction rates given in ref . @xcite agree much better with our reaction rate at higher temperatures if one assumes in ref . @xcite constructive ( destructive ) interference between the resonant and non resonant contributions at energies above ( below ) the resonance energy . we are indebted to j.s . vaagen and j.m . bang for very fruitful comments . this work was supported partially by the the fonds zur frderung wissenschaftlichen forschung in sterreich ( project p10361phy ) and the russian foundation for basic research ( grant no 97 - 02 - 17003 ) . in our calculation of the ground state of @xmath0be the @xmath27 potential is taken in the form @xcite @xmath113-v_a\exp[-(\mu_a\rho)^2]\ ] ] with @xmath114mev , @xmath115fm@xmath97 , @xmath117fm@xmath97 , and @xmath118 , 320 , and 10mev for @xmath119=0 , 2 , and 4 , respectively . the coulomb @xmath27 interaction is also added . the @xmath29 interaction in s , p , and d states is taken into account . as in many previous studies @xcite the s wave repulsive potential @xmath120 $ ] with @xmath121mev and @xmath122fm is used . the initial potential in p and d states @xcite includes central and spin orbit components : @xmath123 with @xmath124mev , @xmath125fm , @xmath126fm , @xmath127mev@xmath128 , @xmath129fm , and @xmath1300.35fm . the parameter @xmath131 is reduced to 39.6mev in the present three body calculation , in order to reproduce the empirical g.s . energy . the three body dynamic equation is written in the form of the faddeev differential equations and each faddeev component is expanded over hyperspherical harmonics and hyperradial basis functions . using the raynal revai rotations of hyperspherical harmonics the matrix elements are reduced analytically to two dimensional integrals . the equations are projected onto subspaces of the basis functions retained that reduces the problem to the algebraic eigenvalue problem . the number of basis functions retained is quite high and ensures the adequate convergence of the calculation . pudliner , v.r . pandharipande , j. carlson , s.c . pieper , and r.b . wiringa , phys . c*56 * , 1720 ( 1997 ) . k. arai , y. ogawa , y. suzuki , and k. varga , phys . c*54 * , 132 ( 1996 ) . woosely , j.r . wilson , g.j . mathews , r.d . hoffmann , and b.s . meyer , astrophys . j. * 433 * , 229 ( 1994 ) . k. takahashi , j. witti , and h.t . janka , astron . and astrophys . * 286 * , 857 ( 1994 ) . s.e . woosely and r.d . hoffman , astrophys . j. * 395 * , 202 ( 1992 ) . j. grres , h. herndl , i.j . thompson , and m. wiescher , phys . c * 52 * , 2231 ( 1995 ) . efros , w. balogh , h. herndl , r. hofinger , and h. oberhummer , z. phys . a * 355 * , 101 ( 1996 ) . h. herndl , r. hofinger , and h. oberhummer , in editors : s. kubono and t. kajino , proceedings of the symposium on origin of matter and evolution of galaxies 97 ( omeg97 ) , nov . 5 - 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a semi microscopic model for the low energy photodisintegration of the @xmath0be nucleus is constructed , and the experimental data are analyzed with its help . the older radioactive isotope data are supported by this analysis . the theoretical photodisintegration cross section is derived . the astrophysical rates for the reaction @xmath1be@xmath2 and the reverse photodisintegration of @xmath0be are calculated . the new reaction rate for @xmath3be@xmath2 is compared with previous estimations .

You are an expert at summarizing long articles. Proceed to summarize the following text: cosmic strings are one dimensional topological defects that may have formed if the vacuum underwent a phase transition at very early times breaking a local @xmath0 symmetry @xcite . the resulting network of strings is of cosmological interest if the strings have a large enough mass per unit length , @xmath1 . if @xmath2 , where @xmath3 is newton s constant and @xmath4 is the speed of light ( i.e. @xmath5g / cm ) then cosmic strings may be massive enough to have provided the density perturbations necessary to produce the large scale structure we observe in the universe today and could explain the pattern of anisotropies observed in the cosmic microwave background @xcite . the main constraints on @xmath1 come from observational bounds on the amount of gravitational background radiation emitted by cosmic string loops ( @xmath6@xcite and references therein ) . a loop of cosmic string is formed when two sections of a long string ( a string with length greater than the horizon length ) meet and intercommute . once formed , loops begin to oscillate under their own tension , undergoing a process of self - intersection ( fragmentation ) and eventually creating a family of non - self - intersecting oscillating loops . the gravitational radiation emitted by each loop as it oscillates contributes to the total background gravitational radiation . in a pair of papers , we introduced and tested a new method for calculating the rates at which energy and momentum are radiated by cosmic strings @xcite . our investigation found that many of the published radiation rates were numerically inaccurate ( typically too low by a factor of two ) . remarkably , we also found a lower bound ( in the center - of - mass frame ) for the rate of gravitational radiation from a cosmic string loop @xcite . our method involved the use of piecewise linear cosmic strings . in this paper we wish to provide greater insight into the behaviour of such loops and , in particular , how they approximate smooth loops by examining the waveforms of the gravitational waveforms of such loops . it has long been known @xcite that the first generation of ground - based interferometric gravitational - wave detectors ( for example , ligo - i ) will not be able to detect the gravitational - wave stochastic background produced by a network of cosmic strings in the universe . the amplitude of this background is too weak to be detectable , except by a future generation of more advanced instruments . however , a recent paper by damour and vilenkin @xcite has shown that the non - gaussian bursts of radiation produced by cusps on the closest loops of strings would be a detectable ligo - i source . while the specific examples studied here do not include these types of cusps the general method developed can be applied to such loops . our space - time conventions follow those of misner , thorne and wheeler @xcite so that @xmath7 . we also set @xmath8 , but we leave @xmath3 explicit . in the center - of - mass frame , a cosmic string loop is specified by the 3-vector position @xmath9 of the string as a function of two variables : time @xmath10 and a space - like parameter @xmath11 that runs from @xmath12 to @xmath13 . ( the total energy of the loop is @xmath14 . ) when the gravitational back - reaction is neglected , ( a good approximation if @xmath15 ) , the string loop satisfies equations of motion whose most general solution in the center - of - mass frame is @xmath16 . \label{x}\ ] ] where @xmath17 here @xmath18 and @xmath19 are a pair of periodic functions , satisfying the gauge condition " @xmath20 , where @xmath21 denotes differentiation with respect to the function s argument . because the functions @xmath22 and @xmath23 are periodic in their arguments , the string loop is periodic in time . the period of the loop is @xmath24 since @xmath25={1\over 2}[{\bf a}(t+\sigma)+ { \bf b}(t-\sigma)]= { \bf x}(t,\sigma ) . \label{periodicity}\ ] ] with our choice of coordinates and gauge , the energy - momentum tensor @xmath26 for the string loop is given by @xmath27 where @xmath28 is defined by @xmath29 with @xmath30 . in terms of @xmath22 and @xmath23 , @xmath31 , \qquad g^{ij } = { \textstyle{1 \over 4 } } [ a'_i b'_j + a'_j b'_i ] , \ ] ] and the trace is @xmath32 .\ ] ] alternatively we may introduce the four - vectors @xmath33 and @xmath34 so that @xmath35 the `` gauge conditions '' are satisfied if and only if @xmath36 and @xmath37 are null vectors . as a consequence of the time periodicity of the loop the stress tensor can be expressed as a fourier series @xmath38 where @xmath39 and @xmath40 the retarded solution for the linear metric perturbation due to this source in harmonic gauge is @xcite @xmath41 \ , e^{i\omega_n ( t - |{\bf x } - { \bf y}| ) } .\ ] ] far from the string loop center - of - mass the dominant behavior is that of an outgoing spherical wave given by @xmath42 \ , e^{i \omega_n \hat { \bf \omega}{\cdot } { \bf y } } , \ ] ] where @xmath43 and @xmath44 is a unit vector pointing away from the source . inserting eq . ( [ tmunu_tilde ] ) into eq . ( [ far_field_metric ] ) we find the field far from a cosmic string loop is @xmath45 \ , e^ { - i \omega_n\bigl [ { 1 \over 2}(u+v ) - \hat { \bf \omega}{\cdot } { \bf x}(u , v)\bigr ] } .\ ] ] the @xmath46 term in this sum corresponds to the static field @xmath47 , \ ] ] @xmath48 as appropriate to a object with mass @xmath49 as may be seen by comparison with the schwarzschild metric in isotropic coordinates ( see , for example , eq . ( 31.22 ) of ref . we denote the radiative part of the field by @xmath50 we may rewrite eq . ( [ far_field_string_metric ] ) as @xmath51 where @xmath52 is a null vector in the direction of propagation and @xmath53 \ , e^ { i{1 \over 2 } \omega_n\bigl[k_\mu a^\mu(u ) + k_\mu b^\mu(v)\bigr]}\ ] ] are polarization tensors . from eq . ( [ gdef ] ) , it is clear that the polarization tensors may be written in terms of the fundamental integrals @xmath54 and @xmath55 in terms of these integrals @xmath56\ ] ] @xmath57\ ] ] @xmath58 + \delta_{ij } \left[i_0 j_0 - { \bf i}{\cdot}{\bf j } \right ] \right\ } , \ ] ] where we have dropped the superscript @xmath59 for clarity . the harmonic gauge condition requires that the polarization tensors satisfy @xmath60 . this is easily verified by noting that @xmath61 and @xmath62 . these equations follow from the identity @xmath63 which is a consequence of periodicity , and the corresponding equation for @xmath64 . the harmonic gauge condition does not determine the gauge completely and we are left with the freedom to make transformations of the form @xmath65 if we make the choice @xmath66\ ] ] and @xmath67 \omega_i + 2 \left [ i_0 j_i + j_0 i_i \right ] \right\}\ ] ] then @xmath68 the spatial components are given by @xmath69 + \delta_{ij } [ i_0 j_0 & - & { \bf i}{\cdot}{\bf j } ] + \omega_i \omega_j [ i_0 j_0 + { \bf i}{\cdot}{\bf j } ] \nonumber \\ & + & i_0[j_i \omega_j + \omega_i j_j ] + j_0 [ i_i \omega_j + \omega_i i_j ] \bigr\ } , \end{aligned}\ ] ] these satisfy the gauge conditions @xmath70 and @xmath71 if we perform a spatial rotation to coordinates @xmath72 where @xmath73 points along the @xmath74-axis then we can write @xmath75 where @xmath76\ ] ] and @xmath77 , \ ] ] define two modes of linear polarization . in terms of the original basis we can write @xmath78 and @xmath79 [ es ] with @xmath80 [ as ] where @xmath81 , @xmath82 and @xmath83 are the euler angles defining the orientation of the frame @xmath72 relative to the original frame ( our conventions follow those of ref . the corresponding linearly polarized waveforms are then defined by @xmath84 recall that @xmath85 is obtained from the full metric perturbation @xmath86 by dropping the @xmath46 term which corresponds to the static ( non - radiative ) part of the field . the power emitted to infinity per solid angle may be written as @xmath87 for convenience we shall now set the length of the loop @xmath88 , and take @xmath89 . these are loops for which the functions @xmath90 and @xmath91 are piecewise linear functions . the functions @xmath90 and @xmath92 may be pictured as a pair of closed loops which consist of joined straight segments . the segments join together at _ kinks _ where @xmath93 and @xmath94 are discontinuous . following the notation of ref . @xcite we take the @xmath95- and @xmath96-loops to have @xmath97 and @xmath98 linear segments , respectively . the coordinate @xmath99 on the @xmath95-loop is chosen to take the value zero at one of the kinks and increases along the loop . the kinks are labeled by the index @xmath100 where @xmath101 . the value of @xmath99 at the @xmath100th kink is denoted by @xmath102 and without loss of generality we set @xmath103 . the segments on the loop are also labeled by @xmath100 , with the @xmath100th segment being the one lying between the @xmath100th and @xmath104th kink . the kink at @xmath105 is the same as the first kink at @xmath106 but , even though @xmath107 and @xmath108 are at the same position on the loop , @xmath103 while @xmath109 . the loop is extended to all values of @xmath99 by periodicity ( with period 1 ) . we denote @xmath110 , and the constant unit vector tangent to the @xmath100th segment by @xmath111 . then we have @xmath112 , \ ] ] and for consistency @xmath113 we have corresponding definitions for the @xmath23-loop and we follow the convention of ref . @xcite by labeling the kinks by the index @xmath114 . it is now elementary to calculate that , for @xmath115 , @xmath116 with a similar equation for @xmath117 . if we insert these expressions into eq . ( [ as ] ) and then into eq . ( [ waveformsum ] ) the sum over @xmath118 for @xmath119 consists of terms of the form @xmath120}\ ] ] which may be performed exactly using the identity @xmath121 this function is extended to other values by periodicity , for example , for @xmath122 we merely replace @xmath123 by @xmath124 in eq . ( [ identity ] ) . such transformations leave the coefficient of @xmath125 unchanged and can only change the coefficient of @xmath123 by a multiple of 2 . as a result when the sum in eq . ( [ piecewise_linear_i ] ) is performed for the coefficient of @xmath125 the sum telescopes and gives zero . thus , _ the waveform of a piecewise linear loop will be a piecewise linear function_. in addition , considering the coefficient of @xmath123 all slopes of the waveform must be a multiple of some fundamental slope . the slope only changes when a ( 4-dimensional ) kink crosses the past light cone of the observer at @xmath126 . these properties are illustrated in the examples below . as our first set of loops we study the loops considered by garfinkle and vachaspati @xcite . the vectors @xmath90 and @xmath91 lie in a plane and make a constant angle @xmath127 with each other where @xmath128 . to be specific , we may take @xmath90 and @xmath91 to be given by @xmath129 @xmath130 it is then straightforward to calculate that , for @xmath115 , @xmath131 and correspondingly @xmath132 [ a_gv ] as described above , the sum over @xmath118 in eq . ( [ waveformsum ] ) may be performed explicitly to yield a piecewise linear function . for example , @xmath133 , @xmath134 is given explicitly by @xmath135 and the waveforms are periodic in @xmath10 with period @xmath136 . the intervals are ordered in the given way for our choice of @xmath133 . @xmath137 is obtained simply by replacing the prefactor by that appropriate to @xmath138 as is clear from eq . ( [ a_gv ] ) . to obtain the waveforms for other angles we may note that the transformation @xmath139 is equivalent to changing the sign of @xmath140 , while the transformation @xmath141 is equivalent to changing the sign of @xmath140 and sign in front of the @xmath142 term in the prefactor in @xmath137 . note that the apparent singularity in the waveforms in the plane of the loop ( @xmath143 ) at @xmath144 and @xmath145 is spurious . this may be seen by noting that the waveform is bounded by the two constant sections of the piecewise linear curve which take on a value which tends to zero in this limit . in fact , the numerator of the prefactor also vanishes in this limit which ensures that the amplitude tends to zero at these points and hence that even the time derivatives ( which determine the power ) are finite . along the axis @xmath146 , eq . ( [ gv_waveform ] ) reduces to @xmath147 waveforms for various angles a plotted in fig . [ gv_plus ] for the case of @xmath148 , corresponding to two lines at right angles . this is the configuration which radiates minimum gravitational radiation for this class of loops , @xmath149 . as our next set of examples we study the set of loops in which @xmath90 lies along the @xmath150-axis and @xmath91 is always in the @xmath123-@xmath151 plane . this class of loops was studied by us in ref . @xcite where we gave an analytic result for the power lost in gravitational radiation by such loops . explicitly @xmath90 is given by @xmath152 it follows that @xmath153 also @xmath154 , so we have @xmath155 it follows immediately that the waveforms vanish along the @xmath150-axis . in ref . @xcite we proved that the minimum gravitational radiation emitted by any loop in this class is given by taking the @xmath156-loop to be a circle : @xmath157 the power emitted in gravitational radiation by this loop is @xmath158 @xmath159 may be determined explicitly as @xmath160 \\ j^{(n)}_2 & = & { 1 \over 2 } \left [ e^{i ( n+1 ) ( \phi - { \pi \over 2 } ) } j_{n+1}(n \sin \theta ) + e^{i ( n-1 ) ( \phi - { \pi \over 2 } ) } j_{n-1}(n \sin \theta ) \right ] \label{circle_j}\end{aligned}\ ] ] this gives the equivalent forms @xmath161 { 1 \over \pi n } e^ { i \pi n \sin^2(\theta/2 ) + i n ( \phi- \pi/2 ) } \nonumber \\ & = & - 2 { \sin\bigl(\pi n \sin^2(\theta/2)\bigr ) \cos(\theta ) \over \sin^2 \theta } j_{n}(n \sin\theta ) { 1 \over \pi n } e^ { i n \phi - i n ( \pi /2)\cos \theta } .\end{aligned}\ ] ] and @xmath162 { 1 \over \pi n } e^ { i \pi n \sin^2(\theta/2 ) + i n ( \phi - \pi/2 ) } \nonumber \\ & = & 2i { \sin\bigl(\pi n \sin^2(\theta/2)\bigr ) \over \sin \theta } j_n'(n \sin\theta ) { 1 \over \pi n } e^ { i n \phi - i n ( \pi /2)\cos \theta } .\end{aligned}\ ] ] the corresponding waveforms for various choices of @xmath81 are plotted in figs . [ circle_plus ] and [ circle_cross ] . ( as the system simply rotates cylindrically with time the choice of @xmath82 is irrelevant , corresponding simply to a shift in @xmath163 . ) in the plane of the @xmath23-loop @xmath134 vanishes so that the wave becomes linearly polarized . on the other hand , as we approach the axis @xmath146 the fundamental mode ( @xmath164 term ) dominates and we have @xmath165 and @xmath166 thus the wave approaches circular polarization but its amplitude vanishes as @xmath167 . as in ref . @xcite we may also consider the case where the @xmath156-loop forms a regular @xmath168-sided polygon . in figs . [ polygon_plus ] and [ polygon_cross ] we compare the waveform for the circle with that for a regular hexagon for which @xmath169 . as mentioned above a change in @xmath82 for the circle - line loop corresponds simply to a shift in @xmath10 , however , this is no longer the case for the polygon for which the waveform will only repeat every @xmath170 . hence in figs . [ polygon_plus ] and [ polygon_cross ] we include hexagon - line waveform for both @xmath171 and @xmath172 ( this choice was made simply to disentangle the two graphs as far as possible ) . it is remarkable that even for such a crude approximation to the circle as a hexagon , the waveform of the hexagon - line loop provides remarkably good piecewise linear approximations to the circle - line waveforms . given the remarkable agreement of the waveforms it is of interest to compare the ` instantaneous power ' defined by @xmath173 in the different polarizations . while this quantity is not gauge invariant its time average is and gives the total power radiated in each polarization . by comparing the function for the polygon - line loops with the circle - line loop we can certainly see that their time averages agree well . as the waveform for a piecewise linear loop is a piecewise linear function , the instantaneous power , which is the square of its derivative , will be piecewise constant . for example , in fig . [ power_fig ] we compare the ` instantaneous power ' in the plus - polarization between the circle - line loop and a regular 24-sided polygon - line loop . the very close agreement between the two curves provides further evidence for the validity of the piecewise linear approximation of string loops used by @xcite . caldwell , `` current observational constraints on cosmic strings '' , in _ proceedings of the fifth canadian general relativity and gravitation conference , 1993 _ , ed . r. mclenaghan and r. mann ( world scientific , new york , 1993 ) . b. allen , _ the stochastic gravity - wave background : sources and detection _ , in proceedings of the les houches school on astrophysical sources of gravitational radiation , eds . marck and j.p . lasota , ( cambridge university press , cambridge , england , 1997 ) .

we obtain general formulae for the plus- and cross- polarized waveforms of gravitational radiation emitted by a cosmic string loop in transverse , traceless ( synchronous , harmonic ) gauge . these equations are then specialized to the case of piecewise linear loops , and it is shown that the general waveform for such a loop is a piecewise linear function . we give several simple examples of the waveforms from such loops . we also discuss the relation between the gravitational radiation by a smooth loop and by a piecewise linear approximation to it .

You are an expert at summarizing long articles. Proceed to summarize the following text: be / x - ray binaries are the most common type of accreting x - ray pulsar systems . they consist of a pulsar and a be ( or oe ) star , a main sequence star of spectral type b ( or o ) that shows balmer emission lines ( see e.g. , slettebak 1988 and coe 2000 for reviews . ) the line emission is believed to be associated with an equatorial outflow of material expelled from the rapidly rotating be star which probably forms a quasi - keplerian disk near the be star @xcite . x - ray outbursts are produced when the pulsar interacts with this disk . be / x - ray binaries typically show two types of outburst behavior : ( a ) giant outbursts ( or type ii ) , characterized by high luminosities ( @xmath4 ergs s@xmath5 ) and high spin - up rates ( i.e. , a significant increase in pulse frequency ) and ( b ) normal outbursts ( or type i ) , characterized by lower luminosities ( @xmath6 ergs s@xmath5 ) , low spin - up rates ( if any ) , and recurrence at the orbital period @xcite . as a population be / x - ray binaries show a correlation between their spin and orbital periods @xcite . on 1998 september 8 a 15.8 s pulsar , which was designated gro j1944 + 26 , was discovered with the burst and transient source experiment ( * ? ? ? * batse ) on the _ compton gamma ray observatory ( cgro)_. gro j1944 + 26 was localized to a @xmath7 error box @xcite . at the same time , the all - sky monitor ( * ? ? ? * asm ) on the _ rossi x - ray timing explorer ( rxte ) _ discovered a new source which they localized to a @xmath8 error box and designated xte j1946 + 274 @xcite . subsequent observations with the proportional counter array ( * ? ? ? * pca ) also on the _ rxte _ on 1998 september 16 revealed 15.8 s pulsations , confirming that batse and _ rxte _ were seeing the same object @xcite . scanning observations with the pca further improved the position to a 2.4 error circle @xcite . observations with _ bepposax _ further improved the x - ray position to r.a . = @xmath9 , decl . = + 2721.5 ( equinox 2000.0 ; error radius 1 at 95% confidence ) @xcite . xte j1946 + 274 lies in the error box of the ariel 5 transient 3a 1942 + 274 ; however , there is a non - negligible probability that this is a chance association @xcite . a likely optical counterpart has been identified as an optically faint @xmath10 mag , bright infrared ( @xmath11 ) be star at r.a . = @xmath12 , decl . = 272200 , which lies in the refined bepposax error circle@xcite . spectroscopic and photometric data indicate a b0 - 1iv - ve star at 8 - 10 kpc . the counterpart s mass is not well constrained , but is expected to be in the @xmath13 range . two previously proposed candidates @xcite lie @xmath14 outside the _ bepposax _ error circle . using data from the 1998 outburst taken with the pca and the high energy x - ray timing experiment ( * ? ? ? * hexte ) on _ rxte _ , @xcite revealed a cyclotron resonance scattering feature or cyclotron line near 35 kev . this feature implies a magnetic field strength of @xmath15 g , where @xmath16 is the gravitational redshift of the emission region . using about 1 year of _ rxte _ asm 2 - 10 kev flux measurements , beginning with the 1998 outburst , @xcite observed 5 outbursts in which they found evidence of an @xmath17 day modulation in the outburst flux . they found that the first outburst in september 1998 was markedly different in rise and decay timescales than the 4 following outbursts and based on this suggested that the first outburst might be classified as a giant outburst . xte j1946 + 274 was observed with the indian x - ray astronomy experiment ( * ? ? ? * ixae ) 1999 september 18 - 30 ( mjd 51408 - 51421 ) and 2000 june 28-july 7 ( mjd 51723 - 51733 ) . they detected 15.8 s pulsations in the 2 - 6 and 6 - 18 kev bands , with similar double peaked profiles in both observations . the pulse period history was consistent with a constant intrinsic spin - up plus an eccentric orbit . the reported pulse periods were @xmath18 s and on mjd 51445.0 and @xmath19 s on mjd 51727.5 . xte j1946 + 274 was active from 1998 september through 2001 july , when it dropped below the detection threshold of the _ rxte _ pca . as of 2002 july , no additional outbursts have been seen with the _ rxte _ asm . table [ tab : obs ] lists the dates and instruments observing xte j1946 + 274 s location . in this paper we will present results from observations of xte j1946 + 274 with batse and _ rxte _ , including histories of pulse frequency and total flux for xte j1946 + 274 . from the pulse frequency history , we derive an orbital solution . applying this orbital solution , we will investigate the observed correlation between spin - up torques and observed flux and discuss implications for the accretion mechanism . we will describe the unusual orbital phasing of the outbursts . using data from the _ rxte _ pca and hexte , we will investigate spectral variations in the initial bright outburst and the last 2 outbursts . we compare our x - ray results to optical h@xmath3 observations . to determine pulse frequencies for xte j1946 + 274 , we performed a grid search over a range of candidate frequencies using data from the large area detectors on batse@xcite . this search technique is described in detail in @xcite and @xcite . we will briefly summarize the technique here . variations of this technique , including searches in pulse period , have been widely used ( * ? ? ? * ; * ? ? ? * for example ) . the pulsar search technique consists of 3 steps ( 1 ) data selection and combination , ( 2 ) 20 - 50 kev pulse profile estimates , and ( 3 ) a grid search in frequency . first , the batse discla channel 1 ( 20 - 50 kev , 1 s time resolution ) data were selected for which the source was visible , the high voltage was on , the spacecraft was outside the south atlantic anomaly , and no electron precipitation events or other anomalies were flagged by the batse mission operations team . the count rates were combined over the 4 lads viewing xte j1946 + 274 , using weights optimized for an exponential energy spectrum , @xmath20 with temperature @xmath21 kev , and grouped into @xmath22 s segments . a segment length of 300 s was used because it was short enough that the background was well - fitted by a quadratic model . the segment boundaries were chosen to avoid earth occultation steps from bright sources@xcite . the second step in this process was estimation of an initial 20 - 50 kev pulse profile for each segment of data . in each segment , the combined rates were fitted with a model consisting of a sixth - order fourier expansion in pulse phase ( representing the 20 - 50 kev pulse profile ) @xmath23 where @xmath24 is the estimated complex fourier coefficient for harmonic where @xmath25 and @xmath26 is the pulse frequency . ] @xmath27 in segment @xmath28 and a spline function with quadratics in time ( representing the background plus mean source count rate ) . a sixth - order fourier expansion was chosen based on the number of harmonics that are significant in a 1-day observation , the typical integration time required to detect xte j1946 + 274 . our initial pulse phase model was of the form @xmath29 , where @xmath30 mhz before mjd 51270 and @xmath31 mhz after mjd 51270 were barycentric frequencies and @xmath32 was the time corrected to the solar system barycenter using the jpl de-200 ephemeris @xcite . the initial pulse frequency was changed after mjd 51270 because continued spin - up of the pulsar moved the observed frequency too far away from the folding frequency . the value and slope of the spline function were required to be continuous across adjacent 300 s segment boundaries , but not across data gaps . ideally we would like to be able to fit a pulse profile to the entire 4-day interval of data for each frequency grid point ; however this is computationally very very expensive . instead we have formulated an equivalent method in which we first fit pulse profiles to short segments of data using a single fixed frequency . then the profiles are combined and shifted by a frequency offset corresponding to each grid point . fitting pulse profiles to short data segments has other advantages in addition to improving computational efficiency . ( 1 ) if the folding frequency chosen was incorrect , the profile will be smeared out . however , the degree of smearing in pulse phase @xmath33 is approximately given by @xmath34 , where @xmath35 is an offset in frequency , @xmath36 is the segment length , and @xmath37 is the number of fourier coefficients . for @xmath38 s , @xmath39 mhz , and @xmath40 , @xmath41 , which corresponds to a drop in the amplitude of the sixth harmonic by 15% . the amplitude of the first harmonic only drops by about 0.4% . using a short segment reduces smearing of the pulse profile . ( 2 ) typically 470 pulse profiles from the 300 s segments are accumulated within each 4-day interval . since we have a large number of profiles we can estimate sample variances for each fourier coefficient . sample variances give us a measure of the noise present in the data and account for aperiodic noise if it is present . the final step in our advanced pulsar search was a grid search in frequency using the set of typically several hundred estimated 20 - 50 kev pulse profiles from 4-day intervals of data . new xte j1946 + 274 pulse frequencies were determined from an initial grid search over 1200 evenly spaced trial barycentric frequencies in the range 62.97 - 63.32 mhz for mjd 48361 - 51270 and 63.10 - 63.45 mhz for mjd 51270 - 51690 . the search range was changed because continued spin - up of the pulsar moved the pulse frequencies outside the original search range . for each frequency offset @xmath35 , a mean pulse profile is computed for each 4-day interval . the mean pulse profile is of the form @xmath42 where @xmath43 , @xmath44 , and @xmath45 is the midpoint barycentric time of segment @xmath28 . the mean coefficients @xmath46 are given by @xmath47 with weights @xmath48 . the typical statistic used for a search in pulse frequency or pulse period is the @xmath49 statistic @xcite given by @xmath50 where @xmath51 is the formal ( i.e. , poisson statistical ) error on the fourier coefficient @xmath46 , respectively . if aperiodic noise is present , either due to xte j1946 + 274 or due to other sources in the large batse field of view , e.g. , cygnus x-1 , the poisson statistical error is an underestimate because the actual underlying noise level is higher than poisson noise this can create a problem with the @xmath49 statistic , causing it to depend on the noise level . to account for aperiodic noise , we used all of the typically @xmath52 pulse profiles from 300 s segments within each 4-day interval to estimate sample variances for the fourier coefficients for each harmonic in the mean profile . each harmonic was treated separately . we then modified the @xmath49 statistic by replacing the poisson variances with our computed sample variances to create a new statistic which we will call @xmath53 after @xcite where it is described in detail . due to the large field - of - view of batse , other pulsars were often also present when we were measuring xte j1946 + 274 . if we limited our statistic to use the first 3 harmonics where xte j1946 + 274 was the brightest , we reduced the chances of contamination of the search results from other pulsars that happen to have harmonics near the higher harmonics of xte j1946 + 274 . the best - fit frequency for each 4-day interval was then determined using the @xmath54 statistic . using a similar method we also generated pulse frequency measurements for _ rxte _ pca observations during the initial outburst in 1998 september - october and during the last 2 outbursts , 2001 march - july . barycentered standard 1 ( 125 ms , no energy resolution ) data were fitted with a model consisting of a constant background plus a 6th - order fourier expansion in pulse phase model , creating an estimated 2 - 60 kev pulse profile for each pca observation . the pulsed phase model consisted of a constant barycentric frequency , @xmath55 mhz for the 1998 september - october observations and @xmath56 mhz for the 2001 observations , estimated from projections of batse measurements . for each set of observations , we searched over a grid of 151 evenly spaced frequencies spanning the range @xmath57 mhz , the same size interval as used for batse . figures [ fig : freqs]a & c show the barycentered pulse frequency history and the 2 - 10 kev total flux history , respectively , for xte j1946 + 274 . in figure [ fig : freqs]a , the trend in the pulse frequencies repeatedly changes from spin - up to spin down and back to spin - up at regular intervals ( every 170 days ) . this repeated pattern is most likely due to the pulsar s orbit . within each 170 day cycle in the pulse frequencies , there are two outbursts . examination of the observed pulse frequency history showed us two things : ( 1 ) a strong orbital signature and ( 2 ) strong intrinsic torque variations . to attempt to extract orbital parameters , we first fitted the observed pulse frequencies @xmath58 with a model consisting of a global orbit and a global polynomial frequency model , i.e. , @xmath59 where @xmath60 is a model of the emitted frequency . the velocity relative to the observer @xmath61 is given by @xmath62 where @xmath63 is the projected semi - major axis of the pulsar s orbit ; @xmath64 is the orbital period ; @xmath65 is the periapse angle ; and @xmath66 is the orbital eccentricity ; @xmath67 is the eccentric anomaly , given by @xmath68 and @xmath69 is the epoch of periastron passage . we fitted models containing polynomials of orders 1 - 10 . all of these models were very poor fits . table [ tab : orb ] lists the orbital parameters obtained using a first and 10th order polynomial for comparison . errors on these parameters have been inflated by a factor of @xmath70 , but are still expected to be considerably underestimated because the @xmath71 values are so large . figure [ fig : poly ] shows the emitted frequency model @xmath72 , @xmath73 the constant term from the emitted frequency model times the velocity relative to the observer to illustrate the amplitude of orbital effects , the full frequency model including the emitted frequency and orbit , and the frequency residuals for the 10th order polynomial model . clearly a simple model such as a global polynomial did not suffice to describe the intrinsic torques . instead we decided to model the intrinsic torques in a piecewise fashion . each outburst was split into 2 - 6 intervals , depending on the length of the outburst , where each interval contained 3 - 4 frequency measurements and spanned 16 - 24 days or 16 - 32 days for the batse and the _ rxte _ pca data , respectively . in each interval , we fitted the observed pulse frequencies with a global orbit plus an independent linear frequency model , equation [ eqn : fobs ] with @xmath74 as a model of the emitted frequency for each time interval @xmath75 . table [ tab : orb ] lists our best fit orbit . figure [ fig : resid ] shows @xmath72 , @xmath76 to illustrate the amplitude of orbital variations ( note @xmath77 was chosen because it lies in near the center of the frequency range . ) , the full frequency model including the emitted frequency and orbit , and the frequency fit residuals . this fit is not formally acceptable either ; however , we believe this is mostly because this model still does not completely describe the intrinsic torques . errors on the individual parameters in table [ tab : orb ] have been inflated by @xmath78 . comparing the three columns of table [ tab : orb ] shows that the orbital parameters appear to be similar , whether we use a simple polynomial model or a more complicated piecewise model for @xmath72 . figure [ fig : freqs]b shows the orbit - corrected spin frequencies . the regular pattern seen in figure [ fig : freqs]a has been removed , while intrinsic torque effects remain in the data . a close look at figure [ fig : freqs ] indicates that the outbursts are not fixed in orbital phase and that we are seeing approximately 2 outbursts per orbit . to study the orbital phase of the outbursts , we fitted the 2 - 10 kev _ rxte _ asm flux in figure [ fig : freqs]c with a quartic polynomial , and used this polynomial to normalize the outbursts . we then computed the orbital position using @xmath79 where @xmath80 is the semi - major axis , estimated from @xmath63 assuming @xmath81 which gives a companion mass of @xmath82 . figure [ fig : orbphase ] shows the normalized intensity versus orbital position , with darker colors indicating higher intensities . the outer loop of the spiral has @xmath83 , the estimated radius assuming @xmath84 . for each successive orbit , we reduced @xmath80 by 10% to offset the orbits . this plot spirals in solely to allow comparison of outburst phases . we are not claiming that the neutron star is spiraling in toward its companion . the outburst peaks clearly are not fixed in orbital phase , although there is typically one peak on each side of the orbit . just before periastron and just before apastron , the flux is typically relatively low . the cause of these low flux periods is unclear . although the fluxes show considerable modulation , they remain detectable with _ rxte _ throughout the period 1998 september - 2001 july . to look for variations in energy spectra during the outbursts observed with the _ rxte _ pca , we first studied hardness ratios . using ftools 5.1 @xcite and standard2 ( 16-s , 129 energy channel ) data , we created light curves in 4 energy bands corresponding to 2 - 5 , 5 - 10 , 10 - 15 , and 15 - 20 kev for both the 1998 and 2001 outbursts . these light curves were background subtracted and their times were corrected to the solar system barycenter using ftools . next we formed 3 hardness ratios from adjacent energy bands . figure [ fig : hi ] shows each of these ratios vs. total pca source count rate in the 2 - 30 kev band . figure [ fig : cc ] shows color - color diagrams . in both figures , the grey - scale denotes intensity , with darker points denoting higher intensities . ( note : this is the measured 2 - 30 kev intensity , not the normalized 2 - 10 kev intensity used in section [ sec : orbph ] and figure [ fig : orbphase ] . ) for all of the hardness ratios , we see a correlation between hardness and intensity in the 2001 outburst that is not present in the 1998 outburst . in the color - color diagrams , the hardness ratios are correlated for both sets of outbursts and have similar slopes . however , during the 2001 outbursts , points move from left to right along the correlation as intensity increases ; while in the 1998 outburst , the points move around in hardness as intensity increases . to better quantify the observed spectral changes , we generated response matrices , count spectra , and background spectra using ftools for pca standard2 data and hexte event mode data for each observation . we then fitted each of the observations in xspec 11.1 @xcite with two models : ( 1 ) an absorbed power law with a high - energy cutoff and an iron line , xspec model : phabs ( powerlaw + gaussian ) highecut@xmath85 and ( 2 ) an absorbed power law with an iron line and a cyclotron absorption feature , xspec model : phabs ( powerlaw + gaussian ) cyclabs@xmath85 . for both models we used only data in the 2.7 - 30 kev energy range for pca data and in the 15 - 50 kev range for hexte . the power law normalizations were allowed to be independent for the pca and each hexte cluster . figure [ fig : highecut ] shows @xmath71 , spectral index , cutoff energy , folding energy , and flux in the fe line vs. 2 - 60 kev flux . figure [ fig : cyclabs ] shows @xmath71 , spectral index , the cyclotron line energy , and the flux in the fe line vs. 2 - 60 kev flux . other spectral parameters did not show significant variations with intensity . a comparison of the two models showed that model 2 generally provided a better fit to the data ( see figures [ fig : highecut ] & [ fig : cyclabs ] , top panel ) . in both models , the power law photon index remains relatively independent of flux for fluxes above about @xmath86 ergs @xmath87 s@xmath5 , however , below that flux level , the power law index softens as the intensity decreases . the integrated flux in the iron line is correlated with the 2 - 60 kev flux in both models , indicating that the iron line is indeed a feature of xte j1946 + 274 s spectrum and not a background feature . in model 2 a cyclotron absorption feature , when it is constrained , tends to be near the value observed by @xcite . to estimate an upper limit to the quiescent flux from xte j1946 + 274 , we fitted pca data from each of 6 observations after pulsations were undetectable in 2001 august - september with an absorbed power law . these observations were consistent with a power law index of @xmath88 , a hydrogen column of about @xmath89 @xmath87 , and a 2 - 30 kev flux of @xmath90 ergs @xmath87 s@xmath5 . since _ rxte _ is not an imaging instrument , we can not distinguish between low - level emission from xte j1946 + 274 and background sources such as the galactic ridge . however , this gives us an upper limit on the flux from xte j1946 + 274 . to estimate a bolometric correction to the _ rxte _ asm measurements , we plotted the 2 - 60 kev fluxes from model 2 in section [ sec : spec ] vs. the 5-day average asm count rate surrounding each pca observation . we then fitted a line to these data , obtaining a bolometric correction of 1 asm ct s@xmath91 ergs @xmath87 s@xmath5(2 - 60 kev ) . the error on this correction has been estimated including systematic effects . figure [ fig : bc ] shows the best fit line to these data . near the peak flux level , the asm fluxes show evidence for a turnover relative to the pca fluxes . we have attempted to account for this turn - over by including a systematic error of 10% on our bolometric correction . using this bolometric correction , we then plotted the average bolometrically corrected asm flux vs. spin - up rate for each interval in our frequency model . figure [ fig : fdotvsflux ] clearly shows a correlation between spin - up rate and flux . such a correlation suggests a disk is present because direct wind accretion is believed to be less efficient at transferring angular momentum @xcite . if enough angular momentum is present , a disk will form . simple accretion theory assumes that the material from the companion star is flowing onto a rotating neutron star with a strong magnetic field . this field determines the motion of material in a region of space called the magnetosphere . the size of this region is denoted by the magnetospheric radius @xmath92 , given by @xcite @xmath93 where @xmath94 is the gravitational constant ; @xmath95 is the mass of the neutron star ; and @xmath96 is the mass accretion rate , which is assumed to be related to the observed bolometric flux @xmath97 by @xmath98 , where @xmath99 is the radius of the neutron star and @xmath100 is the distance to the pulsar . @xmath28 is a constant factor of order 1 . equation [ eqn : rm ] with @xmath101 gives the alfvn radius @xmath102 for spherical accretion and with @xmath103 gives the magnetospheric radius derived by @xcite . the torque applied by accretion of matter onto a neutron star , assuming torques due to matter leaving the system are negligible , is given by @xcite @xmath104 where @xmath105 is the moment of inertia of the neutron star and @xmath106 is the specific angular momentum of the material . if @xmath105 is assumed constant , then @xmath106 is given by @xmath107 where @xmath108 is the spin - up rate . to estimate @xmath106 for xte j1946 + 274 , we assumed typical pulsar parameters , @xmath109 , @xmath110 km , @xmath111 g @xmath112 , typical values of @xmath113 hz s@xmath5 and @xmath114 ergs @xmath87 s@xmath5(see figure [ fig : fdotvsflux ] ) , and a distance @xmath115 kpc @xcite . this yielded @xmath116 @xmath117 s@xmath5 . an accretion disk will form if the specific angular momentum of the material accreted from the be star s disk is comparable to the keplerian specific angular momentum at the magnetospheric radius , i.e. , @xmath118 for xte j1946 + 274 , using the magnetic field measurement @xmath119 g @xcite which implies @xmath120 g @xmath121 , then @xmath122 @xmath117 s@xmath5 for @xmath123 and @xmath124 @xmath117 s@xmath5 for @xmath125 . for xte j1946 + 274 , @xmath126 for @xmath123 and @xmath127 for k = 0.47 , hence an accretion disk is most likely present . in contrast , for the wind - fed system vela x-1 where a disk is not expected to be present , @xmath128 hz s@xmath5 @xcite , @xmath129 ergs s@xmath5 , and @xmath130 g @xmath131 @xcite leading to @xmath132 @xmath112 s@xmath5 and @xmath133 @xmath112 s@xmath5 , i.e. , @xmath134 . further , three - dimensional simulations of wind accretion show that the average specific angular momentum accreted via wind accretion is always smaller than the keplerian value @xcite . since an accretion disk appears to be present , we can use simple accretion theory to derive a distance to xte j1946 + 274 . substituting equations [ eqn : ellrm ] into equation [ eqn : torque ] and solving for @xmath108 gives @xmath135 assuming @xmath123 and the parameters defined earlier , @xmath136 where @xmath137 is the spin - up rate in units of @xmath138 hz s@xmath5 and @xmath139 is the bolometric flux in units of @xmath86 ergs @xmath87 s@xmath5 . we then fitted this model to the data in figure [ fig : fdotvsflux ] . to account for the flux and @xmath108 errors , we computed @xmath71 as @xmath140 where @xmath141 is a free parameter fitted to each flux and @xmath142 is equation [ eqn : nudot ] evaluated at @xmath141 . the solid line shows our best fit , with @xmath143 , which gives us a distance of @xmath144 kpc . this distance is consistent with optical observations of the counterpart . the errors on the distance are computed from the fit itself and take into account only the statistical errors on the flux and spin - up rate and the estimated systematic error on the bolometric correction . however , there is considerable uncertainty in our assumed neutron star parameters . the estimated distance depends on the following combination of neutron star parameters : @xmath145 . using the neutron star equations of state given by @xcite and assuming @xmath146 , we estimate the error on the distance due to uncertainty in the neutron star parameters is about 30% , i.e. , @xmath147 kpc . using ftools and _ rxte _ pca event mode data , we created light curves and corresponding background light curves in 5 energy bands : 2 - 5 , 5 - 10 , 10 - 15 , 15 - 20 , and 20 - 30 kev for each of 29 observations , 12 from the 1998 outburst and 17 from the 2001 outbursts . each light curve was barycentered , background subtracted , and folded at the appropriate frequency from our frequency search . each pulse profile contained 50 phase bins . first we examined profiles from the entire 2 - 30 kev band to search for intensity dependent variations . we aligned all of the pulse profiles by finding the phase of the minimum of the profile using a quadratic interpolation and placing that minimum at phase 0.0 . we found definite variations with intensity in both the 1998 and 2001 outbursts and also evidence that the pulse profile was different at similar intensities during the rise and fall of the initial bright outburst . profiles with similar shapes and similar intensities were averaged to better illustrate the observed shape changes . figure [ fig : intprof]a shows the average 2 - 30 kev pulse profile at 6 different intensities . figure [ fig : intprof]b shows the peak - to - peak pulse fraction versus mean flux for each profile . in the following descriptions and in figures [ fig : intprof]a and b , the pulse profiles are numbered from one to six . ( 1 ) at the lowest intensities ( 6 - 17 counts s@xmath5 pcu@xmath5 , 2 - 30 kev ) , during the 2001 outbursts , the profile s main feature was a deep notch , which we used to align the profiles . the profile consisted of an asymmetric structured main peak that peaked near phase 0.2 . ( 2 ) as the intensity increased ( 23 - 65 counts s@xmath5 pcu@xmath5 ) , a second notch becomes prominent near phase 0.45 , making the profile consist of two main peaks . these peaks were approximately equal in intensity at first , but the second peak brightens as the overall intensity increases . a small peak near phase 0.87 also appears . no _ rxte _ pca observations were taken at intensities between 65 and 178 counts s@xmath5 pcu@xmath5 . ( 3 ) the next profile shape occurs at intensities of 178 - 266 counts s@xmath5 pcu@xmath5 , near the end of the 1998 outburst . this profile is markedly different from that near the peak of the 2001 outburst ( profile 2 ) . both notches are broader , the peak near phase 0.65 is brighter than the peak near phase 0.3 and the small peak at phase 0.87 has disappeared . the main notch appears to be less deep than at lower intensities . ( 4 ) profiles at intensities of 285 - 308 and 330 counts s@xmath5 pcu@xmath5 , all from the rise of the 1998 outburst , show a broader first peak and a less intense shoulder following the second peak than the lower intensity profiles from the fall of the outburst ( profile 3 ) . ( 5 ) at intensities of 322 - 328 counts s@xmath5 pcu@xmath5 , during the decline of the 1998 outburst , the first peak is narrower than at similar intensities during the outburst rise ( profile 4 ) , but broader than at lower intensities during the decline ( profile 3 ) . ( 6 ) at the peak of the outburst , the width of the first peak is at an intermediate width between the profiles from the rise and fall of the outburst . during the 2001 outbursts , the profile shape appeared to depend primarily on intensity , while during the 1998 bright outburst , the profile shape depended on both intensity and whether the profile was from the rise or fall of the outburst . next we looked for variations in the pulse profile versus energy . figure [ fig : profvsenergy ] shows pulse profiles in 5 energy bands from the peak of the 1998 outburst ( left panel ) and near one of the peaks in the pair of outbursts in 2001 ( right panel ) . in the 1998 profiles , the profile consists of two main peaks at lower energies ( @xmath148 kev ) . the first peak is dominant at lower energies and the second peak becomes more dominant as energy increases . also , as energy increases , the notch between the two main peaks fills in . as the 2 - 30 kev intensity decreases , the profile shape " appears to move down in energy . for example , in the right hand panel of figure [ fig : profvsenergy ] , the 2 - 5 kev profile consists of two nearly equal peaks , reminiscent of the 10 - 15 kev profile at the peak of the 1998 outburst . a regular monitoring program was established to study the strength and structure of the h@xmath3 optical emission line during the x - ray observations in 2001 . this line arises from the circumstellar disk surrounding the be star and the line properties are strongly related to the extent and dynamic structure of the disk . data were primarily collected from the 2.5 m isaac newton telescope ( int ) in la palma , spain . the int was equipped with the 235-mm camera and eev#10 ccd . the use of the r1200r grating results in a nominal dispersion of @xmath149 / pixel . intermediate resolution spectroscopy was also performed on 17th july 2001 and 22nd october 2001 using the 1.93-m telescope at the observatoire de haute provence , france . that telescope was equipped with the long - slit spectrograph _ carelec _ and the @xmath150 eev ccd . we used the 1200 ln / mm grating in first order , resulting in a nominal dispersion of @xmath151 / pixel . see table [ tab : ha ] for details . the h@xmath3 profile obtained from each observation is presented in figure [ fig : ha ] . the lines have been shifted vertically by an arbitrary amount in order to present each profile clearly . the data of the 17 july and 7 october 2001 are of a lower standard due to poor weather conditions . nonetheless it is possible to accurately determine the equivalent width of each of the emission lines ; values are presented in table [ tab : ha ] . from these numbers one can see that there is little evidence for any changes in the line flux , assuming a constant continuum . however , from figure [ fig : ha ] , one can see obvious shifts in the line position after 2001 july . if one takes the first spectrum ( 3 may 2001 ) as a base line and subtracts it from each of the other spectra after they have been normalized to the same peak value , then significant changes are apparent . the result of this process is shown in figure [ fig : diff ] . it is apparent that little happens to the profiles ( and , presumably , the circumstellar disk ) until july / august 2001 . starting with the spectrum of 17 july 2001 one can see significant perturbations occurring in the line profile indicative of density changes in the disk structure . between august and september the direction of the perturbation changes , suggesting that the density enhancement / rarefication had changed sides ( or rotated around ) in the disk . the size of the perturbation also seems to have been increasing from july to october , though the separation of the red and blue peaks remains approximately constant at @xmath152340 km / s . xte j1946 + 274 is a very unusual be / x - ray binary system and its behavior does not fit well into the standard normal / giant outburst behavior . instead it showed an extended period of activity from 1998 september - 2001 july when the x - ray flux showed considerable modulation , with two peaks or outbursts per orbit , but the x - ray flux never dropped below the detection threshold of _ this extended period of activity more closely resembles a series of normal outbursts than a single giant outburst ; however , unlike typical normal outbursts , these outbursts shift rapidly in orbital phase , there are two outbursts per orbit , and the x - ray flux does not drop dramatically between outbursts . the initial outburst shows considerable spin - up @xmath153 hz s@xmath5 , like a giant outburst , but this spin - up is not substantially larger than that seen in later outbursts . in addition , the peak flux ( 2 - 60 kev ) of the initial outburst is about @xmath154 ergs @xmath87 s@xmath5 , while the peak flux ( 2 - 60 kev ) of the last two outbursts observed with the pca is about @xmath155 ergs @xmath87 s@xmath5 , only a factor of @xmath156 fainter than the brightest outburst . these fluxes correspond to luminosities of @xmath157 ergs s@xmath5and @xmath158 ergs s@xmath5 , respectively , assuming our best - fit distance of 9.5 kpc . in giant outbursts of be / x - ray binaries , accretion disks are expected to be present and indeed , evidence for an accretion disk , based on correlations between the observed flux and spin - up rate , has been found for several sources during giant outbursts @xcite . independent evidence for an accretion disk based on the detection of quasi - periodic oscillations during a giant outburst has been found for exo 2030 + 375 @xcite and a0535 + 262 @xcite . until recently , normal outbursts were believed to be due to direct wind accretion from the be disk , so significant spin - up was not expected because direct wind accretion is not believed to be very efficient at transferring angular momentum @xcite . if enough angular momentum is present in the accreted material , an accretion disk will form . however , evidence for spin - up during normal outbursts has been observed in gs 0834430 @xcite , 2s 1417624 @xcite , 2s 1845024 @xcite , and in exo 2030 + 375 @xcite . we see a correlation between spin - up and flux for xte j1946 + 274 as shown in figure [ fig : fdotvsflux ] . the spin - up rate and its correlation with bolometric flux during xte j1946 + 274 s outbursts suggest an accretion disk may be present . in addition , our calculations show that the specific angular momentum of the accreted material is comparable to the keplerian specific angular momentum , hence a disk is expected to form in xte j1946 + 274 . further , our fit to the spin - up vs. bolometric flux correlation , which assumed a disk was present , yielded a distance of @xmath2 kpc which is consistent with the distance of 8 - 10 kpc derived from optical observations @xcite . hence an accretion disk is likely to be present in xte j1946 + 274 . between 2001 july 31 ( mjd 52121 ) and 2001 august 9 ( mjd 52130 ) , xte j1946 + 274 dropped below the pca s detection threshold . at approximately the same time , between 2001 june 29 ( mjd 52089 ) and 2001 july 17 ( mjd 52107 ) , the h@xmath3 profile began to change rapidly . the profile was stable for 3 observations on 2001 may 3 , may 10 , and june 29 ( mjd 52032 , 52039 , & 52089 ) that corresponded to x - ray observations during the decline of the second to last outburst , during the low state between the last two outbursts , and near the peak of the last outburst . the coincidence of the change in h@xmath3 with the x - ray turn - off suggests that changes in the be disk caused the x - ray outbursts to cease . to determine whether or not the observed x - ray turn - off was due to centrifugal inhibition of accretion @xcite , we estimate the flux at the onset of this effect by equating the magnetospheric radius to the corotation radius . the magnetospheric radius is given by equation [ eqn : rm ] and the corotation radius is given by @xmath159 where @xmath26 is the spin frequency of the pulsar . setting @xmath160 gives the threshold flux for the onset of centrifugal inhibition of accretion , i.e. , @xmath161 where @xmath162 , @xmath163 , @xmath164 , and @xmath165 are the pulsar s magnetic moment in units of @xmath166 g @xmath131 , mass in units of 1.4 @xmath167 , radius in units of @xmath168 cm , and spin period in units of 15.8 seconds , respectively . using @xmath169 kpc and @xmath170 g @xmath121 yields @xmath171 ergs @xmath87 s@xmath5@xmath172 . for @xmath173 , @xmath174 ergs @xmath87 s@xmath5 . our measured upper limit fluxes are in the range @xmath175 ergs @xmath87 s@xmath5 , consistent with xte j1946 + 274 entering the centrifugal inhibition of accretion regime when the x - rays became undetectable with the pca . we propose that xte j1946 + 274 is a system in which the be star s equatorial plane and the orbital plane are not aligned . this is suggested by : ( 1 ) the optical observations show a relatively narrow fwhm ( 8.6 on 2001 may 10 , which corresponds to @xmath176 200 km s@xmath5 ) for the single - peaked h@xmath3 line , indicating that the be star is viewed from a relatively low inclination angle , i.e. , nearly pole - on . ( 2 ) the orbital signature is quite obvious in the pulse frequencies , indicating that the orbital inclination angle is likely not low : the derived mass function @xmath177 from the piecewise frequency model indicates an inclination angle @xmath178 for the expected mass range of @xmath13 derived from optical observations . figure [ fig : mc ] shows the range of allowed inclinations using the 1-@xmath179 errors on the mass function . we see two outbursts per orbit in xte j1946 + 274 ; however , these outbursts are not fixed in orbital phase . this combination of behaviors is unique to xte j1946 + 274 and is likely not due to a single mechanism . the optical data and mass function suggest that the orbital plane and the be disk are not aligned in this system . such a misaligned system would be expected to produce two outbursts per orbit , each approximately corresponding to the neutron star s passage through the be disk ; hence these outbursts would be expected to be fixed in orbital phase . for xte j1946 + 274 , we are faced with the difficult task of explaining not only why we see the expected two outbursts per orbit , but also why these outbursts are not fixed in orbital phase . we propose that given the relatively high luminosities and spin - up rates in all of the outbursts that perhaps xte j1946 + 274 s unusual outburst behavior can be explained as a giant outburst . in normal outbursts , the be disk is expected to be truncated at a resonance radius by tidal forces from the neutron star s orbit @xcite ; however , in giant outbursts , where much more material is believed to be present in the disk , it is believed that the disk is no longer truncated . hence more material likely means a larger disk around the be star . density perturbations propagating in the be disk could change the contact points of the disk and the orbit . these perturbations would have to be moving quite rapidly in the disk . this idea of a giant outburst combined with density perturbations seems most likely to work in a system where the inclination angle between the be disk and the orbital plane is fairly small . it is not clear that this angle is small in xte j1946 + 274 . we do not claim that the idea of a giant outburst in an inclined system fully explains the behavior observed from xte j1946 + 274 . instead we put the idea forth as a suggestion to those doing simulations of these systems . the phasing of the outbursts of xte j1946 + 274 ( see figure [ fig : orbphase ] ) is difficult to understand . however , it is not the first be / x - ray binary to show shifts in outburst phase . exo 2030 + 375 underwent a fairly sudden ( within 4 orbits ) shift in the peak phase of the outbursts from 6 days after periastron ( phase @xmath180 ) to 2.5 days before periastron ( phase -0.05 ) , followed by a gradual shift in outburst phase to 2.5 days after periastron ( phase 0.05 ) . this shift was believed to be associated with a density perturbation observed in the be disk via h@xmath3 observations @xcite . gs 0834430 also underwent sudden shifts in outburst phase , undergoing 9 outbursts approximately centered on periastron , followed by dramatic shifts to outbursts centered on phases 0.37 and 0.75 @xcite . a companion was not known at the time for gs 0834430 , so we can not confirm that density perturbations in the be disk were responsible for these phase shifts . in xte j1946 + 274 , we have a similar problem , with the companion not observed for much of the time the x - ray outbursts were occurring . in addition , the problem is compounded by the fact that we are likely viewing the be star nearly pole - on ; hence the range of projected rotational velocities is very small , making it difficult to detect perturbed velocities which indicate density perturbations . only large density perturbations , such as the one that coincided with the x - ray turn - off , are likely to be detected . we are grateful to luisa morales for providing one of the optical spectra . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center ( gsfc ) . asm quick - look results were provided by the asm/ _ rxte _ teams at mit and at the gsfc sof and gof . the int is operated on the island of la palma by the isaac newton group in the spanish observatorio del roque de los muchachos of the instituto de astrofsica de canarias . based in part on observations made at observatoire de haute provence ( cnrs ) , france . angelini , l. , stella , l. , & parmar , a.n . 1989 , , 346 , 906 arnaud , k.a . 1996 , astronomical data analysis software and systems v , eds . g. jacoby & j. barnes , asp conf . series , 101 , 17 benensohn , j.s . , lamb , d.q . & taam , r.e . 1997 , apj , 478 , 723 bildsten , l. et al . 1997 , , 113 , 367 blackburn , j.k . 1995 , in asp conf . 77 , astronomical data analysis and software systems iv , ed . r.a . shaw , h.e . payne , & j.j.e . haynes ( san francisco : asp ) , 367 buccheri , r. et al . 1983 , , 128 , 245 campana , s. , israel , g. , & stella , l. 1998 , iau circ . 7039 campana , s. , israel , g. , & stella , l. 1999 , , 352 , l91 coe , m.j . 2000 , in the be phenomenon in early - type stars , iau colloquium 175 , asp conf . 214 , eds . smith & h.f . henrichs , ( san francisco : asp),656 corbet , r.h.d . 1986 , , 220 , 1047 finger , m. h. et al . 1999 , , 517 , 449 finger , m.h . , wilson , r.b . & chakrabarty , d. 1996 , a&as , 120 , 209 finger , m.h . , wilson , r.b . , harmon , b.a . 1996 , apj , 459 , 288 fishman , g. j. et al . 1989 , in proc . gro science workshop , ed . w.n . johnson ( greenbelt : nasa / gsfc ) , 2 ghavamian , p. & garcia , m. 1998 , iau circ . 7022 ghosh , p. & lamb , f.k . 1979 , apj , 234 , 296 hanuschik , r.w . 1996 , , 308 , 170 heindl , w.a . et al . 2002 , , 563 , l35 inam , s.c . & baykal , a. 2000 , , 353 , 617 israel , g.l . , polcaro , v.f . , & covino , s. 1998 , iau circ . 7021 jahoda , k. et al . 1996 , euv , x - ray , and gamma - ray instrumentation for astronomy vii , spie proc . 2808 , ed . sigmund & m.gumm ( bellingham : spie ) , 59 lamb , f.k . , pethick , c.j . , & pines , d. 1973 , , 184 , 271 levine , a.m. , bradt , h. , cui , w. , jernigan , j.g . , morgan , e.h . , remillard , r. , shirey , r. e. , & smith , d.a . 1996 , , 469 , l33 makishima , k. , mihara , t. , nagase , f. , tanaka , y. 1999 , , 525 , 978 okazaki , a.t . & negueruela , i. 2001 , , 377 , 161 parmar , a.n . , white , n.e . , stella , l. , & ferri , p. 1989 , , 338 , 359 paul , b. et al . 2001 , , 370 , 529 pringle , j.e . , & rees , m.j . 1972 , , 21 , 1 quirrenbach , a. et al . 1997 , , 479 , 477 reynolds , a.p . 1996 , , 312 , 872 rothschild , r.e . 1998 , , 496 , 538 ruffert , m. 1997 , , 317 , 793 ruffert , m. 1999 , , 346 , 861 slettebak , a. 1988 , , 100 , 770 smith , d.a . & takeshima , t. 1998 , iau circ . 7014 standish , e. m. , newhall , x. x. , williams , j. g. , & yeomans , d. k. 1992 , orbital ephemerides of the sun , moon , & planets , in explanatory supplement to the astronomical almanac , ed . k. seidelmann ( mill valley : university science books ) , 279 stella , l. , white , n.e . , & rosner , r. 1986 , , 308 , 669 stollberg , m.t . et al . 1999 , , 512 , 313 takeshima , t. & chakrabarty , d. 1998 , iau circ . 7016 verrecchia , f. et al . 2002 , , in press waters , l.b.f.m . & van kerkwijk , m.h . 1989 , , 223 , 196 wilson , c.a . 1997 , apj , 479 , 388 wilson , c.a . , finger , m.h . , wilson , r.b . , scott , d.m . 1998 , iau circ . 7014 wilson , c.a . , finger , m.h . , coe , m.j . , laycock , s. , fabregat , j. 2002 , apj , 570 , 287 wilson - hodge , c.a . 1999 , phd dissertation , university of alabama in huntsville wiringa , r.b . , fiks , v. & fabriocini , a. 1988 , phys . c. , 38 , 1010 lllll 1991 apr 15 - 2000 may 27 & 48361 - 51691 & batse & & + 1996 feb 23 - 2002 apr 11 & 50136 - 52376 & _ rxte _ asm & & + 1998 sep 16 - 1998 oct 14 & 51072 - 51100 & _ rxte _ pca , hexte & 12 & 85.87 + 2001 mar 9 - 2001 sep 25 & 51977 - 52177 & _ rxte _ pca , hexte & 24 & 129.49 + [ tab : obs ] llll @xmath64 ( days ) & @xmath181 & @xmath182 & @xmath183 + @xmath69 & jd@xmath184 & jd@xmath185 & jd@xmath186 + @xmath63 ( lt - sec ) & @xmath187 & @xmath188 & @xmath189 + @xmath66 & @xmath190 & @xmath191 & @xmath192 + @xmath65 ( degrees ) & @xmath193 & @xmath194 & @xmath195 + @xmath196 @xmath197 & @xmath198 & @xmath199 & @xmath200 + @xmath201d.o.f & @xmath202 & @xmath203 & @xmath204 + [ tab : orb ] cccccc 3 may 2001&int&ids+500 camera&r1200r & tek5 & 45.9@xmath1760.6 + 10 may 2001&int&ids+235 camera&r1200r & eev10&41.0@xmath1760.8 + 29 jun 2001&int&ids+235 camera&r1200r & eev10&41.1@xmath1760.3 + 17 jul 2001&ohp&carelec & 1200l / mm&eev & 39.1@xmath1760.4 + 12 aug 2001&int&ids+235 camera&r1200r & eev10&43.9@xmath1760.3 + 26 sep 2001&int&ids+235 camera&r1200r & eev10&43.6@xmath1760.3 + 7 oct 2001&int&ids+235 camera&r1200r & eev10&38.5@xmath1760.5 + 22 oct 2001&ohp&carelec & 1200l / mm&eev & 42.5@xmath1760.3 + [ tab : ha ]

xte j1946 + 274 = gro j1944 + 26 is a 15.8 s be / x - ray pulsar discovered simultaneously in 1998 september with the burst and transient source experiment ( batse ) on the _ compton gamma ray observatory ( cgro ) _ and the all - sky monitor ( asm ) on the _ rossi x - ray timing explorer ( rxte)_. here we present new results from batse and _ rxte _ including a pulse timing analysis , spectral analysis , and evidence for an accretion disk . our pulse timing analysis yielded an orbital period of 169.2 days , a moderate eccentricity of @xmath0 , and implied a mass function of 9.7 m@xmath1 . we observed evidence for an accretion disk , a correlation between measured spin - up rate and flux , which was fitted to obtain a distance estimate of @xmath2 kpc . xte j1946 + 274 remained active from 1998 september - 2001 july , undergoing 13 outbursts that were not locked in orbital phase . comparing _ rxte _ pca observations from the initial bright outburst in 1998 and the last pair of outbursts in 2001 , we found energy and intensity dependent pulse profile variations in both outbursts and hardening spectra with increasing intensity during the fainter 2001 outbursts . in 2001 july , optical h@xmath3 observations indicate a density perturbation appeared in the be disk as the x - ray outbursts ceased . we propose that the equatorial plane of the be star is inclined with respect to the orbital plane in this system and that this inclination may be a factor in the unusual outburst behavior of the system .